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j 2(j k)=p 2k=pkfk k1 and thus (2j=p kvj k1)j2ZZ 2 lq . Besides, P we have k>j+1 kfk k1 j+1 =2; hence, we have on Ej that jf j 2 kj+1 jfk j, that gives 2 j(1 1=p) kvj k1 2qPkj+1 2 (j k)(1 1=p) 2 k(1 1=p)kfk k1 and nally, (2 j(1 1=p) kvj k1)j2ZZ 2 l. Now, we prove (B). P [L1; L1];p Lp is a direct consequence of (A) : we decompose again f into j vj where the functions vj have disjoint supports and we get (since kvj kpp kvj k1kvj kp1 1): kf kpp =
X j
X
kvj kpp (
j
2
j (p
1) kvj kp)1=p (X 2j kvj kp1 )1 1=p < 1: 1 j
Conversely, considering q = p p 1 and L10 the closure of L1 \ L1 in L1, we have [L1; L1]1 ;q = [L1; L10 ]1 ;q Lq (with dense injection), so that the duality theorem gives that Lp [L1; (L10 )0 ]1 ;p = [L1; L1];p (since L1 \ (L10 )0 = L1 \ L1 ). Now, we prove (C). We rst consider f 2 Lp;1. For all t > 0, f can be decomposed into f = gt + ht, with kgtk1 Ct1 1=p and khtk1 Ct 1=p . For > 0, we take t so that 2 Ct 1=p = ; we then de ne E = fx = jf (x)j > g and get that x 2 E ) jgt(x)j > =2, so that (E ) 2Rkgtk1 t = (=2C ) p. if f 2 Lp;, we have to estimate jf (x)j>t =p jf (x)j d = R P Conversely, j 0 2j
XZ j 0
1
+1
2j
+1
jf (x)j d C
X j+1 1=p j 1=p p 0 1 1=p 2 t (2 t ) = C t j 0
Thus, the theorem is proved. We now prove two easy and useful results on pointwise multiplication and convolution in Lorentz spaces. Proposition 2.3: (Pointwise product in Lorentz spaces) Let 1 < p < 1, 1 q 1, 1=p0 + 1=p = 1 and 1=q0 + 1=q = 1. Then pointwise multiplication is a bounded bilinear operator:
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20
Real harmonic analysis
a) from Lp;q L10 to Lp;q ; 0 p;q p ;q b) from L L to L1 ; p;q p ;q 1 1 c) from L L to Lp2 ;q2 for p0 < p1 < 1, q0 q 1, 1=p2 = 1=p + 1=p1 and 1=q2 = 1=q + 1=q1. 0 0 Moreover, if q < 1, then the dual of Lp;q is equal to Lp ;q, the duality bracket being given by the integral of the pointwise product. 0 0 Proof: The duality theorem for real interpolation gives that (Lp;q )0 = Lp ;q if q < 1. Besides, points a) andP b) are obvious. PIn order to prove c), we write, for f 2 Lp;q and g 2 Lp1 ;q1 , f = j fj and g = j gj and we decompose fg into P v +w with v = f P g and w = g P f . Since kg k 2j=p1 2 lq1 , j j j k>j k j j kj k j 1 j j j (1=p+1=p1 ) kvj k1 2 lq2 we get that 2j=p1 Pk>j kgk k1 2 lq1 . Thus, we nd that 2 =p1 1) kvj k1 2 lq2 , which gives P vj 2 Lp2 ;q2 . The same proof and that 2j(1=p+1P j may be used for j wj .
We now consider X = IRd endowed with the Lebesgue measure and consider the properties of convolution between Lorentz spaces: Proposition 2.4: (Convolution of Lorentz spaces) Let 1 < p < 1, 1 q 1, 1=p0 + 1=p = 1 and 1=q0 + 1=q = 1. Then convolution is a bounded bilinear operator : a) from Lp;q (IRd) L10(IR0 d ) to Lp;q (IRd ) ; b) from Lp;q (IRd ) Lp ;q (IRd ) to L1(IRd) ; c) from Lp;q (IRd) Lp ;q (IRd ) to Lp ;q (IRd ) for 1 < p1 < p0 , q0 q 1, 1=p2 = 1=p + 1=p1 1 and 1=q2 = 1=q + 1=q1. p;q Proof: Points a) and Pb) are obvious. P In order to prove c), we write, for Pf 2 L p ;q and g 2 L , fP= j fj and g = j P gj , and we decompose fg into j vj +wj , with vj = fj k<j gk and wj = gj kj fk . Since kgj k1 2j(1=p 1) 2 lq , we P j (1 =p 1) q get that 2 that 2j(1=p+1=p 2)kvj k1 2 k<j kgk k1 2 l . Hence, we getP lq and that 2j(1P=p+1=p 1) kvj k1 2 lq , which gives j vj 2 Lp ;q . We use the same proof for j wj . 1
1
1
2
2
1
1
1
2
1
1
1
1
2
2
2
3. Sobolev inequalities
Proposition 2.4 may be applied to easily get the Sobolev inequalities: Theorem 2.4: (Sobolev inequalities) Let 2 (0; d). Then (p 1) is bounded: d i) from L1 (IRd ) to L d ;1 (IRd); d ;1 d ii) from L (IR ) to L1 (IRd);
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Lorentz spaces iii) from Lp;q (IRd) to Lp ;q (IRd) for 1 < p < d and 1=p1 = 1=p 1
Proof:
21 =d.
d ;1 d This is a convolution operator with kernel jxcj ;d d 2 L d (IR ).
Now, we prove an estimate which will be useful in the study of the Navier{ Stokes equations: Proposition 2.5: (Heat kernel and Lorentz spaces.) Let 2 IR, 0 < < d. Then: (A) If 1 < p < 1 and < d=p and if r1 = 1p d , there exists a constant R Cp; so that, for all f 2 Lp;1 (IRd ), we have 01 ket (p 1) f kLr; dt 1
Cp; kf kLp;1 . (B) If = d=p, then thereR exists a constant Cp; so that, for all f Lp;1 (IRd ), we have the estimate 01 ket (p 1) f k1 dt Cp; kf kLp;1
2
Remark: This proposition means that (p 1) maps Lp;1 to the Besov space B_ L0;r;11 , a slightly more precise result than Theorem 2.4 (since B_ L0;r;11 Lr;1 ). See
Chapters 3, 4, and 5 for more information on Besov spaces. P Proof: The proposition is quite obvious; we write f = n2IN n fn , with kfnk1 Apn , kfnk1 AnRp=1(p 1) and Pn2IN jn j C kf kLp; . By convexity, it is enough to show that 0 ket (p 1) fnkLr; dt Cp; (uniformly with respect to n). Then, we de ne 1 < p1 < p < p2 < r by 1=p1 = 1=2(1 + 1=p) p(pj 1)=pj (p 1) j and 1=pp2 = 1= 2(1=r + 1=p). We have kfnkLpj ; CAp=p . n An Now, ( ) j maps boundedly Lpj ;1 to Lr;1 (r < 1) or L1p (r = 1) for j = d(1=pj 1=r) = d(1=pj 1=p) + . Since, for all 0, ( ) e is a convolution toperator with a kernel k 2 L1 (as proved in the lemma below), we nd that ke (p ) =fnk=pLr; =p= ket(p )2 + j (p ) j fnkLr; j p(pj 1)=pj (p 1) t j CAp=p . We integrate this estimate with j = 2 n An t from t = 0 to t = Tn and with j =p 1 pfrom t = Tn to 1 and we get that pp p p p pp p R 1 t p 1 pp p p + Tn pp Anp p ); we may 0 ke ( ) fnkLr; dt C (Tn An p conclude by choosing Tn = Anp . To nish the proof, we need to prove the following lemma: p Lemma 2.1: For all 0, ( ) e is a convolution operator with a 1 1
1
1
1 2 (1
1 1
1
)
2 2 2
1
( 2(
2) 1)
1 2 1
( 1(
1) 1)
2 2 1
kernel k
2L .
We have k^ () = jj e jj 2 L1; thus k is a continuous bounded function. Moreover, if j > 0, we introduce a function ! 2 D(IRd) so that 0 2= P
Supp ! and j2ZZ !(2 ) = 1. Then jj !() 2 D, and if we write jj !() = Proof:
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2
22 Real harmonic analysis
^ () and = 1 Pj0 !(2j ), we have k^ () = Pj0 2 j ^ (2j )e ( ) j j e jj , hence, 2
kk k1
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X j 0
2 j k
k1kF 1(e
jj2 +
jj2 )k1 + kF 1 (( ) j j e jj2 )k1 < 1:
Chapter 3 Besov spaces and Littlewood{Paley decomposition
In this chapter, we introduce a basic tool for this book: the Littlewood{ Paley decomposition, f = S0f + Pj0 j f , and the Besov spaces Bps;q . Special attention throughout the book will be given to the spaces B1s;1 . We will discuss as well the homogeneous decomposition for distributions modulo the polynomials. Our main references for Besov spaces are the books by Bergh and Lofstrom [BERL 76], by Meyer [MEY 92], and by Peetre [PEE 76], and the paper of Bourdaud [BOU 88a]. Another classical reference is the book by Triebel [TRI 83]. 1. The Littlewood{Paley decomposition of tempered distributions
One of the main tools in this book is the Littlewood{Paley decomposition of distributions into dyadic blocks of frequencies: De nition 3.1: (Dyadic blocks) Let ' 2 D(IRd ) be a non negative function so that j j 12 ) '( ) = 1 and j j 1 ) '( ) = 0. Let be de ned as ( ) = '(=2) '( ). Let Sj and j be de ned as the Fourier multipliers F (Sj f ) = '(=2j )F f and F (j f ) = (=2j )F f . The distribution j f is called the j {th dyadic block of the Littlewood{Paley decomposition of f .
Theorem 3.1: (Littlewood{Paley decomposition) P For all N 2 ZZ and all f 2 S 0 (IRd ), we have f = SN f + jN j f in S 0 (IRd ). This equality is called the Littlewood{Paley decomposition of the distribution If, moreover, limN ! 1 SN f = 0 in S 0 , then the equality f = P f isf .called the homogeneous Littlewood{Paley decomposition of f . j 2ZZ j
P
Proof: Clearly, we have: hSN f + N j
S.
23 © 2002 by CRC Press LLC
24
Real harmonic analysis De nition 3.2: (Distributions vanishing at in nity) We de ne the space of tempered distributions vanishing at in nity as the space S00 (IRd ) of distributions so that limN ! 1 SN f = 0 in S.0 For more general distributions, we cannot recover them from their homogeneous Littlewood{Paley decomposition but modulo polynomials: Proposition 3.1: (Homogeneous decomposition) For all f 2 (S )0 , there is an integer N Pand a sequence of polynomials Pj ; j 2 ZZ of degree less or equalPto N so that j2ZZ (j f + Pj ) converges to f in (S )0 . Thus the equality f = j2ZZ j f holds in S 0 =CI [X1 ; : : : ; Xd ]. Proof: We just have to recall that the Fourier transform S0 f is compactly supported, hence is of nite order K : jhS0 f jgij C Pj jK k @@ g^k1. Now, we recall that S0 is de ned by a Fourier multiplierP' equal to@1 in a neighbourhood of 0; thus, we may write, for all g 2 S , g^ = ( j jK ! @ g^(0))'() + gK (). Then, P clearly, limj! 1 hSj f jgK i = limj! 1 hS0 f jSj gK i = 0. Thus, the series j 2ZZ Tj , where the distributions Tj are de ned by hTj jg i = hj f jgK i, converges in S 0 to a limit T1. But Tj j f are polynomials, and so is T1 f (since its Fourier transform is supported by f0g). When dealing with the Littlewood{Paley decomposition, it is convenient to introduce the functions '~() = '(=2) and ~() = '(j=4) '(4) as well as the operators S~j and ~ j de ned by F (S~j f ) = '~(=2 )F f and F (~ j f ) = ~(=2j )F f (so that Sj = S~j Sj and j = ~ j j ). Using these operators, we obviously get the Bernstein inequalities: Proposition 3.2: (Bernstein inequalities) For all 2 INd and 2 IR, for all j 2 ZZ, for all 1 p q 1 and for all f 2 S 0 (IRd ), we have: @ S f k k @ F 1 '~k kS f k 2j j j (a) k @x j p 1 j p @x @ f k k @ F 1 ~k k f k 2j j j (b) k @x j p 1 j p @x p (c) k( ) j f kp kF 1(j j ~)k1 kj f kp 2j P (d) kj f kp dl=1 kF 1( jjl ~)k1 k @x@ l j f kp 2 j (e) kSj f kq kF 1'~kr kSj f kp 2j(d=p d=q) with 1=r = 1 (1=p 1=q). (f) kj f kq kF 1 ~kr kj f kp 2j(d=p d=q) with 1=r = 1 (1=p 1=q). Another important tool in Littlewood{Paley analysis is the paraproduct operator introduced by J.M. Bony [BON 81]: Proposition 3.3: (Paraproduct operator) d ). Then: Let f; g 2 S 0 (IRP (a) The series j2IN Sj 2 f j g converges in S 0 . Its sum, (f; g), is called 2
the paraproduct of f and g.
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P
Besov spaces 2 f j g converges in S 0 =CI [X1 ; : : : ; Xd].
(b) The series j2ZZ Sj _ (f; g), is called the homogeneous paraproduct of f and g.
25 Its sum,
Since the Fourier transform of a product is the convolution of the Fourier transforms, we easily1 check that the spectrum of the product Sj 2 f j g is supported on P the set f = 4 2j jj 4 2j g. Moreover, f is tempered, so that we may write f^ = j jN @ f with jf ()j C (1 + jj)M for some constants C and M and some measurable functions f , and similar estimates for g^, and we get, for j 0, F (Sj 2 f j g) = Pj jN 0 @ f ;j with jf ;j ()j C 0(1 + jj)M 0 for some constants C 0 , N 0 and M 0 which do not depend on j . Thus, (f; g) is well de ned in S 0 . For j < 0, we similarly get F (Sj 2 f j g) = Pj jN 0 @ f ;j , with estimates jf ;j ( )j C 00 j j M 00 for some constants C 00 , N 00 and M 00 which do not depend on j . This gives that there is an integer KPand a sequence of polynomials Pj ; j < 0, of degree less or equal to K so that j<0 (Sj 2 f j g + Pj ) converges in S 0 . Proof:
2. Besov spaces as real interpolation spaces of potential spaces
In this section, we introduce the potential spaces Hp and the Besov spaces
Bp;q :
De nition 3.3: (Potential spaces) For 2 IR and 1 p 1, the potential space Hp is de ned as the space (Id ) =2 Lp, equipped with the norm kf kHp = k(Id )=2 f kp.
We need a precise description of the convolution operator (Id )=2 , which we obtain by using the Littlewood{Paley decomposition: Proposition 3.4: (Bessel potentials) For 2 IR, let k 2 S 0 (IRd ) be the distribution de ned by F k ( ) = (1 + jj2 )=2 and (Id )=2 be the convolution doperator with kd . Then: (a) (Id )=2 operates boundedly on S (IR ) and on S 0 (IR ). (b) For all j 2 ZZ, Sj k 2 S (IRd ) and j k 2 S (IRd ). (c) For all N 2 IN, there exists a constant C;N so that for all j 0, we have for all x 2 IRd jj k (x)j C;N 2j(d+) (1 + 2j jxj) N (d) If < 0, then k 2 L1 (IRd ). More precisely, for all N exists a constant D;N so that we have for all x 2 IRd
jk (x)j D;N ! (x)(1 + jxj)
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N
2 IN,
there
26 with ! (x) = jxj d
Real harmonic analysis if d + > 0, = 1+ln+ 1 if d + = 0 and 1 if d + < 0. jxj
(a) is obvious since F k is a smooth function whose derivatives of any order are slowly increasing functions. (b) is obvious as well since the Fourier transforms of Sj k and of j k are compactly supported smooth functions. To prove (c), we may assume j 1 so that we have Proof:
Z
Z
(2)dkj k k1 (1+jj2)=2 j (=2j )j d (1+2=2) jj j ()j d 2j(d+) p
N We estimate in a similar way kj xjN j k k1 d max1ld kxN l j k k1 pN R @N 1 d 2 = 2 j d max1ld ( 2 ) j @xN ((1 + j j ) (=2 ))j d and get kjxjN j k k1 = l O(2j(d+ N ) ). P We now prove (d). We write k = SP 0 k + j0 j k . Since S0 k 2 S , S0 k ful lls the desired size estimate. For j0 j k , we consider two cases : if jxj > 1, we write for any M > d + that jj k (x)j CM; 2j(d+ M ) jxj M and just sum over j 2 IN ; if jxj 1, we x M > d + and write jj k (x)j CM; 2j(d+) for 2j jxj 1 and jj k (x)j CM; 2j(d+ M ) jxj M for 2j jxj > 1 ; the summation over the j -s such that 2j jxj > 1 gives an estimate O(jxj d ) ; the summation over the j -s such that 2j jxj 1 gives an estimate O(jxj d ) if d + > 0, 0(1 ln jxj) if d + = 0 and O(1) if d + < 0. We may now de ne the Besov spaces: De nition 3.4: (Besov spaces) For 1 p 1, 2 IR, 1 q 1, the Besov space Bp;q is de ned as Bp;q = [Hp ; Hp ];q where 0 < < 1 and = (1 )0 + 1 . We easily prove that this de nition does not depend on the choice of 0 and 1 by giving the Littlewood{Paley decomposition of distributions in Bp;q : Proposition 3.5: (Littlewood{Paley decomposition of Besov spaces) For 1 p 1, 2 IR, 1 q 1, N 2 ZZ and f 2 S 0 (IRd), the following 0
1
assertions are equivalent : (A) f 2 Bp;q (B) SN f 2 Lp , for all j N j f 2 Lp and (2j kj f kp)jN 2 lq . Moreover, if > 0, if we de ne for k 2 IN the Sobolev space W k;p as d @ k;p p W = ff 2 L = 8 2 IN with j j k; @x f 2 Lp g, if k0 < < k1 , = (1 )k0 + k1 , then Bp;q = [W k0 ;p ; W k1 ;p ];q .
Proof: (B ) ) (A) is quite obvious. Indeed, if (B) holds for some N , it holds as well for N = 0. Now, we write f = S0f + Pj0 j f ; then S0f belongs to Hp1 Hp0 and so does j f . Thus, it is enough to check that, for some positive 6= 1, j J (j ; j f ) 2 lq . We start from inequality kj f kHpi =
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Besov spaces 27 k~ j ki j f kp Ci 2ji kj f kp (according to Proposition 3.4) and write, for j = j J (j ; j f ) = max( j kj f kHp ; j(1 )kj f kHp ) that j C max( j 2 j( ) ; j(1 ) 2j(1 )( ) )2j kj f kp ; we then conclude by taking = 2 ( ). (A) ) (B). We write, for some positiveP 6= 1, f = P We now prove j f with J (j ; fj ) 2 lq . We then compute j f = l2ZZ j fl and j 2ZZ jP SN f = l2ZZ SN fl . We have kfl kH l l and kfl kH l( 1) l , with (l ) 2 lq . Since fl = k i (kji fl), we nd (using Proposition 3.4) that, for j N , kj fl kp Ci ;N 2 i kfl kH i and kSN fl kp Ci ;N kflkH i . Since P min( kfl kH ; kfl kH ) < 1, we nd that SN f 2 Lp . Moreover, we have l j 2 kj flkp C min(l2j( ); ( 1)l2j( 1)( )) l and we conclude by taking = 2 ( P ): we obtain, for 0 < =P(1 ) < 1 < = , that 21j kq j f kqp C l2ZZ min( j l ; j l )l with l2ZZ min( l ; l ) < 1. Since l l l , we have nished to prove (A). We now prove that Bp;q = [W k ;p; W k ;p];q . We use the reiteration theorem for the real interpolation method: it is enough to check that Bpk;1 W k;p Bpk;1 . But this is a direct consequence of the Bernstein inequalities. The duality theorem for the real interpolation method combined with the Riesz theorem gives that the Besov spaces are dual spaces: Proposition 3.6: (Duality and Besov spaces) For 1 p 1, 20 IR, 1 q 1, let p0 ; q0 be the conjugate exponents 0 ;q ;q ~ of p and q and let Bp0 be the closure of S in Bp0 . Then Bp;q is the dual 0 space of B~p0 ;q. 0
1
1
0
1
1
0
0
0
0
1
1
1
1
0
1
0
0
0
1
The case p > 1 is a straightforward application of the duality theorem. Clearly, S is dense in Hp0i for 0 < 0 < 1 , and this gives (for = 01 + (1 )0 ) that S is dense in Bp0;q if q0 < 1 while the dual of Bp0;q = [Hp0 ; Hp0 ];q0 is equal to [Hp ; Hp ];q = Bp;q . The case q = 1 follows the same lines,;1since l1(IN) is the dual of c0: it is 0then enough to check that f belongs to B~p0 (p0 < 1) if and only if S0 f 2 Lp and (2 j kj f kp0 )j0 2 0c0. The case p = 1 requires only a slight modi cation: for q0 < 1 we have B~1;q = [(Id ) =2C0; (Id ) =2C0];q0 (and a similar result for q0 = 01 changing l1 into c0 ); the duality theorem gives that the dual space of B~1;q is equal to [(Id ) =2 M; (Id ) =2M ];q (where M is the space of nite complex Borel measures on IRd). We easily nd that f 2 [(Id ) =2 M; (Id ) =2 M ];q if and only if S0f 2 M and (2j kj f kM )j0 2 lq ; but S0f = S~0(S0 f ) and j f = ~ j (j f ) and S~0 and ~ j map M to L1; since for g 2 M \0 L1, we have g 2 L1 and kgkM = kgk1, this gives that the dual space of B~1;q is B1;q . Chapter 7 demonstrates that for 1 < p < 1 the space Hp is easily characterized by the Littlewood{Paley decomposition as well: Proof: 0
1
0
0
0
1
1
1
0
© 2002 by CRC Press LLC
1
28
Real harmonic analysis Proposition 3.7: (Littlewood{Paley decomposition of potential spaces) For 1 < p < 1, 2 IR, N 2 ZZ and f 2 S 0 (IRd ), the following assertions are equivalent: (A) f 2 Hp (B) SN f 2 Lp , for all j p L (dx).
N j f 2 Lp
and
(PjN 4j jj f j2)1=2 2
We now prove the following corollary of this result: Corollary: For 1 < p < 1 and 2 IR, we have Bp;max(p;2) .
Bp;min(p;2)
Hp
For 2 p < 1, we use the Fubini theorem and the inclusion l2 lp to get the inclusion Hp Bp;p : we just have to write the inequalities Proof:
X
(
j N
X
2jp kj f kpp)1=p = k(
j N
X
2jp jj f jp )1=p kp k(
j N
4j jj f j2 )1=2kp:
To get Bp;2 Hp , it is enough to use the Minkovski inequality in Lp=2: X
k(
j N
4j jj f j2)1=2 kp = k
X
j N
X
4j jj f j2 k1p==22 (
j N
4j k(j f )2kp=2)1=2 :
The result for p 2 follows by duality. 3. Homogeneous Besov spaces
In this section, we are interested in distributions modulo polynomials. We start from this easy proposition: Proposition 3.8: (Distributions modulo polynomials) R (A) The space S1 = ff 2 S (IRd )=8 2 INd P x f (x) dx = 0g is closed in S . (B) f belongs to S1 if and only if f 2 S and j2ZZ j f converges to f in S . (C) The orthogonal (S1 )? in (S )0 is the space of polynomials CI [X1 ; : : : ; Xd]. Thus, (S1 )0 is the space of tempered distributions modulo the polynomials S 0 =CI [X1 ; : : : ; Xd ]. 0 (D) P For all f 2 (S ) , there is a sequence0 of polynomials Pj ; j 2 ZZPsuch that j 2ZZ (j f + Pj ) converges to f in (S ) . Thus the equality f = j 2ZZ j f holds in S 0 =CI [X1 ; : : : ; Xd ]. Proof: (A) is obvious. For (B), we have only to check that Sj f goes to 0 in S as j goes to 1 when jf belongs to S1 . Using the Fourier transform, we have to check that '(=2 )f^() goes to 0 in C 1 whenever f^ vanishes at
© 2002 by CRC Press LLC
Besov spaces 29 0 @with in nite order; this is easy to check since, for all 2 INd, we have j @ ['( 2j )]j C j j j j and lim!0 supjj j j j j f^( ) = 0. (C) is easy as well: it is enough to check that f 2 S 0 is orthogonal to S1 if and only if f^ is supported by f0g. (D) was proved in Proposition 3.1. We may now de ne homogeneous Besov spaces in the following way: De nition 3.5: (Homogeneous Besov spaces) Let 1 p 1, 2 IR and 1 q 1. Then, the homogeneous Besov space B_ p;q is de ned as the Banach space of distributions f 2 S 0 =CI [X1 ; : : : ; Xd] such that for all j 2 ZZ j f 2 Lp and (2j kj f kp )j2ZZ 2 lq (ZZ). The duality theorem has a homogeneous counterpart: Proposition 3.9: (Duality and homogeneous Besov spaces) For 1 p 1, 2 IR, 1 q 1, let p0; q0 be the conjugate exponents 0 ;q 0 of p and q and let B~_ p0 be the closure of S1 in B_p0;q . Then B_p;q is the dual ;q0 space of B~_ p0 : a distribution f 2 S 0 belongs to B_p;q if and only if there exists +
a constant C so that for all ! 2 S1 we have jhf j!ij C k!kB_
;q0. p0
This is a direct consequence of the duality theorem for non homogeneous Besov spaces. This proposition should be modulated in the following manner: the duality bracket in0 Proposition 8 is the bracket between S1 and its dual; but for a given , B_p0;q contains a bigger subspace of S than S1 , and the duality bracket should be the one between this subspace and its dual. We thus begin with the following lemma: R Lemma 3.1: f 2 S (IRd ) belongs to B_p0;1 if and only if x f (x) dx = 0 for 0 all 2 INd so that j j < d=p. Similarly, f belongs to B_p0;q with q0 < 1 if R and only if x f (x) dx = 0 for all 2 INd such that j j d=p. Proof:
R
Proof: Let N be an integerPso that x f (x) dx = P 0 for all 2 INd so that @ f j j < N . We write f^( ) = j j=N ! @x (0) '( ) + j j=N +1 f^ ( ) where ' is the function associated to the Littlewood{Paley operator S0P and where f belongs to S . Then, for j 2, wePhave j f = 2j(d+N ) f (2j x)+ j j=N +1 f (2j(d+N +1) (2j x)) with F f = j j=N ! @@x f (0) () and F = (). This gives kj f kp0 = 2j(N +d=p)k f kp0 + O(2j(N +1+d=p) .
This lemma leads us to the following alternative de nition of homogeneous Besov spaces:
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Real harmonic analysis
De nition 3.6: (Realization of homogeneous Besov spaces) The realization B_ p;q of the homogeneous Besov space B_ p;q is de ned in the following manner: we de ne N (; p; q) as the smallest integer N so that N > d=p if q > 1 (or so that N d=p if q = 1); then, a distribution f 2 S 0 belongs Rto B_ p;q if and only if there exists a constant C so that for all ! 2 S with x ! dx = 0 for all 2 INd with j j < N (; p; q) we have jhf j!ij C k!kB_ p0;q0.
If P < d=p (or if = d=p and q = 1), then for all f 2 B_p;q , the decomposition j2ZZ j f converges in S 0 ; in that case, we can easily check that the realization of the Besov space B_p;q is equal to B_ p;q = B_p;q \ S00 and P that the mapping f 7! j2ZZ j f is an isomorphism from B_p;q to B_ p;q . More generally, B_ p;q , for 2 IR, is a space of distributions modulo polynomials of degree less than N (; p; q). Example:
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Chapter 4 Shift-invariant Banach spaces of distributions and related Besov spaces
We shall often use Banach spaces of distributions whose norm is invariant under translations kT (x x0)k = kT k and on which dilations operate boundedly. This chapter is devoted to the basic de nitions and results in this setting. 1. Shift-invariant Banach spaces of distributions
De nition 4.1: (Shift-invariant Banach spaces of distributions.) A) A shift-invariant Banach space of test functions is a Banach space E such that we have the continuous embeddings D(IRd) E D0 (IRd ) and so that: (a) for all x0 2 IRd and for all f 2 E , f (x x0 ) 2 E and kf kE = kf (x x0 )kE . (b) for all > 0 there exists C > 0 so that for all f 2 E f (x) 2 E and kf (x)kEd C kf kE . (c) D(IR ) is dense in E B) A shift-invariant Banach space of distributions is a Banach space E which is the topological dual of a shift-invariant Banach space of test functions E () . The space E (0) of smooth elements of E is de ned as the closure of D(IRd ) in E .
An easy consequence of hypothesis (a) is that a shift-invariant Banach space of test functions Ed satis es S (IRd) E and a consequence of hypothesis (b) is that E S 0 (IR ). Similarly, we have for a shift-invariant Banach space of distributions E that S (IRd) E (0) E S 0 (IRd). In particular, E (0) is a shift-invariant Banach space of test functions. Remark:
Shift-invariant Banach spaces of distributions are adapted to convolution with integrable kernels: Proposition 4.1: (Convolution in shift-invariant spaces of distributions.) If E is a shift-invariant Banach space of test functions or of distributions and ' 2 S (IRd), then for all f 2 E we have f ' 2 E and kf 'kE kf kE k'k1 .
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Real harmonic analysis
Moreover, convolution may be extended into a bounded bilinear operator from E L1 to E and we have for all f 2 E and all g 2 L1 the inequality kf gkE kf kE kgk1.
Proof: It is enough to check that for f; g 2 S (IRd) the Riemann P d k 1 k N d k2ZZd g ( N )f (x N ) converge to f g in S (IR ) as N goes to 1.
sums
Due to this proposition, we very often use very often convolution operators with an integrable kernel. We thus introduce a convenient notation: De nition 4.2: (Convolution operators) If k 2 L1 (IRd ) and m = k^, we de ne the operator m(D) as the convolution operator m(D)f = f k and we de ne the norm kjm(D)jk1 as kjm(D)jk1 = kkk1. A special case of convolution operator is the j {th dyadic block of the Littlewood{Paley decomposition j f = (D=2j )f : we obtain the estimates kSj f kE kjS0 jk1kf kE and kj f kE kj0 jk1 kf kE . Due to the Littlewood{ Paley decomposition, we may easily compare E to a Besov space: Proposition 4.2: Let E be a shift-invariant Banach space of distributions. Let C1=2 be the operator norm in L(E; E ) of the operator f 7! f (x=2). Then, E B1ln2 (C1=2 );1 .
Since the norm of E is shift-invariant and since E S 0 , we nd that 1j maps E to L1. Moreover, since dilations operate boundedly on E and on L , we may write Dj (f ) = f (2j x) and j f = Dj 0 D j f which gives that for j 0 j maps E to L1 with an operator norm O((C1=2 )j ). As for the case of Lebesgue norms, we have the Bernstein inequalities: Proposition 4.3: (Bernstein inequalities) Let E be a shift-invariant Banach space of distributions. Then, for all 2 INd and 2 IR, for all j 2 ZZ and for all f 2 S 0 (IRd ), we have (whenever the right-hand sides are well-de ned): @ S f k k @ F 1 '~k kS f k 2j j j (a) k @x j E 1 j E @x @ f k k @ F 1 ~k k f k 2j j j (b) k @x j E 1 j E @x p (c) k( ) j f kE kF 1(j j ~)k1 kj f kE 2j P (d) kj f kE dl=1 kF 1( jjl ~)k1 k @x@ l j f kE 2 j Proof:
2
Another interesting example of convolution operator that will be considered throughout this book is the convolution with the heat kernel. We have a semigroup of operators et with the following regularity estimates:
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Shift-invariant spaces (Heat kernel)
33
Proposition 4.4: The heat kernel et satis es jket ekj1 C j ln t ln j. Hence: a) If E is a shift-invariant Banach space of test functions and if f 2 E , then etf 2 Cb ([0; 1); E ), i.e., the map t 7! etf is continuous and bounded from [0; 1) to E . b) If E is a shift-invariant Banach space of distributions and if f 2 E , then etf 2 C ([0; 1); E ), i.e., the map t 7! et f is continuous from (0; 1) to E and continuous at t = 0 for the *-weak topology (E; E () ) on E .
We write et e = Rt es ds; hence, jket ekj1 Rt jkekj1 dss . For the continuity at t = 0, it is enough to check that for f 2 S (IRd) t 7! etf is continuous from [0; 1) to S . Sometimes, we consider a special case of shift-invariant spaces of distributions: spaces of local measures, where the spaces are requested to remain invariant through pointwise multiplication with bounded continuous functions: De nition 4.3: (Shift-invariant spaces of local measures) A shift-invariant Banach space of local measures is a shift-invariant Banach space of distributions E so that for all f 2 E and all g 2 S (IRd ) we have fg 2 E Proof:
and kfgkE CE kf kE kgk1, where CE is a positive constant (which depends neither on f nor on g).
As a matter of fact, we have a more precise result on pointwise multiplications: Lemma 4.1: Let E be a shift-invariant Banach space of local measures. Then: a) the elements of E are local measures (i.e. are distributions of order 0). 1 of uniformly locally nite More precisely, they belong to the Morrey space Muloc measures. b) E , its predual E () and the space of smooth elements E (0) in E are stable under pointwise multiplication by bounded continuous functions.
a) is obvious: if f 2 Ed and if K is a compact subset of IRd, then we consider a function K 2 D(IR ) which is equal to 1 on K , and we write for all ! 2 D(IRd) with support included in K : jhf j!ij = jhf!j K ij kf kE k K kE k!k1 Thus, f is a local measure d. Moreover, taking K = dx0 + [ 1; 1]d and K = [ 1;1]d (x x0 ), we nd that supx 2IRd jj(x0 + [ 1; 1] ) < k [ 1;1]d kE kf kE , 1 . and thus f 2 Muloc b) is easy. We notice that E () dis the closure of D in E 0. Now, if K is a d compact subset of IR , if d K 2 D(IR ) satis es 1K K 1 everywhere,1 then we have for all ! 2 D(IR ) with support included in K and all g 2 C (IRd) Proof:
( )
0
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( )
34 Real harmonic analysis that k!gkE0 = k!g K kE0 k!kE0 kgk1. Thus, pointwise multiplication may be extended to a bounded bilinear operator from E () Cb to E () . 2. Besov spaces
In this section, we introduce the potential spaces over a shift-invariant Banach space of distributions: De nition 4.4: (Potential spaces) Let E be a shift-invariant Banach space of distributions. Then, for 2 IR, the space HE is de ned as the space (Id ) =2 E , equipped with the norm kf k;E = k(Id )=2 f kE . We now de ne Besov spaces over a shift-invariant Banach space of distributions, in pa very similar way as for the Besov spaces Bps;q based on the Lebesgue spaces L : De nition 4.5: (Besov spaces) Let E be a shift-invariant Banach space of distributions. For 2 IR, 1 q 1, the Besov space BEs;q is de ned as BEs;q = [HE ; HE ];q where 0 < < 1 and = (1 )0 + 1 . Just like for the Besov spaces Bps;q , we may give an easy characterization s;q of the Besov spaces BE through the Littlewood{Paley decomposition: Proposition 4.5: (Littlewood{Paley decomposition of Besov spaces) Let E be a shift-invariant Banach space of distributions. For 2 IR, 1 q 1, N 2;qZZ and f 2 S 0 (IRd), the following assertions are equivalent: (A) f 2 BE (B) SN f 2 E , for all j N j f 2 E and (2j kj f kE )jN 2 lq . Moreover, if > 0, if we de ne for k 2 IN the Sobolev space W k;E = ff 2 @ f 2 E g, if k < < k , = (1 )k + k , E = 8 2 INd with j j k @x 0 1 0 1 s;q k ;E k ;E then BE = [W ; W ];q . Proof: This is just the same proof as for the spaces Bps;q , since we only used a convolution estimate with the integrable kernel of j k . We now give a simple example of application of the Littlewood{Paley decomposition: pointwise multiplication is a bounded bilinear mapping on regular enough Besov spaces. Theorem 4.1: (Pointwise multiplication in Besov spaces) 0
0
1
1
Let E be a shift-invariant Banach space of local measures. Then: (a) E \ L1 is an algebra for pointwise multiplication. (b) For 1 q 1 and > 0, BE;q \ L1 is an algebra for pointwise multiplication.
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Shift-invariant spaces 35 (c) For 1 q 1, > 0 and 2 ( ; 0), the pointwise multiplication is ;1 ) (B ;q \ B ;1 ) to B + ;q . a bounded bilinear operator from (BE;q \ B1 1 E E The proof is based on a basic lemma: Lemma 4.2: Let E be a shift-invariant Banach space of distributions, 2 IR, 1 q 1 and let (fj )j2IN a sequence of elements of E so that (2j kfj kE )j2IN 2 lq (IN). Then: a) If for some constants 0 < A < B < 1, wePhave for all j 2 IN that f^j is supported by f 2 IRd = A2j j j B 2j g, then j2IN fj converges in S 0 to a distribution f 2 BE;q . ^ b) If for some constant 0 < B < 1, we have for P all j 2 IN that fj is0 d j supported by f 2 IR = j j B 2 g and if > 0, then j2IN fj converges in S to a distribution f 2 BE;q . Proof lemma: The lemmaPis obvious. The convergence of the series P fofisthe obvious in case b) since j2IN kfj kEP< 1; in case a), we may write j 2IN j N fj = gj with 2N + > 0 and we nd that j2IN kgj kE < 1. Moreover, if k 2 IN is such that 2 k+1 < A < B < 2k 1, we have: P P { in a), j f =P j klj+k j fl, hence 2j kj f kE P C j klj+k 2l kflkE { in b), j f = 0lj+k j fl, hence 2j kj f kE C 0lj+k 2(j l) 2l kflkE , and in both cases we may conclude that f 2 BE;q .
We write fg = (f; g) + (g; f ) + R(f; g) where is the paraproduct operator: P 8 (f; g) = Pj2IN Sj 2 f j g > < (g; f ) = j2IN Sj 2 gj f R ( f; g ) = S fS g + P2f 0g + P1 f 0g + 1f 1g + (f; g) > 0 0 : (f; g) = j2IN j f ( j 2kj+2 k g) We may now easily conclude, applying Lemma 4.2 to (df; g), (g; f ) jand (f; g): Sj 2 f j g has its spectrum contained in f 2 IR = j j 5 2 1 g and kSj 2f j gkE CE kSj 2f k1kj gkE ;1; moreover, we have f 2 L1 , j supj0 kSj f k1 < 1 and, for < 0, f 2 B1 , supj0 2 kSj f k1 < 1. As a corollary, if E is a shift-invariant Banach space of local measures and if E B1;1 for some 0, then BE;q is a Banach algebra for pointwise multiplication for + > 0 and 1 q 1, or for = > 0 and q = 1. Proof of Theorem 4.1:
3. Homogeneous spaces
We now de ne homogeneous shift-invariant spaces:
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Real harmonic analysis (Homogeneous spaces)
De nition 4.6: A) A homogeneous shift-invariant Banach space of test functions is a Banach space E so that we have the continuous embeddings S1 (IRd ) E S 0 (IRd )=CI [X1 ; : : : ;dXd] and so that: (a) for all x0 2 IR and for all f 2 E , f (x x0 ) 2 E and kf kE = kf (x x0 )kE . (b) for all > 0, there exists C > 0 so that for all f 2 E f (x) 2 E and kf (x)kE d C kf kE . (c) S1 (IR ) is dense in E B) A homogeneous shift-invariant Banach space of distributions is a Banach space E that is the topological dual of a homogeneous shift-invariant Banach space of test functions E () . The space E (0) of smooth elements of E is de ned as the closure of S1 (IRd ) in E .
We sometimes need homogeneous spaces modulo polynomials of a given degree. As with homogeneous Besov spaces, we de ne the realization of the homogeneous space of distributions through a duality setting: Lemma 4.3: Let E be a homogeneous shift-invariant Banach space of test functions. Then, there exists an integer NR 2 IN such that the space SN = ff 2 S (IRd ) = 8 2 INd such that j j < N x f (x) dx = 0g is continuously embedded in E .
The continuous embedding of S1 into E gives the inequality, for some constants C 0 and M 2 IN, kf kE C Pj jM Pj jM k @@ f^k1. If f 2 SM +1 , we easily check that, using the operators Sj of the Littlewood{Paley decomposition, thePapproximation f Sj f of f satis es f Sj f 2 S1 and P @ limj! 1 j jM j jM k @ ('( 2j )f^)k1 = 0, hence that f 2 E . De nition 4.7: (Realization of a homogeneous Banach space of distributions) Proof:
Let E be a homogeneous shift-invariant Banach space of distributions and let N be the greatest integer so that SN E () . Then, the realization Er of E is the space of distributions f 2 S 0 (IRd ) so that there exists a constant C so that for all 2 SN we have jhf j ij C k kE() .
We may now extend the notion of homogeneous Besov spaces to Besov spaces based over shift-invariant Banach spaces of distributions: De nition 4.8: (Homogeneous Besov spaces) Let E be a shift-invariant Banach space of distributions or a homogeneous shift-invariant Banach space of distributions, 2 IR and 1 q 1. Then, the homogeneous Besov space B_E;q is de ned as the Banach space of distributions f 2 S 0 =CI [X1 ; : : : ; Xd ] so that for all j 2 ZZ j f 2 E and (2j kj f kE )j2ZZ 2 lq (ZZ). Those homogeneous Besov spaces are indeed homogeneous shift-invariant Banach spaces of distributions:
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Shift-invariant spaces (Duality and homogeneous Besov spaces)
37
Proposition 4.6: Let E be a homogeneous shift-invariant Banach space of distributions, 20 ;q IR, 1 q 1, and let q0 be the conjugate exponent of q. We de ne B~0_ E() ;q as the closure of S1 for the norm k2 j kj f kE() klq0 (ZZ) . Then, B~_ E() is a homogeneous shift-invariant Banach space of test functions and B_E;q is the dual ;q0 space of B~_ E() : a distribution f 2 S 0 belongs to B_E;q if and only if there exists a constant C so that for all ! 2 S1 we have jhf j!ij C k!k ~_ ;q0 . BE()
As for the usual Besov spaces over Lebesgue spaces, we shall write B_ E;q for the realization of the homogeneous Besov space B_E;q .
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Chapter 5 Vector-valued integrals We often consider statements such as f (t; x) 2 Lp ((0; T ); E ), where E is a Banach space of distributions on IRd . We do not want to get into deep considerations on measurability of vector valued functions. Therefore, we introduce here some ad hoc de nitions for such integrals in our setting, putting a special emphasis on the spaces Lp (Lq ) for which we shall need a duality theorem.
1. The case of Lebesgue spaces We consider here two measured space (X; ) and (Y; ), where the measures and are positive { nite measures. In particular, we may apply the Fubini theorem to integrate over the product space (X Y; ).
De nition 5.1: (Spaces Lp (Lq )) For 1 p 1 and 1 q 1, we de ne the space Lp (X; Lq (Y )) as the space of measurable functions on X Y so that k kf kLq (d (y)) kLp (d(x)) < 1. Remark: For p = 1, this is not the usual de nition. We have the following easy result on Lp (X; Lq (Y )) spaces:
Proposition 5.1: (A) If 1 p; q 1, if f is a measurable function on X Y , then kkf kLq (d (y))kLp(d(x)) = 0 if and only if f is equal to 0 almost everywhere on X Y . (B) If 1 p; q 1, the space of measurable functions on X Y so that kkf kLq (d (y))kLp(d(x)) < 1, quotiented by the space of functions which are equal to 0 almost everywhere, is a Banach space. Proof: This is very easy. Indeed, (A) is a direct consequence of the Fubini theorem. To get point (B), we consider a sequence of functions fn so that P 0 0 k f k have nite measures, we n n2IN P Lp (Lq ) < 1. Now, if X X and Y Y P get that n2IN kfnkL1 (X 0 Y 0 ) < 1. This proves that n2IN fn (x; y) converges forP almost every (x; y) 2 X Y to a measurable function f and the convergence of fn to f in Lp (Lq ) is then easy to check. 39 © 2002 by CRC Press LLC
40
Real harmonic analysis The main result we shall use on Lp (Lq ) is the following duality theorem:
Theorem 5.1: (Duality for Lp (Lq ) spaces) p q For 1 0 p < 1 0 and 1 q < 1, the dual space of L (X; L (Y )) is the space Lp (X; (Lq (Y )) with 1=p +0 1=p0 =0 1=q + 1=q 0 = 1. More precisely, if f 2 Lp (X; Lq (Y )) and if g 2 Lp (X; (Lq (Y )) then fg 2 L1 (X Y ) and to every bounded linear functional L on Lp (X; Lq (Y )) we may associate a p0 (X; (Lq0 (Y )) so that for all f 2 Lp (X; Lq (Y )) we have L(f ) = unique g 2 L RR X Y f g d(x) d (y ). Proof: We want to identify the bounded linear functionals L on Lp (Lq ). If r = max(p; q), we see that for all X 0 X and Y 0 Y with nite measures, Lr (X 0 Y 0 ) Lp (X 0 ; Lq (Y 0 )) (with a dense embedding) so that 0 L is given on Lp (X 0 ; Lq (Y 0 )) by the integration against a function g0 2 Lr (X 0 Y 0 ), due to the Riesz representation theorem. Thus, we easilyR Rconclude that there is a measurable function g on X Y so that L(f ) = X Y f g d(x) d (y) (at least when f 2 Lr (X Y ) and Suppf is contained in a product X 0 Y 0 with (X 00) <0 1 and (Y 0 ) < 1). The point is now to prove that g belongs to Lp (Lq ). We x X 0 X and Y 0 Y with nite measures and N 0. If p; q > 1, we de ne E = f(x; y)=x 2 X 0 ; y 2 Y 0 ; jg(x; y)j 0N g, p R q 1
= 1E (x; y)jgj and f = sgn(g(x; y)) (x; y) q 1 ( Y (x; z ) q 1 d (z )) q0 1 . p0 R R 0 Then we have jL(f )j = X ( Y (x; z )q d (z )) q0 d(x) kLkopkf kLp(Lq ) = p0 R R kLkop( X ( Y (x; z )q0 d (z )) q0 d(0x))1=p . Thus, we have proved the inequality p R R 0 0 ( X ( Y 1E (x; y) jg(x; y)jq d (y)) q0 d(x))1=p kLkop. Letting N go to 1, X 0 go to X and Y 0 go to Y gives the desired estimate on g. The case p = 1; q > 1 may be dealt with in the same way. We de ne E andR as we did in the previous case; then, we de ne for1 A 0 F = fx = Y (x; y)q0 d (y) > Aq0 g and f = sgn(g(x; y)) (x; y) q 1 1F (x). We nd that 0
jL(f )j = F ( Y R (x;Rz )q0 d (z )) d(x) kLkopkf kL (Lq ) = kLRkop(R F ( Y (x; z )q0 d (z )) q d(x)) kLkop ((F )) q0 ( F ( Y (x; z )q0 d (z )) d(x)) q This gives (F ) = 0 for A > kLkop. R
Aq (F )
R
1
1
1
1
The case of q = 1 is harder (except for p = q = 1 where we have the usual duality between L1 (X Y ) and L1(X Y )). We are going to show that for almost every x 2 X , the function g(x; y) is essentially bounded on Y . For this, we write X = ["n2IN Xn with (Xn ) < 1. We x A a positive real number, and we de ne the set E (depending on n and A) as E = f(x; y)=x 2 Xn ; y 2 Y; jg(x; y)j < Ag and the function as = 1E (x; y)jgj. Moreover, we write Y = ["k2IN Yk , we x > 0 and de ne for x 2 X the set Ex = fy 2 Yk = (x; y) > (1 )k (x; :)kL1(Y ) g. We then de ne h =
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Vector valued integrals
41 0
0
sgngk (x; :)kpL1(1Yk ) 1(EExx ) . We have khkLp(L1 ) = k kpLp=p0 (L1(Yk )) whereas L(h) = RR 0
(x; y)k (x; :)kpL1(1Yk ) 1E(xE(xy)) d (y) d(x) which gives that jL(h)j (1 0 )k kpLp0 (L1(Yk )) Thus, we get k kLp0 (L1(Yk )) 1 1 kLkop. We let rst go to 0, then we let Yk go to Y (and check that for a measurable function f on Y we have kf kL1(Y ) = supk2IN kf kL1(Yk ) ) ; the monotone convergence theorem gives us that k kLp0 (L1(Y )) kLkop. We may let Xn go to X and thus get that kgAkLp0 (L1(Y )) kLkop where gA (x) = g(x; y) if jg(x; y)j < A and gA = 0 otherwise. Since g is nite almost everywhere on X Y , we know that g(x; :) is nite almost everywhere on Y for almost every x 2 X , so that almost everywhere on X we have limA!1 kgA kL1(Y ) = kgkL1(Y ) , hence we get by the monotone convergence theorem that limA!1 kgAkLp0 (L1) = kgkLp0 (L1) kLkop. We complete this section with a density result:
Proposition 5.2: If X and Y are locally compact {compact metric spaces and if and are regular Borel measures on X and Y , then Ccomp(X Y ) is dense in Lp (X ; Lq (Y )) for 1 p < 1 and 1 q < 1. Proof: It is obvious (by the monotone convergence theorem) that, for r = max(p; q), Lrcomp(X Y ) is dense in Lp (Lq ). The proposition is then obvious, since Ccomp (X Y ) is dense in Lrcomp (X Y ).
2. Spaces Lp (E ) A nice space of test functions on (0; T ) IRd is the space T ((0; T ) IRd ) of the smooth functions, which are compactly supported in time and have rapid decay in space. We write (0; T ) = ["n2IN [an ; bn]; then T is the inductive limit of the Frechet spaces Tn = ff 2 T =Suppf [an ; bn ] IRd g, where the space Tn is equipped with the semi-norms
@ @p sup jx p f (t; x)j; 2 INd; 2 INd ; p 2 IN: @x @t an
We are thus able to take a partial Fourier transform with respect to the spatial variable: Z F f (t; ) = f (t; x)e ix: dx:
IRd
The space T 0 will then be the space of distributions ! on (0; T ) IRd so that for all [a; b] (0; T ) there exist C 0 and N 2 IN such that for all ' 2 D((0; T ) IRd , with Supp' [a; b] IRd , we have the inequality jh!j'ij C Pj jN Pj jN PpN kx @x@ @t@ pp f (t; x)k1 . The Fourier transformation may then be de ned on T 0 as usual by duality, through the formula
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Real harmonic analysis
hF !(t; x)Rj'(t; x)i = h!(t; x)jF '(t; x)i. (We use the Hermitian duality bracket hT j'i = T (t; x) '(t; x) dt dx.) P @ . Then Now, we write for the spatial Laplacian operator = di=1 @x 2 j d for all 2 CI the operator (Id ) operates on S (IR ), hence on T ((0; T ) IRd ) and on T 0 ((0; T ) IRd ). 2
We may now de ne the spaces Lp (E ) in use throughout the book:
De nition 5.2: (Space Lp (E )) s;1 Let E be a shift-invariant Banach space of distributions and let E B1 for some s 2 IR. Let M 2 IN, M > s=2. Then for 1 p 1 and T 2 (0; 1], we de ne Lp ((0; T ); E ) as the subspace of T 0 ((0; T ) IRd ) of distributions f such that: a) (Id ) M f 2 L1loc ((0; T ) IRd ); b) for almost all t 2 (0; T ), f (t; :) 2 E ; c) kkf (t; x)kE kLp ((0;T )) < 1.
Let us make some comments on this de nition. (Id ) M operates on 0 T , hence (a) is meaningful. Then, using the Fubini theorem, we get that for almost all t 2 (0; T ) (Id ) M f (t; :) belongs to L1loc (IRd ) and that for all test function ' 2 D((0; T ) IRd) we have the equality hf j'i(0;T )IRd = RT M M 0 h(Id )0 f (t; :)j(Id ) '(t; :)iIRd dt (where h j i is the duality bracket between D ( ) and D( )). We thus have given sense to f (t; :) in D0 (IRd) for almost all t 2 (0; T ). Thus, assertion b) is meaningful. In order to give sense to assertion c), we prove that t 7! kf (t; :)kE is Lebesgue measurable:
Lemma 5.1: Let E be a shift-invariant Banach space of distributions. Then, under the hypotheses a) and b) of De nition 5.2, we have: i) For all 2 D(IRd ), the map t 7! hf (t; :)j iIRr is Lebesgue measurable; ii) For all e 2 E , the map t 7! kf (t; :) ekE is Lebesgue measurable.
Proof: We know that for some M , we have f (t; x) = (Id )M g(t; x) where g 2 Lp ((0; T ); L1). Thus, f (t; :) is de ned for almost every t as f (t; :) = (Id )M g(t; :). If we want to compute kf (t) ekE , we write kf (t) ekE = sup 2H jhf (t) ej ij where H is a countable set of test functions (H S (IRd)) which is dense in the unit ball of the closure of S in E () (the existence of H is easy to check: for instance, the linear span of the Meyer wavelet basis ('(x k))k2ZZd ; (2jd=2 (2j x k))12d 1;j2IN;k2ZZd [MEY 92] is dense in E () ). Since hf (t) ej i = hg(t)j(Id )M i hej i, we see (using the Fubini theorem) that t 7! hf (t) ej i is measurable, hence t 7! kf (t) ekE is measurable. We now give a characterization of our spaces in terms of Bochner vector integration:
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Vector valued integrals
43
Theorem 5.2: (Bochner integrability) Let E be a shift-invariant Banach space of distributions. Then: (A) For 1 p < 1, D((0; T ) IRd ) is dense in Lp ((0; T ); E (0) ) and p L ((0; T ); RE (0)) is the space of strongly measurable f with values f (t; :) in E (0) such that 0T kf kpE dt < 1. (B) If 1 p 1 , if 1=p + 1=q = 1, then for all f 2 Lp ((0; T ); E ) and R g 2 Lq (0; T ), we have 0T g(t)f (t; x) dt 2 E . If f belongs more precisely to R Lp ((0; T ); E (0)), then 0T g(t)f (t; x) dt 2 E (0) . To prove (A), we use the following characterization of Bochner integrability:
Lemma 5.2:
A function f de ned on (0; T ) with values in a separable Banach space E is strongly measurable if and only if for all e 2 E the function kf (t) ekE is measurable.
Proof of the lemma: This lemma is easy to prove. IfPf is strongly measurable, i.e., if there is a sequence of simple functions fn = 1kNn 1Xk;n (t) ek;n which converges almost everywhere to f , then it is obvious that the function kf (t) ekE is measurable. Conversely, since E is separable, for all n 2 IN , there is a sequence (xk;n )k2IN so that E = [k2IN B (xk;n ; 1=2n). Then, we de ne the set Yk;n = ft 2 (0; TP) = f (t) 2 B (xk;n ; 1=2n) and 8p < k f (t) 2=PB (xp;n ; 1=2n)g. We have (0; T ) = k2IN Yk;n . Then, we choose K (n) so that kK (n) jYk;n \ (0; n)j P min(T; n) 21n . By construction, the sequence kK (n) 1Yk;n (t) xk;n converges almost everywhere to f . Thus, Lemma 5.2 is proved. Proof of Theorem 5.2: We now prove the theorem. If f 2 Lp ((0; T ); E (0) ) and p < 1, it is easy to see that we may apply lemma 5.2 ; E (0) is separable since S is dense in E (0) and since the imbedding of S into E (0) is continuous. We now apply Lemmas 5.1 and 5.2 to conclude that f is strongly measurable. Now, we know that we may approximate f by a simple function gn so that Yn = Supp(gn ) (0; min(T; n)), jYn j min(T; n) 21n Rand supt2Yn kf (t) gn (t)kE < 21n . We nd that kf gn kLp((0;T );E 2nn + ( t=2Yn kf (t)kpE dt)1=p . Since p is nite, we nd that gn converges to f in Lp((0; T ); E (0) ). Now, when g is a simple function, it is obvious that g can be approximated in Lp ((0; T ); E (0) ) by a sequence of test functions. Thus, (A) is proved. (B) is quite obvious. If f 2 Lp ((0; T );RE ) and g 2 Lq ((0; T ); E ), then we write f = (Id )M h and we see that 0T f (t; x)g(t) dt is de ned in S 0 R R as (Id )M 0T g(t)h(t; x) dt; moreover, we have h 0T f (t; x)g(t) dtj (x)i = RT (by the Fubini theorem). Thus, f (t; x)g(t) belongs to 0 hf (t; x)j (x)i g(t)R dt T 1 L ((0; T ); E ) and jh 0 f (t; x)g(t) dtj (x)ij kf (t; x)g(t)kL1 ((0;T );E) k kE() .
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Real harmonic analysis R
This gives that 0T f (t; x)g(t) dt belongs to E . If f 2 Lp ((0; T ); E (0), we approximate f (t; x)g(t) in L1 ((0; T );RE (0)) by a sequence of test functions n 2 D((0; T ) IRd); weRthen nd that 0T n dt is a Cauchy sequence in E (0) which converges in S 0 to 0T f (t; x)g(t) dt.
3. Heat kernel and Besov spaces Since throughout the book we use the heat kernel operating on shiftinvariant Banach spaces of distributions, it will be very useful to characterize the action of the heat kernel on the Besov spaces associated with such spaces.
Theorem 5.3: (Heat kernel and Besov spaces) Let E be a shift-invariant Banach space of distributions. For 2 IR, 1 q 1, t0 > 0, 0 so that > and f 2 S 0 (IRd), the following assertions are equivalent : (A) f 2 BE;q p (B) for all t > 0, et f 2 E and (t =2 ( t) pet f )0
p p Proof:p We rst notice that ( t) et = ( t) et=2 et=2 ; the operator ( t) et=2 is a convolution operator with a kernel td=1 2 k ( pxt ) where p k 2 L1 , thus k( t) et f kE C ket=2 f kE . P (A) ) (B ): We may assume t0 = 1. We write f = S0 f + j0 j f with S0 f 2 p E and kj f kE = 2 j j with (j )j2IN 2 lq . We estimate the norm kt =2 ( t) etf kE by p p kt =2 ( t) etS0 f kE t( )=2 k( ) S~0 S0 f kE C t( )=2 kS0 f kE p since ( ) S~0 is a convolution operator with an integrable kernel. Similarly, writing j f = using the integrability of the kernel of the p~ j j f and ~ 0 , we get convolution operator ( ) p kt =2 ( t) et j f kE C t( )=2 2j kj f kE : p p p we write ( t) etj f = t( N )=2 ( p t)N et( )
Then, and, owing to the integrability of the kernel of ( N 0,
p kt =2 ( t) et j f kE C ;N t(
N ~ j j f
~ 0 , we get that, for ) N
N )=2 2j ( N ) k
j f kE :
For = 0 and N > 0, we thus have proved that et f 2 E for all t > 0. Now, if t 1, we choose that 1=4 < 4j0 t 1 and we choose N > ; we obtain p j0 so t = 2 that kt ( t) e f kE is bounded by j0 where
X X j0 = C (2 j0 ( ) kS0f kE + 2(j j0 )( ) j ) + C ;N 2(j j0 )( N ) j : j>j0 j j0
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Vector valued integrals
=2 p
45
We have kt ( t) et f )kLq ((0;1); dtt ;E) (ln 4)1=q k(j )j2IN klq (IN) and we may conclude since the sequence (j = 2j( ) if j 0; = 2j( N ) if j > 0) belongs to l1 (ZZ), hence (j ) (j ) 2 lq (IN). (B ) ) (A): We may assume t0 = 1. We get that S0 f 2 E by writing S0 f = e S0 e f and using the integrability of the p kernel of thep convolution operator e S0 . Similarly, we write j f = e t( pt) j ( t) etf . For 1=4 < 4j t 1, the convolution operator e t ( t) j has an integrable kernel k ;j;t withp kk ;j;t k1 C , so that we nally obtain 2j kj f kE max(1; 2 )C t =2 k( t) et f kE for 1=4 < 4j t 1. This theorem may be easily applied to the case of p homogeneous Besov spaces. In order to avoid any problem of de nition for ( t) et f , we shall consider only the case = 0 and < 0:
Theorem 5.4: (Heat kernel and homogeneous Besov spaces) Let E be a shift-invariant Banach space of distributions. Let 2 IR, < 0, 1 q 1, and f 2 S 0 (IRd). Then: (A) If, for all t > 0, et f 2 E and if t =2 et f 2 Lq ((0; 1); dtt ; E ), then f belongs to B_ E;q \ S00 (IRd ) ; (B) Conversely, if f belongs to B_ E;q \ S00 (IRd ), then, for all t > 0, et f 2 E and t =2 et f 2 Lq ((0; 1); dtt ; E ). Moreover, in that case, the norms kt =2 et f kLq ((0;1); dtt ;E ) and kf kBE;q are equivalent.
;1 . Proof: Let us rst remark that S0 maps E to L1 , hence maps B_E;q to B_ 1 ;q ;q d 0 _ _ Thus,Pwe may de ne the realization BE S (IR ) of BE by the isomorphism f 7! j2ZZ j f . The proof of Theorem P 4 is exactly the same as for non homogeneous spaces, since we have etf = j2ZZ etj f for f 2 S00 . In particular, if t =2 etf 2 P Lq ((0; 1); dtt ; E ), then f belongs to B_ E;q ; thus, we may write f = j2ZZ j f + P = f0 + P with P 2 C[ I X1 ; : : : ; Xd ]; since f0 2 B_ E;q \ S00 (IRd ), we nd that = 2 t q t e f0 2 L ((0; 1); dtt ; E ), hence t =2 et P 2 Lq ((0; 1); dtt ; E ) ; but P = t e S0 etP , hence P R2 E ; now, this would give that, for any ! 2 D, we should have supx0 2IRd j !(x x0 )P (x) dxj < 1 and this implies that P is a constant P = p 2 C; I but in that case we have etP = P and we nd that = 2 q t P 2 L ((0; 1); dt ; E ), hence that P = 0 and f belongs to S00 . t
We give an example of application with the Sobolev embeddings:
Proposition 5.3: (Heat kernel and Sobolev embeddings) Let 2 IR, 0 < < d. Then : (A) If 1 < p < 1, 1 q 1 and < d=p and if 1r = 1p d , then the operator 1 p;q 0 _ 0;q (p ) is continuous from L to BLr;1 \ S0 .
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Real harmonic analysis
In particular, there exists a constant Cp; so that, for all f 2 Lp;1 (IRd ), we R1 have 0 ket (p 1) f kLr;1 dt Cp; kf kLp;1 . 0;q . (B) If = d=p, then the operator (p 1) is continuous from Lp;q to B_ 1 In particular, Rthere exists a constant Cp; so that, for all f 2 Lp;1 (IRd ), we have the estimate 01 ket (p 1) f k1 dt Cp; kf kLp;1
Proof: We use the Sobolev embeddings for 1=p1 = 1=p and 1 = d and 1=p2 = 1=p + and 2 = + d and for 1=r = 1=p =d = 1=pi i =d to get that (p 1) maps Lp1 ;1 to H_ Ldr;1 and Lp2 ;1 to H_ Lr;d1 , hence by interpolation 1 d=p maps Lp;q to B_ 1 0;q . Lp;q to B_L0;qr;1 . The same proof works to get that (p ) Moreover, if < d=p, we know that (p 1) maps Lp;q to Lr;q S00 . If 1 d=p maps Lp;1 to L1 ; we = d=p and q = 1, we already know that (p ) easily check that test functions are dense in Lp;1 and that for f 2 S we have 1 1 p;q 0 (p )d=p f 2 C0 , (p )d=p maps more precisely L to C0 S0 . For q > 1, we 1 d=p maps the Hardy space H1 to Lp=(p 1);1 , shall see in Chapter 6 that (p ) 0;1 ; this is enough to get that p 1 d=p maps hence maps Lp;1 to BMO B_ 1 ( ) 0;q of B_1 0;q . Lp;q to the realization B_ 1 The nal P estimates on the heat kernel are then provided by the equality et f = j2ZZ j et f for f 2 S00 .
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Chapter 6 Complex interpolation, Hardy space, and Calderon{Zygmund operators 1. The Marcinkiewicz interpolation theorem and the Hardy{Littlewood maximal function In this section, we begin by recalling a basic theorem of modern analysis, namely, the Marcinkiewicz interpolation theorem stated without proof by Marcinkiewicz in 1939 [MAR 39] and proved by Zygmund [ZYG 56] in 1956.
De nition 6.1: (Weak Lp space) For 1 p < 1, the space Lp; (which is called the weak Lp space) is de ned by
2 Lp; , 9C 0 8 > 0 (fx = jf (x)j > g) C
f
p
A map T de ned from Lp (U; d) to Lp; (V; d ) is a sublinear operator if it satis es jT (f + g )j jjjT (f )j + jjjT (g )j. A sublinear operator T is bounded from Lp (U; d) to Lp; (V; d ) if there exists a constant C so that for all > 0 and all f 2 Lp (U; d) we have (fx = jT f (x)j > g) C kf kpp p . (If p = 1, we shall say that T is bounded from Lp (U; d) to Lp; (V; d ) if T is bounded from L1 to L1 ).
Theorem 6.1: (The Marcinkiewicz interpolation theorem) If 1 p0 < p1 1 and if T is a bounded sublinear operator from Lpi (U; d) to Lpi ; (V; d ), then T is bounded from Lp (U; d) to Lp (V; d ) for 1 1 p = p0 + p1 ; 0 < < 1. P
Proof: We just write j T f j pp j2ZZ 2jp (fx=jT f j > 2j g) and decompose f into fj + gj where fj = f 1fx=jf j>2j g . If p1 = 1, we choose k 2 ZZ so that 2k kT kop(L1;L1) 1=2 and we write fx=jT f j > 2j g fx=jT fj+k j > 2j 1 g kT kpop0 (Lp0 ;Lp0; ) kfj+k kpp00 2(1 j)p0 and thus we get for M0 = kT kop(Lp0 ;Lp0; ) and M1 = kT kop(L1;L1) :
j T f j pp M0p
Z
0
jf (x)jp
0
U
X
j 2ZZ;2j+k <jf (x)j
2jp 2(1 j)p0 d(x) CM0p0 M1p p0 kf kpp
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Real harmonic analysis
If p1 < 1, we write Ci = kT kop(Lpi ;Lpi; ) and we choose k 2 ZZ so that (p p0 )p1 (p1 p)p0 2k(p0 p) C0p0 2k(p1 p) C1p1 C0 p1 p0 C1 p1 p0 . We write : fx=jT f j > 2j g fx=jT fj+k j > 2j 1 g + fx=jT gj+k j > 2j 1 g and get
fx=jT f j > 2j g C0p0 kfj+k kpp00 2(1 j)p0 + C1p1 kgj+k kpp11 2(1 j)p1 which gives
j T f j pp and thus
8 <
:+
C0p0 U jf (x)jp0
R
P
jp 2(1 j )p0
d(x)
C1p1 U jf (x)jp1
R
P
jp 2(1 j )p1
d(x)
j 2ZZ;2j+k <jf (x)j 2 j 2ZZ;2j+k jf (x)j 2
j T f j pp C (2k(p p) C0p 0
0
+ 2k(p1 p) C1p1 )kf kpp
Remark: The Marcinkiewicz theorem remains valid for an operator on vector valued functions: for instance, if X = Lq0 (E0 ) and Y = Lq1 (E1 ), if T satis es kT (f )kY = jjkT (f )kY , kT (fp+ g)kp Y C (kT (f )kY + kT (g)kY ) and (fv 2 V = kT (f )(v)kY > g) Ci i kf kLipi (U;;X ) pi , then T is bounded from Lp (U; ; X ) to Lp (V; ; Y ) for p0 < p < p1 . Now, we prove two classical covering lemmas: the lemmas of Vitali and Whitney. We consider a separable quasimetric space (X; d), i.e., X is a separable metrizable space, the function d is continuous on X X and satis es: i) d(x; y) 0 and d(x; y) = 0 , x = y ii) for all x 2 X the family B (x; ) = fy 2 X = d(x; y) < g is a basis of neighbourhoods of x iii) there exists a constant K such that, for all x; y; z 2 X , d(x; y) K (d(x; z ) + d(y; z )).
Proposition 6.1: (The Vitali covering lemma) Let E X be decomposed as a union of balls E = [ 2A B (x ; r ), where (B (x ; r )) 2A is a family of balls so that sup r < 1. Then there exists a (countable) subfamily of balls (B (x ; r )) 2B (B A) so that 6= ) B (x ; r ) \ B (x ; r ) = ; and so that E [ 2B B (x ; 5K 2r ). Proof: We construct B by trans nite induction. Let be the rst uncountable cardinal number; then we de ne (B ) in the following way: for choosing B , we consider whether there exist balls B (x ; r ) that do not meet O = [< B . If so, we choose so that B (x ; r ) \ O = ; and r > 1 2 supfr = B (x ; r ) \ O = ;g; in that case, we choose B = B (x ; r ); otherwise, we choose B = ;. It is easy to check that B = ; : X is separable and the balls B are disjoint open sets. For 2 A, we de ne =
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Hardy space
49
minf = B (x ; r ) \ O 6= ;g ; then, we have B (x ; r ) \ B (x ; r ) 6= ; and r 2r . If y 2 B (x ; r ) \ B (x ; r ) and z 2 B (x ; r ), we have d(x ; z ) Kd(x ; y) + K 2(d(x ; y) + d(x ; z )) 5K 2r .
Remark: An equivalent proof may be done without introducing trans nite induction : let E be the set of of families of balls (B (x ; r )) 2F so that the balls of F are disjoints and such that for all 2 A either B (x ; r ) does not meet [ 2F B (x ; r ) or there exists 2 F with B (x ; r ) \ B (x ; r ) 6= ; and r 2r ; then E is inductive (with respect to usual set ordering) and a maximal element of E (whose existence is granted by Zorn's lemma) satis es the conclusion of Proposition 1. Proposition 6.2: (The Whitney covering lemma) For all M > 1P , if G is an open subset of X then we may write G as a disjoint union G = n2IN Gn where the sets Gn are Borelian and satisfy (when not empty) diam Gn M1 dist(Gn ; X nG). Proof: Let D be a countable dense subset of X . If x 2 G, there exists y 2 D so that d(x; y) < M11 dist(x; X nG). Let r 2 Q\]d(x; y); M11 dist(x; X nG)[. Then, x 2 B (y; r) with diam(B (y; r)) 2Kr while for z 2 B (y; r) and w 2 X nG, we ) 2Kr (M1 =K 2K )r. Thus, it is enough to choose have d(z; w) d(x;w K 2 M1 so that M12K2K 2 < M1 . Finally, we introduce the main result of this section: the boundedness of the Hardy{Littlewood maximal function on Lp . We consider a separable quasimetric space (X; d) and a Borelian measure on X . The Hardy{Littlewood maximal function Mf for a measurable function f isR de ned ( almost everywhere) by Mf (x) = supr>0; 0<(B(x;r))<1 (B(1x;r)) B(x;r) jf (y)j d(y). Similarly, the modi ed Hardy{Littlewood maximal R function of f is de ned by M~ f (x) = supr>0; 0<(B(x;5K 2 r))<1 (B(x;15K 2 r)) B(x;r) jf (y)j d(y).
Theorem 6.2: (Integrability of the Hardy{Littlewood maximal function) (a) If has the doubling property, then the map f ! Mf is bounded from L1 (X; ) to L1; (X; ) and from Lp (X; ) to Lp (X; ) for p > 1. (b) If does not have the doubling property, the same conclusion holds for ~f. the modi ed Hardy{Littlewood maximal function M Proof: The case p = 1 is obvious. Therefore, it is enough to consider the case p = 1. We de ne EN = fx = 9r < N so that 0 < (B (x; 5K 2 r) < R 1 1 and (B(x;5K 2 r) B(x;r) jf (y)j d(y) > g and E = fx = Mf (x) > g; then E is the (monotone) union of the sets EN . The Vitali covering lemma gives 1X (EN ) (B (x ; 5K 2r )) 2B 2B X
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Z
B (x ;r )
jf (x)j d(x)
Z
1 jf j d X
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Real harmonic analysis
R and this is enough to get (E ) 1 X jf (x)j d(x).
2. The complex method in interpolation theory The complex method in interpolation theory was introduced by Calderon [CAL 64], as a generalization of the well-known interpolation theorem of RieszThorin. We make little use of complex interpolation. The reader may nd a precise description of this theory in Bergh and Lofstrom [BERL 76].
De nition 6.2: For 0 < < 1, we de ne the complex interpolation space [A0 ; A1 ] of A0 and A1 as follows: Let F (A0 ; A1 ) be the space of bounded continuous functions f from fz 2 CI =0
Theorem 6.4: (Lp spaces) [Lp0 ; Lp1 ] = Lp for 0 < < 1 and p1 = 1p0 + p1 with equality of norms. Theorem 6.5: (Weighted Lebesgue spaces) Let (X; ) be a measured space and w0 ; w1 two positive weights on X . Then, for 1 p < 1 we have [Lp (w0 d); Lp (w1 d)] = Lp (w01 w1 d) for 0 < < 1 with equality of norms. Proof: Lp [Lp0 ; Lp1 ] and Lp (w01 w1 d) [Lp (w0 d); Lp (w1 d)] is obvious. Indeed, ifp f 2p Lp , with kf kp = 1, we de ne for > 0 F (z ) = 2 2 e(z ) jff j jf j(1 z) p0 +z p1 . For z = + i, we nd that F (z ) 2 Lq with 2 2 2 1=q = (1 )=p0 + =p1 and kF(z )kq = e( ) ; this shows that f 2
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51
[Lp0 ; Lp1 ] and that kf k[Lp0 ;Lp1 ] e( ) ; then, letting go to 0, we nd that kf k[Lp0 ;Lp1 ] kf kp. Similarly, if f 2 Lp (w d) where w = w01 w1 d, 2 2 we de ne for > 0 F (z ) = e(z ) w1wzwz 1=p f ; then, for z = + i, we nd 0 1 2 2 2 that F (z ) 2 Lp (w01 w1 d) and kF (z )kLp(w d) = e( ) kf kLp(w d) ; p p this shows that f 2 [L (w0 d); L (w1 d)] and that kf k[Lp(w0 d);Lp(w1 d)] kf kLp(w d) . The converse inequalities are quite as easy. We consider f 2 [Lp0 ; Lp1 ] , f = F () with F 2 F (Lp0 ; Lp1 ). We de ne q0 ; q1 ; q as the conjugate exponents of p0q; p1 ;qp. If g is a simple function with kgkq 1, we Rde ne G(z ) = g (1 z) q0 +z q1 . For every z , G is a simple function,thus I (z ) = F (z )G(z ) d jgj jgj is well de ned. This is a continuous bounded function on fz 2 CI =0
2
We now mention the duality theorem for complex interpolation:
Theorem 6.3: (Duality) If A0 \ A1 is dense in A0 and in A1 and if A0 or A1 is re exive, then [A0 ; A1 ]0 = [A00 ; A01 ] for 0 < < 1 with equality of norms. Proof: See the original paper of Calderon [CAL 64] or the book of Bergh and Lofstrom [BERL 76].
3. Atomic Hardy space and Calderon{Zygmund operators In this section, we study the interpolation between the Hardy space and the Lebesgue spaces Lp .
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H1
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Real harmonic analysis
De nition 6.3: (Atomic Hardy space) d ) is de ned as: f 2 H1 if and only if f The (atomic) Hardy space H1 (IRP may be decomposed into a series f = j 2IN j aj where Supp aj is contained in R P a ball B (xj ; Rj ), j aj j 1 (Rj ) d , aj dx = 0 and jj j < 1. The function aj is called an atom carried by the ball B (xj ; Rj ). P
P
It is normed by kf kH1 = inf f j jj j = f = j j aj g where the minimum runs over all the atomic decompositions of f . Then, it is easy to see that H1 is a Banach space which is continuously embedded in L1.
Theorem 6.7: [H1 ; Lp ] = Lq where q1 = 1 + p and 1 < p < 1. Proof: Since H1 L1, we have obviously [H1 ; Lp ] [L1; Lp ] = Lq . In order to prove the reverse inclusion, we begin by the construction of atomic decompositions in Lq . For f 2 Lq , 1 < p < 1, we de ne f its HardyLittlewood maximal function and k = fx= f (x) > 2k g. k is an open set with nite measure (since q < 1). Now, we call Qk the family of dyadic cubes Q = l=2j + 1=2j [0; 1]d k so that the double cube 2 Q = l=2j + 2=2j [0; 1]d is still included in k but not the quadruple cube 4 Q = l=2j + 4=2j [0; 1]d; we then P have k = [Q2Qk Q and j k j = Q2Qk jQj. Now, for k 2 ZZ and Q 2 Qk , we introduce the function Q;k = 1Q (f
mQ f
X
R2Qk+1
1R (f
mR f )) where mR f =
Z
1 jRj R f (x) dx:
We rst notice that for any two dyadic cubes Q; R, 1Q 1R 6= 0 if and only if Q R or R Q; for R 2 Qk+1 and Q 2 Qk , we notice that R k and 2 R
k , so that 1Q 1R 6= 0 if and only if R P Q; this givesPthat Supp k;Q Q and R dx = 0. Clearly, we have that Q2Qk Q;k = P Q2Qk 1Q (f mQ f ) P Q;k 1 ( f m f ). Let b be the function b = R k k R2Qk+1 R Q2Qk 1Q (f mQ f ) P and gk = f bk = f 1IRd k + Q2Qk 1Q mQ f . By construction, jgk j C 2k and jbk j C f 1 k , hence:
f=
X
k2ZZ
gk+1
gk =
X X
k2ZZ Q2Qk
Q;k
Moreover, we have j k;Q j C 2k : since the family P Qk is essentially disjoint, we may write Q;k = 1Q f 1IRd k+1 1QmQ f + R2Qk+1 1Q 1R mR f and get the required estimate. P P We thus got an atomic decomposition f = k2ZZ Q2Qk Q;k . Now, we P P z de ne for
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Hardy space P
1 1
k2ZZ 1=q 1=p .
P
kq=p 1Q (x)
Q2Qk 2
C (f (x))q=p .
This gives Lq
53
[H1 ; Lp ]
for =
We have proved another interesting result concerning H1 .
Proposition 6.3: (p-atoms) Let HPp1 be the space of functions f , which may be decomposed into a series f = j 2IN j aj where Supp aj is contained in a ball B (xj ; Rj ), j aj j p R P 1 for all p 2 (1; 1). (Rj )d=p d , aj dx = 0 and jj j < 1. Then Hp1 = H1 Proof: It is enough to prove that a p-atom supported by the ball B (0; 1) belongs to H1 with a bounded norm. We thus take a 2 Lp with Supp a R B (0; 1), j aj p 1 andP a dx = 0. We use the atomic decomposition of a P P described above: a = = Q;k k2ZZ Q2Qk k gk+1 gk . But we better write P a = g0 + k2IN gk+1 gk . Indeed, the function h = a g0 is supported by
0 which is bounded (since a (x) (kxkC 1)d for kxk 2). Moreover, we have R P P h = k2IN Q2Qk Q;k and khkH1 Pk2IN PQ2Qk 2k jQj C 0 a dx R C j 0 j(p 1)=p ka kp . Thus, h dx = 0 and nally g0 = a h is a bounded function with a bounded support and a zero integral ; thus, g0 2 H1 and this concludes the proof. We now give an easy application of the preceding theorem to the analysis of some classical operators. Let us recall the de nition of (generalized) Calderon{ Zygmund operators given by Coifman and Meyer [COIM 78] [MEY 97]:
De nition 6.4: (Calderon{Zygmund operators.)
A Calderon{Zygmund operator is a bounded linear operator T on L2 (IRd ) (T 2 L(L2 ; L2 )) so that there exists a function K (x; y ) de ned on IRd IRd (where is the diagonal set x = y ) which satis es: i) There exists a positive constant C0 so that
8x 8y jK (x; y)j jx C0yjd ii) There exists two positive constants and C1 so that 8x 8y 8z 2 B (0; 12 jx yj) jK (x + z; y) K (x; y)j C1 jx jzyj jd+
iii) There exists two positive constants and C1 so that 8x 8y 8z 2 B (0; 12 jx yj) jK (x; y + z ) K (x; y)j C1 jx jzyj jd+
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Real harmonic analysis
iv) For all f
2 L2comp and all g 2 L2comp,
Suppf \ Suppg = ; ) hT f jgi =
Z Z
K (x; y)f (y)g(x) dx dy
Theorem 6.8: Let T be a Calderon{Zygmund operator. Then, T is bounded from H1 (Hardy) to L1 , from L1 to BMO and from Lp to Lp for 1 < p < 1. Proof: We notice that the estimates on T are invariant under transposition, under translation and under dilation. Then, the proof is easy. Just test T on an atom supported by B (0; 0). T (a) is locally integrable, since it belongs R to L2 . Since a dx = 0, we use the regularity assumption on K to get that T (a) is O(kxk d ) at in nity. Then, we use translations and dilations to deal with a general atom. This gives the boundedness from H1 to L1 . Interpolation between H1 and L2 gives the result for 1 < p 2 and transposition gives the result for p > 2 (since BMO = (H1 ) ). Example: A typical example of a Calderon{Zygmund operator is the Riesz transform Rj = p@j . The boundedness of Rj on L2 (IRd) is obvious: using the Fourier transform, we see that Rj is a Fourier multiplier with a bounded function F (Rj f )( ) = ijjj F f ( ). The Lp boundedness (for 1 < p < 1) is then a consequence of the Calderon{Zygmund theory since p@j Æ is a homogeneous distribution with homogeneity degree d on IRd . We may also use atomic decomposition and complex interpolation to get an elementary proof of the Sobolev inequalities :
Theorem 6.9: (Sobolev inequalities.) a) For 0 < < d, the operator ( p 1 ) is bounded from the Hardy space H1 to L d d and from Ld= to BMO = (H1 ) . b) For 1 < p < 1 and 0 < < n=p the operator ( p 1 ) is bounded from Lp to Lq where 1q = p1 d . Proof: It is enough to check that the transform ( p 1 ) (a) of an atom a de ned d on B (0; 1) belongs to L d with a bounded norm. Then, using the invariance through translations and the homogeneity of the kernel, we see that this gives d the boundedness from H1 to L d , then self-adjointness gives the boundedness d from L d = (L d ) to (H1 ) . Finally, we conclude by interpolation (using Theorem 6.6 and the duality theorem). To estimate the transform of an atom a, we note that the transform ( p 1 ) is a convolution operator with a kernel jxjcd ; this kernel is uniformly locally
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Hardy space
55
integrable, and thus ( p 1 ) a is bounded. Moreover, if jxj > 10, we may R write ( p 1 ) a(x) = c jyj1 a(y) jx y1jd jxjd1 dy, thus ( p 1 ) a(x) is 1 ). We thus get that ( p 1 ) a belongs to Lr for all r > d . In O( jxjd+1 d+1 d d particular, ( p 1 ) a belongs to L d (and even to L d ;1 ). The proof gives a better statement for the space H1 :
Proposition 6.4: For 0 < < d, the operator ( p 1 ) is bounded from the d Hardy space H1 to the Lorentz space L d ;1 and from Ld= ;1 to BMO = (H1 ) .
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Chapter 7 Vector-valued singular integrals In this chapter, we recall the basic result of the Calderon{Zygmund theory on vector-valued singular integrals. Then, we give two applications : the characterization of Lp by the the Littlewood{Paley decomposition and the maximal Lp (Lq ) regularity of the heat kernel.
1. Calder on{Zygmund operators We consider three locally compact {compact metric spaces X; X1; X2 , regular Borelian measures ; 1 ; 2 on those spaces and three numbers p; p1; p2 2 (1; 1). We de ne E = Lp1 (X1 ; 1 ) and F = Lp2 (X2 ; 2 ). We consider a linear operator T bounded from Lp (X; ; E ) to Lp(X; ; F ). We assume that there exists a continuous function L(x; y; x1 ; x2 ) de ned on (X X )X1 X2 (where is the diagonal set of X X ) so that, whenever x 2= Suppf , i.e., when there exists r > 0 so that f is equal to 0 R R 1 -almost everywhere on B (x; r) X1 , then T f (x) is given by T f (x; x2 ) = X X1 L(x; y; x1 ; x2 )f (y; x1 ) d(y) d1 (x1 ). We de ne L(x; y) the operator Rfrom E = Lp1 (X1 ; 1 ) to F = Lp2 (X2 ; 2 ) given by the integral L(x; y)f (x2 ) = X1 L(x; y; x1 ; x2 )f (x1 ) d1 (x1 ). The Calderon{ Zygmund theory gives suÆcient condition to extrapolate from the Lp boundedness to the Lq boundedness for other values of q:
Theorem 7.1: (Calderon{Zygmund operators) We consider a quasi-distance d on X (de ning the topology of X ), satisfying the quasi-triangular inequality d(x; y ) K (d(x; z )+ d(z; y )), such that there exists positive real numbers n and C such that for all x 2 X and all r > 0 (B (x; r)) C rn . (A) Assume that T is bounded from Lp (X; ; E ) to Lp (X; ; F ): Z
Z
kT (f )(x)kpF d(x) C0p kf (x)kpE d(x) R
Assume that whenever x 2= Suppf we have T (f )(x) = L(x; y ) f (y ) d(y ) where the function L is continuous from X X to L(E; F ) and satis es for some positive : (
(CZ )
kL(x; y)kop(E;F ) C1 d(x;y1 )n (z;yn) d(z; y) 21K d(x; y) ) kL(x; y) L(x; z )kop(E;F ) C2 d(dx;y ) +
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Real harmonic analysis
Then, T is bounded from L1 (X ; E ) to L1; (X ; F ) and, for 1 < q < p, from Lq (X ; E ) to Lq (X ; F ). (B) If, moreover, L satis es
(CZ )
d(x; z )
1 d(x; z ) d(x; y) ) kL(x; y) L(z; y)kop(E;F ) C2 2K d(x; y)n+
then T is bounded from Lq (X ; E ) to Lq (X ; F ) for p < q < 1.
Remark: We give here a proof derived from the recent paper of Nazarov, Treil and Volberg [NAZTV 98]. In contrast to classical proofs (such as in Coifman and Meyer [COIM 78], Coifman and Weiss [COIW 77], Stein [STE 93],: : :), no doubling property is assumed on the measure . Proof: We consider > 0 and f 2 Ccomp(X ; E ) and we need to estimate (fx = kT f (x)kF > g). We de ne G = fx = kf (x)kE > g, f = f 1G and g = f f . We rst notice that G is an open set (since f is continuous) and has a nite kf k measure: (G ) L1 (E) . Now, we de ne X = (fx = kT f(x)kF > =2g) and Y = (fx = kT g(x)kF > =2g). We have clearly (fx = kT f (x)kF > g) (X ) + (Y ). Since g 2 Lp (X ; E ), weRmay easily estimate the measure of Y by (Y ) 2kT g kLp(X;F ) p (2C )p kf kE d . Besides, we have (X ) (G ) + 0 P (X G ). We now use a Whitney decomposition ofPG : G = i Gi with diam(Gi ) 21K d(Gi ; X nG ). PWe may write f = i fi in Lp (X ; E ) with fi = f 1Gi , so that T (f) = i T (fi) in P Lp (X ; F ); thus, we may nd a sequence (Nk ) so that T (f )(x) = limNk !1 iNk T (fi )(x) almost everywhere. For each Gi , we choose a point xi 2 Gi and then we write for x 2= G Z
T (fi )(x) = (L(x; y) L(x; xi ))fi (y) d(y) + L(x; xi ) Thus, we get kT (f)(x)kF
A(x) =
Z
fi (y) d(y):
A(x) + B (x) where
X Z
i
k (L(x; y) L(x; xi ))fi (y) d(y)kF
R
P
and B (x) = k i L(x; xi ) fi (y) d(y)kF . We then write X G [ A [ B with A = fx 2= GR = A(x) > =4g and B = fx 2= G = B (x) > =4g. kf kE d We have (G ) < . We easily estimate (A ) by estimating I = R A ( x ) d ( x ). Indeed, we have: X nG
I C2
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XZ
i
d(xi ; y) 1 kf (y)k d(y) d(x) d(x; xi ) d(x; xi )n i F
Singular integrals
59
For y 2 Gi , x 2= G and di = diamGi , we have d(xi ; y) di 21K d(x; xi ) while for l 2 IN we have Z 1 di d(x) C 2n (2K ) 2 l n d ( x; x ) d ( x; x ) l l +1 i i 2 di2Kd(x;xi)<2 di
n P R and nally, I R C2 C12 2(2K ) i Gi kf kE d. This gives the estimate (A ) < 4C2 C 2n (2 K ) kf kE d . 1 2 R kf kE d We now estimate (B ). We begin by checking that, if (X ) > , we may nd disjoint sets Ei and positive real numbers Æi so that (writing Bi = B (xi ; Æi ) and B i = Bcl (xi ; Æi ) for the open and closed balls with center xi and radius Æi ) we have : R
kf k d : Bi [ji Ej and (Ei ) = Gi R kf k d We build Ei by induction. We have (X [j Gi . Then, we k f k d Gi de ne Æi by Æi = min g. We thus have R fd = (B (xi ; d) [j k f k d (B i [j
Bi;
Lemma 7.1: If A; B are two measurable sets in X and r > 0, if moreover A B and (A) r (B ), then there exists C so that A C B and (C ) = r. This lemma is easy. We may assume that (A) < r < (B ). Let D =
fd0 ; d1 ; : : :g be a countable subset of X . Let > 0. Then X = [i2IN B (di ; ). We x N the smallest N so that (A [ ([iN (B (di ; ) \ B ))) is bigger than r. Then, AN = (A [ ([i
R
Now, we write Gi f d = 1Ei d ui with kui kE = 1. Thus, we have
L(x; xi )
Z
Gi
f d = L(x; xi )
The next step is to compare in some way X
T(
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i
1Ei ui ) =
XZ
i
P
Z
1Ei ui d
i L(x; xi )
R
fi (y) d(y) to
L(x; y)1Ei (y)ui d(y)
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Real harmonic analysis
Indeed, we have
R
X (4C0 )p kf kE d (fx= kT ( 1Ei ui )kF > g) (4C0 )p k 1Ei ui kpLp(X ;E) 4 i i Alas, the estimate we need is Rnot so direct. We look only at the case x 2= R P kf kE d . Let (x) = i L(x; xi ) fi (y)d(y) E1 = [i Ei since R P ([i Ei ) and (x) = Ri 1X nB(xi ;2KÆi ) (x) L(x; y)1Ei (y) ui d(y). In order to estimate , we write X nE1 k kF d U + V where X
U=
X
and
i
V=
Z
Z
X nB (xi ;2KÆi )
kL(x; xi ) L(x; y)kop(E;F )1Ei (y) d(y) d(x)
XZ
Æi d(x;xi)2KÆi
i
kL(x; xi )kop(E;F ) k
Z
R
Gi
f dkE d(x)
On Ei , d(xi ; y) Æi , thus X nB(xi ;2KÆi ) kL(x; xi ) L(x; y)kop(E;F )d(x) C C2 , R which gives U C C2 R(E1 ) C C2 kf kE d. We get a similar estimate for V : V C1 C (2K )n kf kE d since for Æi d(x; xi ) 2KÆi we have
kL(x; xi )kop(E;F ) C1 d(x;1x )n C1 C (2K )n (B (x 1; 2KÆ )) i
i
i
To estimate (B ), we may write B E1 [ C [ D where we have C = fx 2= E1 = k kF > =8g and D = fx 2= E1 = k kF > =8g. We know that we have a good control on (E1 ) and on (C ), so R that we now have to deal with (D ). We estimate (D ) by estimating k kpF d. We know that Z
(
Z
k kpF d)1=p = supf j hgj iF 0 ;F dj =
Z
p
kgkFp 0 d = 1 g 1
Thus, we consider g 2 Lp=(p 1) (X ; F 0 ) and we write Z
hgj iF 0 ;F d =
with
XZ
i
Z
h
Ei X nB (xi ;2KÆi )
L (x; y)g(x)d(x)jui iE0 ;E d(y) = X + Y
P R R X = R iR Ei h X nB(y;3Æi ) L (x; y)g(x) d(x)jui iE0 ;E d(y) P Y = i Ei h B(y;3K 2 Æi )nB(xi ;2KÆi ) L (x; y)g(x) d(x)jui iE0 ;E d(y)
R Let T () be the operator T ()f (y) = supr>0 k X nB(y;r) L(x; y)f (x) d(x)kE0 ; we estimate jX j by
jX j
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Z
E1
T ()g(y) d(y) Cp kT ()gk (p p 1) ;1 (E1 )1=p
Singular integrals
61
(since k1E1 kLp;1 Cp k1E1 k11=p k1E1 k11 1=p = Cp (E1 )1=p ). The estimate on Y is easy: for y 2 Ei and x 2= B (xi ; 2KÆi) we have d(x; y) (1=K )d(x; xi ) d(xi ; y) Æi 1 1 n C1 C (15K 4)n and thus kL(x; y)kop(F 0 ;E0 ) C1 d(x;y ) (B (y;15K 4 Æi ) , hence R we get that jY j C1 C (15K 4)n E1 M~ kgkF 0 (y)d(y) and nally the estimate jY j C1 C (15K 4)n kM~ kgkF 0 k p p 1 ((E1 ))1=p . We easily control Y , since we already know that g 7! M~ kgkF 0 is bounded p p from L p p1 (X ; F 0 ) to L p p1 . To control X , we shall prove that T () is bounded from L p 1 (X ; F 0 ) to L p 1 ;1 . RWe will then conclude that we have (D ) 8k kLp(X;F ) p C(E ) C kf kE d . 1 To get a control on T ()g,, we establish the following inequality: for almost every y we have T ()g(y) C (M~ kT gkE0 (y) + (M~ kgkp=0(p 1) (y))1 1=p ). F Indeed, let y 2 Supp and r > 0. We de ne rj = (5K 2 )j r and we consider the smallest integer k 1 so that (B (y; rk+1 )) < 4 (25K 4)n (B (y; rk 1 )). (Such (B (y;(25K 4 )l r)) C rn < 1). We a k exists since for 2l < k we have 4l (25 (B (y;r)) K 4 )lnR(B (y;r)) take R = (5K 2)k 1 r and begin by comparing X nB(y;r) L(x; y)g(x) d(x) to R X nB (y;5K 2 R) L (x; y )g (x) d(x): R k B(y;5K 2R)nB(y;r) L(x; y)g(x) d(x)kE0 R Pkj=1 B(y;rj )nB(y;rj 1 ) C1 d(x;y1 )n kg(x)kF 0 d(x) Mg (y) Pkj=1 C1 (B (y; rj+1 )) rj n1 = I We know the following estimates: { (B (y; rk+1 )) 4 (25K 4)n (B (y; rk 1 )) (4 (25K 4)n )1 (B (y; rk )); { (B (y; rk )) (4 (25K 4)n )1=2 (B (y; rk )); { (B (y; rk 1 )) (4 (25K 4)n )0 (B (y; rk )); { for j k 2, (B (y; rj ))P (4 (25K 4)n ) 1 (B (y; rj+2 )). Thus, we get I C1 M~ g (y) kj=1 2jR k 4(125K 3)n C 8 (125K 3)n C C1 M~ g (y). The next step is to compare X nB(y;5K 2 R) L (x; y)g(x) d(x) to R (y) = 1 R norm in E 0 is bounded by kRkE0 (B (y;R)) B (y;R) T g (z ) d(z ) (whose R 2 n 4 (25K ) M~ kT gkE0 (y)).We write X nB(y;5K 2R) L(x; y)g(x) d(x) R (y) = H + K with R R H = (B(1y;R)) B(y;R)( X nB(y;5K 2 R) (L (x; y) L (x; z ))g(x) d(x)) d(z ) 1 R K= (B (y;R)) B (y;R) T (g 1B (y;5K 2 R) )(z ) d(z ) R It is enough to write that k X nB(y;5R) (L (x; y) L(x; z ))g(x) d(x)kE0 CC2 Mg (y) for d(y; z ) < R to see that kH kE0 is controlled by M~ kgkF 0 (y). Finally, to estimate kK kE0 , we take ! 2 E with k!k = 1 and we write hK j!iE0 ;E = (B (1y; R)) hg1B(y;5K 2R) jT (1B(y;R)!)iL p p 1 (X;F 0 );Lp(X;F )
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Real harmonic analysis
which nally gives
kK kE0 (4 (25K 4)n )1 1=p kT kop(Lp(X;E);Lp(X;F ) (M~ kgkp=F 0 p (
1)
(y))1 1=p
and thus (A) is proved. is then easy : if T satis es (CZ ), we nd that T is bounded from 0 (B) 0 0 q 0 q L (F ) to L (E ) for 1 < q0 < p p 1 and by duality T is bounded from Lq (E ) to Lq (F ) for p < q < 1. Theorem 7.1 is proved. We have proved a usable estimate:
Corollary : Under the same hypotheses (T is bounded from Lp(X; E ) to ~ Lp (X; F ), T satis es R (CZ ) and (CZ )), the maximal operator T qde ned by ~ T f (x) = supr>0 k d(x;y)r L(x; y)f (y) d(y)kF is bounded from L (X ; E ) to Lq (X ) for all q 2 (1; 1). Proof: Indeed, (switching the role of T and T ),r we proved that when 1 < r r < 1 and T is bounded from L r 1 (X ; F 0 ) to L r 1 (X; E 0 ) then T~ is bounded from Lr (X; E ) to Lr;1. Besides, Theorem 7.1 proved that T is bounded from L (X; F 0 ) to L (X; E 0 ) for all 1 < < 1. . We then conclude using the Marcinkiewicz theorem by choosing 1 < r0 < q < r1 < 1. The boundedness Lri (X; E ) to Lri ;1 then gives the boundedness Lq (X; E ) to Lq .
2. Littlewood{Paley decomposition in Lp Let us now recall the de nition of the Littlewood{Paley decomposition :
De nition 7.1: (Littlewood{Paley decomposition) Let ' 2 D(IRd ) be such that j j 12 ) '( ) = 1 and j j 1 ) '( ) = 0. Let be de ned as ( ) = '(=2) '( ). Let Sj and j be de ned as the Fourier multipliers F (Sj f ) = '(=2j )F f and F (P j f ) = (=2j )F f . Then for all N 2 ZZ and all f 2 S 0 (IRd ) we have f = SN f + j N j f in S 0 (IRd ). This equality is called the Littlewood{Paley decomposition of the distribution f . The following result is then a direct consequence of Theorem 7.1:
Theorem 7.2: (Littlewood{Paley decomposition of Lp (IRd)) Let f 2 S 0 (IRd ) and 1 < p < 1. Then the following assertions are equivalent:
(A) f 2 Lp (IRd ). P (B) S0 f 2 Lp (IRd ) and ( j 2IN jj f (x)j2 )1=2 2 Lp (IRd ). P P (C) f = j 2ZZ j f and ( j 2ZZ jj f (x)j2 )1=2 2 Lp (IRd ).
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Singular integrals
63
Moreover, the followingPnorms are equivalent on Lp :P kf kp, kS0 f kp + k( j2IN jj f (x)j2 )1=2 kp and k( j2ZZ jj f (x)j2 )1=2 kp .
Proof: We prove the equivalence of (A) and (C), the proof for (A) and (B) is similar. If f 2 L2 , it is obvious (by the dominated convergenceP theorem applied to F (Sj f )) that limj! 1 kSj f k2 = 0. Thus, we have f = j2ZZ j f . The Fubini and Plancherel theorems give X
k(
j 2ZZ
jj f (x)j2 )1=2 k22 = ( 21 )d
Z
jf ( )j2 (
X
j 2ZZ
j (=2j )j2 ) d kf k22
since there P exists two positive constants 0 < A0 < A1 so that for 6= 0 we have A0 j2ZZ j (=2j )j2 A1 . Thus, the mapping f 7! (j f )j2ZZ is bounded from L2 (IRd ) = L2 (IRd ;C) I to L2 (IRd ; l2 (ZZ)). We may then apply the theory of singular integrals to this mapping (interpreting CI as l2 (f0g)). Indeed, the kernel L(x; y) is the operator z 2 CI 7! (2jd F 1 (2j (x y))z )j2ZZ 2 l2 . We have
k @x@ j L(x; y)kop =
kL(x; y)kop = k @y@ j L(x; y)kop =
( j2ZZ 4jd jF 1 (2j (x y))j2 )1=2 P ( j2ZZ 4j(d+1) j( @x@ j F 1 )(2j(x y))j2 )1=2 P
ky 2 ! (y )k1 ) we check and (writing 2j j!(2j x)j min(2j k!k1; 2 j jxj 2P d d easily that for ! 2 S (IR ), > 0 and x 2 IR we have ( j2ZZ 4j j!(2j x)j2 )1=2 C ;! jxj . Thus, Theorem 7.1 gives that kj f kLp(l2 ) Cp kf kp for 1 < p < 1. Conversely, let ~ 2 D(IRd ) so that = Supp ~ and ~ = . Let P 0 2 j ~ ~ ~ j = (D=2 ). Then, we have f = j2ZZ j j f . Next, look at the operP ~ j fj . This operator is bounded from L2 (l2 ) to L2 : since ator (fj )j2ZZ 7! j2ZZ ~ j fj and ~ l fl 0 2= Supp ~, there is a number M so that if jj lj > M then have disjoint spectrums (i.e., the Fourier transforms have disjoint supports), so P ~ j fj k22 (2M + 1)(Pj2ZZ k ~ j fj k22 ) (2M + 1)k ~k21 k(fj )k2L2 (l2 ) . that k j2ZZ We again apply the theory of singular integrals. Indeed, the kernel L~ (x; y) of the operator is the operator
(j )j2ZZ 7!
X
j
j 2jd F 1 ~(2j (x y)):
P We have kL~ (x; y)kop = ( j2ZZ 4jd jF 1 ~(2j (x y))j2 )1=2 , k @x@ j L~ (x; y)kop = k @y@ j L~ (x; y)kop = (Pj2ZZ 4j(d+1) j @x@ j F 1 ~(2j (x y))j2 )1=2 and the theory of P ~ j fj kp Cp k(fj )kLp(l2 ) . Calderon{Zygmund operators gives that k j2ZZ
We now extend this theorem to the potential spaces Hp :
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Real harmonic analysis
Proposition 7.1: (Potential spaces) Let f 2 S 0 (IRd ), 2 IR and 1 < p < 1. Then the following assertions are equivalent: (A) f 2 Hp (IRd ). P (B) S0 f 2 Lp (IRd ) and ( j 2IN j4j j f (x)j2 )1=2 2 Lp (IRd ). Moreover, the following norms are equivalent on Hp (IRd ) : k(Id )=2 f kp and kS0 f kp + k(Pj2IN j4j j f (x)j2 )1=2 kp .
Proof: (Id )=2 is an isomorphism between Hp and Lp . Thus, it is enough to check that we may apply the theory of vector-valued Calderon{Zygmund operators toPthe operators f 7! (2j j ((Id ) =2 f ))j0 and (fj )j0 7! ~ j (2 j fj ). This is easily done using the estimates on the (Id )=2 j0 Bessel potentials proved in Chapter 3: if k is the kernel of the convolution operator (Id )=2 , then, for all N 2 IN, there exists a constant C;N so that for all j 0 we have jj k (x)j C;N 2j(d+) (1 + 2j jxj) N for all x 2 IRd .
3. Maximal Lp (Lq ) regularity for the heat kernel In this section, we prove the maximal Lp (Lq ) regularity theorem for the heat kernel:
Theorem 7.3: (Maximal Lp (Lq ) regularity for the heat kernel.) R t (t s) The operator A de ned by f (t; x) 7! Af (t; x) = 0 e f (s; :) ds is bounded from Lp ((0; T ); Lq (IRd )) to Lp ((0; T ); Lq (IRd )) for every T 2 (0; 1], 1 < p < 1 and 1 < q < 1. Proof: We may suppose T = 1 (if T < 1, we may extend f by f = 0 on (T; 1) ; this is harmless since Af (t; x) depends only from the values of f on (0; t) IRd). Moreover, we extend f and Af to negative values of t by f = Af = 0 on ( 1; 0).Then, Theorem 7.3 is proved in three steps. Step 1: A is bounded on L2(L2 ). kxk2 Indeed, let W be the kernel of e: W (x) = (4) d=2 e 4 and let
de ned by (t; x) = td=1 2 (W )( pxt ) for t > 0 and by (t; x) = 0 for R R t < 0. Then, we have Af (t; x) = s2IR x2IRd t 1 s (t s; x y)f (s; y) ds dy. Thus, A is a convolution operator on L2 (IR IRd ). We compute the Fourier transform (in t and x) of 1t : the Fourier transform in x gives for t > 0 R 2
(t; x)e ix: dx = j j2 e tjj and then the Fourier transform in t gives R 1 2 tjj2 it F ( ; ) = 0 j j e e dt = jjj2+j2i . Since jF ( ; )j 1, we nd that A is bounded on L2(IR IRd ).
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Step 2: A is bounded on Lp (Lp ) for 1 < p < 1. We interpret A as a Calderon{Zygmund operator on IR IRd endowed with the Lebesgue measure on IRd+1 and with the quasi-distance d((t; x); (s; y)) = (jx yj4 +jt sj2 )1=4 . We have for all (t; x) 2 IRIRd and all r > 0 jB ((t; x); r)j = C rd+2 . The kernel is given by L((t; x); (s; y)) = 1(0;1) (t s)
x y 1 (W )( p ): (t s)(d+2)=2 t s
p k1 If jx yj t s, we write jL((t; x); (s; y))j jt ksjW yj (d+2)=2 , while if jx pt s, we write jL((t; x); (s; y))j kjzjd+2 W (z)k1 ; thus, we obtain that jx yjd+2 C jL((t; x); (s; y))j d((t;x);(s;y))d+2 . In the same way, we obtain the estimates j @t@ L((t; x); (s; y))j d((t;x);(Cs;y))d+4 and j @x@ j L((t; x); (s; y))j d((t;x);(Cs;y))d+3 . This gives for (; h) small with respect to d((t; x); (s; y)) that jL((t; x); (s; y)) L((t+; x+h); (s; y))j C ( d((t; x);j(s;j y))d+4 + d((t; x);j(hs;j y))d+3 ) which gives 0
jL((t; x); (s; y)) L((t + ; x + h); (s; y))j C dd((((t;t;xx);)(; t(s;+y;))xd+3+ h)) We have similar estimates for the regularity of L with respect to (s; y) and thus we get the boundedness of A on Lp (IR IRd) = Lp (IR; Lp (IRd )).
Step 3: A is bounded on Lp (Lq ) for 1 < p < 1 and 1 < q < 1. We now interpret A as a Calderon{Zygmund operator on IR. The kernel L(t; s) is now e(t s) and we have kLkop(Lq ;Lq ) = tCs and k @t@ Lkop = k @s@ Lkop = k2 e(t s)kop = (t Cs)2 . Thus, the Lq (Lq ) boundedness implies the Lp (Lq ) boundedness.
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Chapter 8 A primer to wavelets Wavelet theory was introduced in the 1980's as an eÆcient tool for signal analysis. We do not give a detailed presentation of the theory or discuss its applications (see the excellent books of Daubechies [DAU 92] or Mallat [MAL 98]). We are interested in wavelets as tools for getting basic results of real harmonic analysis, in the spirit of the books of Coifman and Meyer [MEY 97] or Kahane and Lemarie-Rieusset [KAHL 95].
De nition 8.1: (Wavelet bases) A wavelet basis of L2 (IRd ) is a family of functions ( ;j;k )12d 1;j 2ZZ;k2ZZd
such that i) they are derived through dyadic dilations and translations from a nite set of functions ( )12d 1: ;j;k (x) = 2
jd=2
jx
(2
k)
ii) the family is a Riesz basis of L2 (IRd ), i.e., the mapping
(;j;k )12d 1;j2ZZ;k2ZZd 7!
X
12d 1;j2ZZ;k2ZZd
;j;k ;j;k
is an isomorphism between l2 (f1; : : : ; 2d 1g ZZ ZZd ) and L2 (IRd ). iii) same structure : for all f 2 L2 we have f = P The dual basis has the jd=2 (2j x k ). h f j ;j;k i ;j;k with ;j;k (x) = 2 12d 1;j2ZZ;k2ZZd
The associated projection operators Qj are de ned by
Qj f =
X
12d
1;k2ZZd
hf j
;j;k i ;j;k :
The range Wj of Qj is the closed linear span of the functions ;j;k ; 1 2d 1; k 2 ZZd , and its kernel Wj is the closed linear span of the functions d 1; k 2 ZZd . ;j;k ; 1 2 Scaling functions '; ' associated with the wavelets are (if they exist) functions in L2 so that the family ('(x k))k2ZZd is a Riesz basis of V0 = 67 © 2002 by CRC Press LLC
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L
L (' (x k))k2ZZd isPa Riesz basis of V0 = j<0 Wj and, de ning the projection operator Pj = l<j Ql (with range Vj and kernel Vj ), we have P P0 f = k2ZZd hf j' (x k)i '(x k). When ' = ' and = for 1 2d 1, we shall speak of orthogonal scaling functions and of orthogonal wavelets.
j<0 Wj ,
1. Multiresolution analysis We now construct wavelet bases on IR, preceded by a loose introduction to the construction of wavelet bases. Assume that '; ' are scaling functions associated with some biorthogonal wavelet basis. The multiresolution analysis associated to the scaling function ' is the family of closed linear subspaces Vj = Im Pj (i.e., Vj is the range of the projection operator Pj ) ; Vj satis es the following properties: i) Vj Vj+1 ii) \j2ZZ Vj = f0g and [j2ZZ Vj is dense in L2 iii) f 2 Vj if and only if f (2x) 2 Vj+1 iv) V0 has a shift{invariant Riesz basis ('(x k)k2ZZ ). In particular, we have, since ' 2 V0 V1 , (8:1)
'(x) =
X
k2ZZ
k '(2x k)
for some sequence (k ) 2 l2 (ZZ). Thus, taking the Fourier transform of (8:1), we get that '^( ) = m ^(=2) where m0 2 L2 (IR=2ZZ) is the periodical P0 (=2)' 1 function m0 ( ) = 2 k2ZZ k e ik . Similarly, since ' and ' play symmetrical roles, we nd that '^ ( ) = m0 (=2)'^ (=2) for some periodical function m0 ( ) 2 L2 (IR=2ZZ). Now, we use the following Poisson summation formula: if !; 2 L2(IR), P ^ + 2k) = P h!(x)j (x k)ie ik: then the equality k2ZZ !^ ( + 2k) ( k2ZZ holds for almost C (!; )( ) for the periodical function P every 2 IR. We writeik: C (!; )( ) = k2ZZ h!(x)j (x k)ie . Since '^( ) = m0 (=2)'^(=2) and '^ ( ) = m0 (=2)'^ (=2), we nd that
( )C ('; ' )( ) + m0 ( + )m 0 ( + )C ('; ' )( + ) C ('; ' )( ) = m0 ( )m 2 0 2 2 2 2 2 thus, the duality between the system ('(x k)k2ZZ and the system (' (x k)k2ZZ gives that C ('; ' ) = 1 and hence that m0 ( 2 )m 0 ( 2 )+ m0 ( 2 + )m 0 ( 2 + ) = 1. In order to construct 'P ,' , we thus start with a pair P of univariate trigonometric polynomials m0 = k0 kk1 ak e ik and m0 = l0 ll1 l e il with real{valued coeÆcients such that:
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j) m0 (0) = m0 (0) = 1 and m0 ( )m 0 ( ) + m0 ( + )m 0 ( + ) = 1. jj) m0 and m0 have no zero on [ = 2 ; + = 2]. Q Q1 jjj) the functions '^ = 1 j =1 m0 ( 2j ) and '^ = j =1 m0 ( 2j ) are square integrable on IR. P
Lemma 8.1: Let m0 be a trigonometric polynomial m0 = k0 kk1 ak e ik Q with m0 (0) = 1 and let '^ = 1 j =1 m0 ( 2j ). Then '^ is the Fourier transform of a compactly supported distribution ' with support contained in [k0 ; k1 ]. Proof: Since m0 (0) = 1, we have jm0 ( ) 1j km00 k1 j j and thus the in nite product is convergent for all 2 IR.0 Moreover, for j j , we have Q Q1 km0 k1 j n supN 1 N j =1 jm0 (=2 )j Qj =1 (1 + 2j ) = < 1, while for 2 N n +1 j n +1 B j j 2 we have supN 1 j=1 jm0 (=2 )j km0k1 AQj j with A = km0 k1 and B = ln km0k1 = ln 2. Thus, the nite products Nj=1 m0 (=2j ) converge to '^ in S 0 ; taking the inverse Fourier transforms gives that ' is the limit in P S 0 of the convolution products 2(2x) 4(4x) : : : 2N (2N x) with (x) = k0 kk1 ak Æ(x k). The support of is fk0 ; : : : ; k1 g [k0 ; k1 ] and thus the support of the convolution 2(2x) 4(4x) : : : 2N (2N x) is contained in [(1=2 + 1=4 + : : : + 1=2N )k0 ; (1=2 + 1=4 + : : : + 1=2N )k1 ] (1 2 N )[k0 ; k1 ]. Lemma 8.2: Let m0 and m0 satisfy hypothesis j). Then, if p and p are de ned as
^p ( ) = 1[ 2p;2p ] ( )
p Y m0 ( j ) and ^p ( ) = 1[ 2p;2p ]( ) m0 ( j ) 2 2 j =1 j =1 p Y
we have p 2 L2 (IR), p 2 L2 (IR) and hp (x)jp (x k )i = Æk;0 . If moreover m0 and m0 satisfy hypotheses jj) and jjj), then p converges to ' and p converges to ' in L2 (IR) as p goes to 1, and thus h'(x)j' (x k)i = Æk;0 .
Proof: We use again the Poisson summation formula to estimate C (!; )( ) = P ik: . Since ^p ( ) = m0 (=2)^p 1 (=2) and ^ ( ) = p k2ZZ h! (x)j (x k )ie m0 (=2)^p 1 (=2), we nd that the function Cp ( ) = C (p ; p )( ) satis es the following recursion formula : Cp ( ) = m0 ( )m 0 ( )Cp 1 ( ) + m0 ( + )m 0 ( + )Cp 1 ( + ) 2 2 2 2 2 2 thus, by induction, Cp = 1 and we have hp (x)jp (x k)i = Æk;0 . If we assume that hypotheses jj) and jjj) are satis ed, then , we know that on [ 2p; 2p ] we have '^( ) = ^p ( )'^(=2p ) while on [ ; ] the continuous function '^( ) does not vanish (since none of the terms m0 (=2j ), 1 j ,
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vanishes and since the in nite product is convergent in CI ); thus, the function ^p ( ) satis es j^p ( )j Aj'^( )j with A = maxjj j'^(1)j and Lebesgue's dominated convergence theorem gives that p converges to ' in L2 as p goes to 1. Similarly, p converges to ' in L2 as p goes to 1 and this gives that h'(x)j' (x k)i = Æk;0 . Lemmas 8.1 and 8.2 allow us to construct scaling functions ', ' . In order to get wavelets , , we need to have a simple criterion to check the unconditionality of wavelet bases. This is performed through the vaguelettes lemma of Y. Meyer [MEY 92] (see also Kahane and Lemarie-Rieusset [KAHL 95]):
Proposition 8.1: (The vaguelettes lemma) A) Let ! be a compactly supported square integrable function on IR (! 2 2comp(IR)). Then, there is a constant C! such that for all (k )k2ZZ 2 CI ZZ , LP k k2ZZ k !(x k)k22 C! Pk2ZZ jk j2 . B) Let !; ! be compactly supported square integrable functions on IR such that h! (x)j! (x k )i = Æk;0 (= 0 if k 2 ZZnf0g and = 1 if k = 0). Then, there are two constants C! and D! so that D!
X
k2ZZ
jk j2 k
X
X k !(x k)k22 C! jk j2 k2ZZ k2ZZ
C) [The vaguelettes lemma] Let ! be a compactly supported R square integrable function on IR so that ! 2 H 1 (i.e. ! 0 2 L2 ) and ! dx = 0. 2 Then, there exists a constant E! so that for all (j;k )(j;k)2ZZ2 2 C I ZZ , we have P P k (j;k)2ZZ2 j;k 2j=2 !(2j x k)k22 E! (j;k)2ZZ2 jj;k j2 . D) Let !; ! be compactly supported square integrable functions on IR such 0 0 that h2j=2 ! (2j x k )j2j =2R! (2j x R k 0 )i = Æj;j 0 Æk;k0 . Assume that ! and ! belong to H 1 and satisfy ! dx = ! dx = 0. Then, there are two constants E! and F! so that
F!
X
(j;k)2ZZ
2
jj;k j2 k
X
(j;k)2ZZ
2
j;k 2j=2 !(2j x k)k22 E!
X
(j;k)2ZZ
2
jj;k j2
Proof: (A) is obvious; we just use the Cauchy-Schwarz inequality and write P j Pk2ZZ k !(x k)j Pk2ZZ jk j2 j!(x P k)j2 1=2 k2ZZ 1Supp ! (x k) 1=2 ; thus, we nd the result with C! = k!k22 k k2ZZ 1Supp ! (x k)k1 . P P P To prove (B), we write jkjK jk j2 = h k2ZZd k !(x k)j jkjK k ! (x k)i P P P and using (A), we get jkjK jk j2 k k2ZZd k !(x k)k2 k jkjK k !(x P P k)k2 k k2ZZd k !(x k)k2 C!1=2 jkjK jk j2 1=2 , which gives (B) with D! = 1=C! .
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To prove (C), it is enough to prove it in the case where all the j;k 's but nitely many are equal to 0. Then, when we have obtained the estimate with a constant E! independent from the number of non vanishing coeÆcients, we may conclude that the whole series converges in L2 and satis es the required estimate. For a nite j;kP , we may write (de ning !j;k = P numbers of coeÆcients P 2j=2 !(2j x k)) k (j;k)2A j;k !j;k k22 = (j;k)2A (j0 ;k0R)2A j;k j0 ;k0 h!j;k j!j0 ;k0 i and then we integrate by parts: since ! 2 L2comp and ! dx = 0, we may nd d , so (de ning = 2j=2 (2j x k ) a function 2 L2comp so that ! = dx j;k j x k )) we have h!j;k j!j 0 ;k0 i = 2j 0 j h j;k j j 0 ;k0 i. Then, and j;k = 2j=2 d! (2 dx applying (A), we get for j; j 0 2 ZZ :
j
X
X
sX
(j;k)2A (j0 ;k0 )2A
0 j;k j0 ;k0 h!j;k j!j0 ;k0 ij C (!)2 jj j j
k2ZZ
jj;k j2
sX
jj0 ;k0 j2
k0 2ZZ
This is enough to provePassertion (C), since for any P non negative sequence j P 0j j j j we have the inequality j2ZZ j0 2ZZ 2 j j0 3 j2ZZ 2j . (D) is then a direct consequence of (C), just as (B) is a direct consequence of (A).
Remark: The vaguelettes lemma may be easily extended to the multivariate case: let ! be a compactly supported square integrable function on IRd so that R @ 2 1 ! 2 H (i.e. @xi ! 2 L for i = 1; : : : ; d) and ! dx = 0; then, we have k P(j;k)2ZZZZd j;k 2jd=2 !(2j x k)k22 E! P(j;k)2ZZZZd jj;k j2 . The proof is the same as for d = 1, except that we have to now use d functions for the R integration by parts: since ! 2 L2comp and ! dx = 0, we may nd d functions P !i 2 L2comp such that ! = di=1 @x@ i !i . Recollecting the above results, we obtain the following theorems on the construction of wavelets :
Theorem 8.1: (Orthogonal wavelets) P Let m0 be a trigonometric polynomial m0 = k0 kk1 ak e ik with real-
valued coeÆcients so that: i) m0 (0) = 1 and jm0 ( )j2 + jm0 ( + )j2 = 1. ii) m0 has no zero on [Q=2; +=2]. De ne ' by '^ = 1 de ned by ^( ) = j =1 m0 ( 2j ). Then the function i= 2 e m 0 (=2 + )'^(=2) generate an orthonormal wavelet basis ( j;k )j2ZZ;k2ZZ of L2 (IR) with associated scaling function '. The wavelet and the scaling function ' have compact supports: Supp ' [k0 ; k1 ] and Supp [(k0 k1 + 1)=2; (k1 k0 + 1)=2].
Theorem 8.2: (Bi-orthogonal wavelets) P Let m0 , m0 be trigonometric polynomials m0 = k0 kk1 ak e ik and P m0 = l0 ll1 l e il with real-valued coeÆcients so that:
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i) m0 (0) = m0 (0) = 1 and m0 ( )m 0 ( ) + m0 ( + )m 0 ( + ) = 1. ii) m0 and m0 have no zero on [ =2; += 2] . Q Q1 iii) the functions ' and ' de ned by '^ = 1 j =1 m0 ( 2j ) and '^ = j =1 m0 ( 2j ) belong to H 1 . Then the functions and de ned by
^( ) = e i=2 m 0 (=2 + )'^(=2) and ^ ( ) = e i=2 m 0 (=2 + )'^ (=2)
generate a wavelet basis ( j;k )j 2ZZ;k2ZZ of L2 (IR) and its dual wavelet basis )j2ZZ;k2ZZ , with associated scaling functions '; ' . ( j;k Moreover, the wavelets ; and the scaling functions '; ' have compact supports; more precisely, Supp ' [k0 ; k1 ], Supp ' [l0 ; l1 ], Supp [(k0 l1 +1)=2; (k1 l0 +1)=2] and Supp [(l0 k1 +1)=2; (l1 k0 +1)=2].
Proof: Step 1: We rst check that, under the hypotheses of Theorem 8.1 (orthonormal scaling lters), ' belongs to L2 . We already know (from the proof of Lemma 8.2) that p belongs to L2 and that kp k2 = 1. Since p is weakly convergent to ' (the Fourier transform of p is bounded by 1 and converges pointwise to '^, hence p converges to ' in S 0 ), this gives that ' 2 L2 . Thus, we may apply Lemma 8.2 and get that the family ('(x k))k2ZZ is orthonormal. Step 2: Let Vj the closed linear span of the family (2j=2 '(2j x k))k2ZZ . Then we are going to show that: k) Vj Vj+1 kk) \j2ZZ Vj = f0g and [j2ZZ Vj is dense in L2 kkk) f 2 Vj if and only if f (2x) 2 Vj+1 Indeed, P kkk) is obvious and k) is a direct consequence of the equality '(x) = ak 2'(2x k). If V = [j2ZZ Vj , we nd that its closure is invariant under any dyadic translation, hence under any translation. A classic result of harmonic analysis states that there exists a Borel set E so that f 2 V if and only if Supp f^ E . Since '^ is analytic, E must be equal to IR. Now, point B) of Proposition 8.1 proves that Pthere exists a positive constant A such that for all f 2 V0 we P have kf k22 A k2ZZ jhf j' (x k)ij2 . Thus, if f 2 \j2ZZ Vj , kf k22 A P minj2ZZ k2ZZ jhf j2j=2 ' (2j x k)ij2 . But the operators Uj de ned by Uj g = k2ZZ jhgj2j=2 ' (2j x k)ij2 are equicontinuous on L2 and it is easy to check that for g 2 L2comp we have limj! 1 Uj g = 0 (indeed, for , there exists a K 2 IN so that for all j 0 we have Uj g = P such a gj= 2 ' (2j x k)ij2 , while it is obvious that for each xed k 2 ZZ we jh g j 2 jkjK have limj! 1 hgj2j=2 ' (2j x k)i = 0). Step 3: In this step, we are looking for a basis P for W0 = V1 \ (V0 )? . f belongs to V1 if and only if f may be decomposed as f = k2ZZ k '1;k for some sequence (k ) 2 l2 (ZZ). Equivalently, f belongs to V1 if and only f^ = (=2)'^(=2) for some 2{periodical locally square-integrable function . On the other hand, f
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belongs to (V0 )? if and only hf j' (x k)i = 0 for all k 2 ZZ, or equivalently (using the Poisson summation formula for F (x) = hf (y)j' (y x)i) if and only if P ^ ^ k2ZZ f ( +2k )'^ ( +2k ) = 0 almost everywhere. Writing f = (=2)'^(=2) and '^ = m0 (=2)'^ (=2), we get (=2)m 0 (=2) + (=2 + )m 0 (=2 + ) = 0. We then write ( ) = ( ) m0 ( )m 0 ( ) + m0 ( + )m 0 ( + ) and get ( ) = m 0 ( + )( ( + )m0 ( ) + ( )m0 ( + )) = e i m 0 ( + )A(2 ) for some 2-periodical function A. This gives that W0 is the closed linear span of the functions (x k), and a similar proof gives that W0 = V1 \ (V0 )? is the closed linear span of the functions (x k) and that h (x)j (x k)i = Æk;0 . Step 4: We may now easily conclude. We know that the family ( j;k ) is orthonormal (under the hypotheses of Theorem 8.1) or that the family ( j;k ) is the Riesz basis of some closed subspace of L2 (from point D) of Proposition 8.1). We need to now show that the closed linear span of the family ( j;k ) is the whole of L2 . We write Pj for the projection operator with range Vj and kernel (Vj )? and Qj for the projection operator with range Wj and kernel (Wj )? , whereLWj is the closed linear span of the family ( j;k )k2ZZ ; we have Pj1 Pj0 = j0 jj1 1 Qj . For all f 2 L2, limj!+1 kf Pj f k2 = 0 (by equicontinuity ofpthe family Pj and density of [j2ZZ Vj ). Moreover, since kPj f k2 is equivalent to Uj f (where Uj is de ned in Step 2), we get that, for all f 2 L2 , limj! 1 kPj f k2 = 0.
2. Daubechies wavelets We now prove the existence of scaling functions and of wavelets with high regularity.
Theorem 8.3: (Daubechies wavelets [DAU 92]) For N 2 IN , let m0 ( ) be the trigonometric polynomial de ned by : i) the degreeP of m0 is equal to 2N 1 ii) m0 ( ) = 0k2N 1 ak e ik with ak 2 IR iii) m0 (0) = 1 dk m ( ) = 0 iv) For 0 k N 1, d k 0 v) jm0 ( )j2 + jm0 ( + )j2 = 1. Q Then, the function '^( ) = 1 j =1 m0 ( 2j ) is the Fourier transform of an orthonormal scaling function '. Moreover, ' is compactly supported with support [0; 2N 1] and, for N 2, ' is continuous. More precisely, ' belongs to C N =3) with N = ln(4 2 ln 2 N + o(N ) as N goes to 1. Proof: Due to hypothesis iv), we may factorize (1 + e i )N in m0 and thus get N jm0 ( )j2 = ( 1+cos ) for some polynomial P . Hypothesis v) then gives 2 ) P1 (cos 1+ X N that ( 2 ) P (X )+( 2X )N P ( X ) = 1. The Bezout identity grants that there exists one and only one solution P with a degree to N 1. We write less or equal P N + k N 1 1 = ( 1+2X + 1 2X )2N 1 and nd P (X )= k=0 2N 1 1+2X k 1 2X N 1 k .
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Another way to nd P is to de ne Y = ( 1 2X ) and to write P (1 2Y ) = 1 N P (2Y 1) (1 Y )N Y (1 Y )N ; then a Taylor expansion gives that P (X ) = P (1 2Y ) = PN 1 N + k 1 Y k. k=0 N 1 The polynomial P is clearly non negative on [ 1; 1]. Then the Riesz lemma states that P (cos ) is the square modulus jQ( )j2 of a trigonometric polynomial PdegP Q( ) = k=0 k e ik with real-valued coeÆcients k . (Q P is not uniquely k de ned. If we require that Q(0) = 1 and that all the roots of degP k=0 k z = 0 have a modulus less or equal to 1, then Q is uniquely de ned.) We choose Q with Q(0) = 1 and then de ne m0 ( ) = (1 + e i )N Q( ). Since m0 does not vanish on ( ; ), we may apply Theorem 8.1 and check that m0 is the scaling lter associated to a Hilbertian basis of compactly supported wavelets. We shall now estimate the regularity of the scaling function Q i i j Q '. We write m0 ( ) = ( 1+e2 )N Q( ), and thus '^ = ( 1 ie )N 1 j =1 (=2 ). K K +1 For 2 j j < 2 , we write 1 Y j'^( )j ( j2 j )N kQkK1 sup j Q( 2j )j jj2 j=1 kQk1
p
and get, for j j 1, that j'^( )j C j j N + log 2 . But kQk1 = P ( 1) = N N N N 1 2N 1 ; for N 2, we have 2 2N 1 = 2N 1 + 2N 1 < kQk1 < 1 so that '^ is integrable. (1 + 1)2N 1 , thus we nd that N + loglog 2 In order to get better information on the regularity of ', we notice that for a compactly supported function ', the regularity exponent = maxf = ' 2 C g and the decay exponent for its Fourier transform = maxf = j j '^ 2 L1g are related by: 1 . Thus, we shall try to estimate . Since m0 does not vanish on ( ; ), we nd that '^(2=3) 6= 0 and that j'^(2L 2=3)j = jQ(2=3)j Lj'^(2=3)j, hence N log jQ(2=3)j . Moreover, for 2K j j < 2N log 2 K +1 2 , we write K 1 Y Y j'^( )j ( j2 j )N j Q( 2j )j sup j Q( 2j )j jj2 j=1 j =1 log
s
1=K log . We and get, writing = lim supK !1 k K j =1 Q( 2j )k1 , N log 2 shall now prove that 3jQ(2=3)j; as a matter of fact, a more precise result of Cohen and Conze states thatp = jQ(2=3)j. We introduce = max(sup 1=2X 1 P (X ); sup 1X 1=2 P (X )P (2X 2 1)). We get, estimating P (cos( 2j )) alone if cos( 2j ) > 1=2 or estimating P (cos( 2j ))P (cos( 22j )) in Q )j2 P ( 1) K 1 and thus p . We the other case, that j K j j =1 Q( 2 2 N 2 2 write jQ(2=3)j = P ( 1=2) N 1 ( 34 )N 1 (which is the last term of Q
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P expressed as the Taylor expansion of 1 ( 11 X )N ). On the other hand, P 2 is decreasing on [ 1 ; 1] and thus sup P (X ) = P ( 1=2). In order to 1 = 2 X 1 p 2 estimate P (X )P (2X 1) for 1 X 1=2, we write for X < 0 NX1
N +k 1 P (X ) = N 1 k=0
N X1 1 X k 1 X k 2 N 2 N 1 k=0 2 2
and thus get P (X ) 2NN 12 ( 1 2X )N 1 ( 1 XX ). If X 1=8, this gives more simply P (X ) 9 2NN 12 ( 1 2X )N 1 . We then consider two domains for X : p 1 X 47 : 2X 2 1 1=8 so that 2 9 2 N 2 2 P (X )P (2X 1) P ( 1)P ( 1=8) 81 N 1 1N 1( )N 1 16
p7
2 1 are less than 1=8, so 4 < X 1=2: in that case, X and 2X 2 that we have P (X )P (2X 2 1) 81 2NN 12 ( 1 2X )N 1 (1 X 2)N 1 . Since p x 7! (1 x)(1 x2 ) increases on [ 47 ; 1=2], we obtain 2 3 3 P (X )P (2X 2 1) 81 2NN 12 ( )N 1 ( )N 1 4 4
Since
2N 2 ( 3 )N 1 N 1 4
jQ(2=3)j2, this gives 3jQ(2=3)j.
ishes the proof, since we have proved that
9 2NN 12 ( 34 )N 1 . tion:
This n-
2N 2 ( 3 )N 1 jQ(2=3)j2 N 1 4
Another useful way of constructing wavelets is integration and di erentia-
Proposition 8.2: (Di erentiation of wavelets) Let (Vj ), (Vj ) be a bi-orthogonal multi-resolution analysis of L2 (IR) with compactly supported dual scaling functions ', ' and associated scaling lters m0 , m0 . Assume that ' belongs to the Sobolev space H 2 . Then: i) The derivative '0 of ' can be written as '0 (x) = '~(x) '~(x 1) where '~ is a compactly supported scaling function associated to the scaling lter m ~ 0 ( ) = 2 i m0 ( ). 1+e
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ii) The primitive x1 ' (t) dt of ' satis es xx+1 ' (t) dt = '~ (x) where '~ is a compactly supported scaling function with associated scaling lter m ~ 0 ( ) = 1+ei m0 ( ). 2 iii) '~ and '~ are compactly supported dual scaling functions for a biorthogonal multiresolution analysis (V~j ), (V~j ). Moreover, the projection operators Pj onto Vj in the direction of (Vj )? and P~j onto V~j in the direction of (V~j )? satisfy: R
R
d d Æ Pj = P~j Æ : dx dx iv) The bi{orthogonal wavelets and associated with ', ' and the wavelets ~ and ~ associated with '~, '~ satisfy:
(8:2)
(8:3)
Z x (t) dt: ~ = 1 d and ~ = 4 4 dx 1
Proof: We easily prove this proposition. First, we prove m0 () = m0 () = 0: P
If g belongs to H 1 (IR) then k2ZZ jg( + 2k)j2 converges uniformly on [ ; ]: we write H 1 C0 , hence, using a smooth function ! equal to 1 on [ ; ] and supported in [ 2; 2], we nd the inequality kgkL1([(2k 1);(2k+1)]) C kg(x)!(x 2k)kH 1 C 0 kgkL2([(2k 2);(2k+2)]) + kg0 kL2([(2k 2);(2k+2)]) ). We apply this estimate to the Fourier transforms '^, '^ of the compactly supported square{integrable dual scaling function ', ' . We nd that the sum P k2ZZ '^( + 2k )'^ ( + 2k ) is equal to 1 for every 2 IR (we already know by the Poisson sumation formula and the bi{orthogonality of the scaling functions that the sum is equal to 1 almost everywhere; the uniform convergence of the series gives us that the sum is a continuous function, hence that it must be identically equal to 1). Hence, we
such that P nd that there exists a positive P constant ( + 2k)j2 for every 2 IR we have k2PZZ j'^( + 2k)j2 and j ' ^ P k2ZZ 2 2
. We k2ZZ j'^(4k )j , B = k2ZZ j'^(2 + 4k )j and P then write for A2 = 2 C = k2ZZ j'^( + 2k)j the equalities A = jm0 (0)j (A + B ) = A + B , hence 0 = B = jm0 ()j2 C , and (since C 6= 0) m0 () = 0. We obtain m0 () = 0 by the same way. We now prove the proposition. Using m0 () = 0, we obtain that, for any L 2 ZZ with L 6= 0, '^(2L) = 0: write L = 2N (2k + 1) and '^(2N (2k + 1)) = m0 (0)N 1 m0 ()'^((2k + 1)) = 0. Then,using the Poisson summation formula, we get X X '(x k) = '^(2k)e2ikx = 1: k2ZZ k2ZZ
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P P Thus, k2ZZ '0 (x k) = 0. We then de ne '~(x) = k0 '0 (x k); we have of course '0 (x) = '~(x) '~(x P 1). Moreover, '~ is clearly equal to 0 on x < 0 min Supp' and (since '~(x) = k<0 ' (x k ) as well) on x > max Supp' 1. Since this locally nite sum of functions belonging to H 1 belongs locally to H 1 and is compactly supported, we nd that '~ belongs to H 1 . Moreover, we have i '^( ) = (1 e i )'^~( ), which gives
2 2i 1 e i m0 ( )'^~( ) = m ( )'^~( ): '^~(2 ) = 2 i i 1 e 1 + e i 0 R Similarly, '~ (x) = xx+1 ' (t) dt is a compactly supported function (with support contained in [min Supp' 1; max Supp' ]) which belongs to H 1 and i we have i '~^ ( ) = (ei 1)'^ ( ), giving '~^ (2 ) = 1+2e m0 ( )'~^ ( ). MoreP over, we clearly have '~^ '^~ = '^ '^, hence k2ZZ '~^ ( + 2k)'^~( + 2k) = P k2ZZ '^ ( + 2k )'^( + 2k ) = 1. Thus, '~ and '~ are dual scaling functions. d Æ P = P~ Æ d . It is We still have to prove the commutation formula dx j j dx enough to prove it for j = 0 (by homogeneity of the di erentiation operator). d Æ P f by an Abel transformation of the series We compute dx 0 X
k2ZZ
hf j' (x k)i('~(x k) '~(x k
1))=
X
k2ZZ
hf j(' (x k) ' (x k+1)i'~(x k)
then we integrate by parts
d d hf j(' (x k) ' (x k + 1)i = hf j dx '~ (x k)i = h f j'~ (x k)i: dx
3. Multivariate wavelets The quickest way to construct a wavelet basis for L2(IRd) is to start from d univariate multiresolution analysis and to construct by tensorization a d-variate multiresolution analysis :
Proposition 8.3: (Separable wavelets) Let (Vj(i) )j 2ZZ , (Vj(i) )j 2ZZ (1 i d) be d bi-orthogonal multiresolution analyses of L2 (IR) with associated scaling functions '(i) , '(i) and associated wavelets (i) , (i) . Then: ^ :::
^ Vj(d) (the closure of Vj(1) : : : Vj(d) in L2 (IRd )) i) The space Vj = Vj(1)
de nes a multi-resolution analysis: a) Vj Vj +1 b) \j 2ZZ Vj = f0g and [j 2ZZ Vj is dense in L2 c) f 2 Vj if and only if f (2x) 2 Vj +1 d) V0 has a shift-invariant Riesz basis ('(x k )k2ZZd )
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with '(x) = '(1) : : : '(d) (x) = '(1) (x1 ) : : : '(d) (xd ). ^ :::
^ Vj(d) de nes a multi-resolution analii) Similarly, the space Vj = Vj(1)
ysis and V0 has a shift-invariant Riesz basis (' (x k )k2ZZd ) with ' = '(1)
: : : '(d) . iii) h'(x k )j' (x k 0 )i = Æk;k0 . iv) Wj = Vj +1 \ (Vj )? has as Riesz basis ( ;j;k )12d 1;k2ZZd , where ;j;k is de ned by ;j;k (x) = 2jd=2 (2j x k ) and by: '(i);0 = '(i) , '(i);1 = (i) P and, for = di=0 i 2i 1 the dyadic decomposition of , = '(1);1 : : :
'(d);d . )12d 1;k2ZZd , where v) Similarly, Wj = Vj+1 \ (Vj )? has as Riesz basis ( ;j;k jd=2 (2j x k ) and = '(1);1 : : : '(d);d ;j;k (x) = 2 vi) The family ( ;j;k )12d 1;j 2ZZ;k2ZZd is a wavelet basis of L2 (IRd ) with dual )12d 1;j2ZZ;k2ZZd . basis ( ;j;k
Proof: This is obvious.
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Chapter 9 Wavelets and functional spaces
We may now begin the description of functional spaces through wavelet coeÆcients. We do not attempt to give optimal statements (see Kahane and Lemarie-Rieusset [KAHL 95] for the case of univariate wavelets), since wavelets are only a tool to prove some useful estimates. We consider very optimistic hypotheses, wasting a lot of regularity in our assumptions, but this is harmless since we have shown that there exists compactly supported wavelets with arbitrarily high regularity. Thus, we shall only consider N -regular wavelets :
De nition 9.1: (N -regular wavelet bases) A wavelet basis of L2 (IRd ) ( ;j;k )12d 1;j2ZZ;k2ZZd is N -regular (N 1) when it satis es that: i) this is a separable wavelet basis ii) the wavelets and the dual wavelets are compactly supported functions ; iii) they are associated to compactly supported scaling functions ', ' ; iv) the dual scaling function ' is continuously di erentiable ; v) the scaling function ' is N times continuously di erentiable.
1. Lebesgue spaces A well-known property of wavelet bases is that they provide Riesz bases to the Lebesgue spaces Lp (IRd ):
Theorem 9.1: (Lebesgue spaces) Let ( ;j;k )12d 1;j2ZZ;k2ZZd be a 1-regular wavelet basis (with associated dual wavelet and associated scaling functions ', ' ). Then: i) for 1 < p < 1, the family ( ;j;k )12d 1;j2ZZ;k2ZZd is a Riesz basis of Lp (IRd ); more precisely, there exists two positive constants 0 < Ap Bp < 1 so that for all f 2 Lp we have, writing (x) = 1[0;1]d (x),: Ap kf kp k
X X
X
j 2ZZ k2ZZd 12d 1
ij2 (2j x k)1=2 kp Bp kf kp 2jd jhf j ;j;k
ii) for 1 < p < 1, the family ('(x k))k2ZZd [ ( ;j;k )12d 1;j2IN;k2ZZd is a Riesz basis of Lp (IRd ); more precisely, there exists two positive constants
79
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Real harmonic analysis
0 < Ap Bp < 1 so that for all f Ap kf kp Np (f ) Bp kf kp where
Np (f )=
X
k2ZZd
jhf j'0;k ijp
1=p
+k
2 Lp we have, writing (x) = 1[0;1]d (x),
X X
X
ij2 (2j x k)1=2kp 2jd jhf j ;j;k
j 2IN k2ZZd 12d 1
Proof: We choose a function ! 2 D(IRd ) equal to 1 on [0; 1]d, we write !j;k (x) = 2jd=2 !(2j x k), and we introduce the operator f 7! A(f ) =
X X
X
j 2ZZ k2ZZd 12d 1
jhf j
2 ! 2 1=2 j;k
;j;k ij
which we associate to the vector-valued kernel (y)!j;k (x)) K (x; y) = ( ;j;k j 2ZZ ;k2ZZd ;12d and the operator
f
1
Z
7! Bf (x) = K (x; y) f (y) dy:
2 , The vaguelettes lemma gives usPthat B is bounded from L2 (IRd ) to L2xlj;k; P P 2 2 2 since kBf kL2l2 = j2ZZ k2ZZd 12d 1 jhf j ;j;k ij k!k2. Moreover, B is a vector-valued Calderon{Zygmund operator:
kK (x; y)kl
2
~ xK (x; y)kl2 + kx yk kr ~ y K (x; y)kl2 + kx yk kr
C kx yk
d
and we may thus extrapolate its L2; L2 l2 boundedness to the Lp ; Lp l2 boundedness for 1 < p < 1. Thus, we have kAf kp = kBf kLpl2 Bp kf kp for 1 < p < 1. Since !2 , this gives that for 1 < p < 1
k
X X
X
j 2ZZ k2ZZd 12d 1
ij2 (2j x k)1=2 kp Bp kf kp 2jd jhf j ;j;k
We have, of course, a similar estimate with the dual system: for 1 < q < 1
k
X X
X
j 2ZZ k2ZZd 12d 1
2jd jhf j ;j;k ij2 (2j x k) 1=2 kq Bq kf kq
2 Lp \ L2 and g 2 Lq \ L2 (1=p + 1=q = 1), we have R P P P f g dx = P P j2P ZZ k2ZZd 12d 1 hf j ;j;k i Rh ;j;k jg i jd j = j 2ZZ P k2ZZd P 12d 1 hf j ;j;k i h ;j;k jg i 2 (2 x k ) dx RP = j 2ZZ k2ZZd 12d 1 hf j ;j;k i j;k (x) h ;j;k jg i j;k (x) dx
Now, for f
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with j;k (x) = 2jd=2 (2j x k). We then use the Cauchy{Schwarz inequality 2 inR lj;k; and the Holder inequality for the spaces Lp (dx); Lq (dx) to get that j f g dxj kA(f )kp kA(g)kq kA(f )kpBq kgkq . We thus get that for f 2 Lp \ L2 we have kf kp Bq kA(f )kp . This is easily extended to the whole space Lp : if f 2 Lp we choose f 2 Lp \ L2 with kf f kp < and we write: kf kp kfkp + Bq kA(f )kp + Bq kA(f )kp + Bq kA(f f)kp + Bq kA(f )kp + (Bp Bq + 1). Thus, (A) is proved. The proof of (B) is exactly the same.
2. Besov spaces In order to analyze spaces of regular functions, we shall need a lemma on regular wavelets:
Lemma 9.1: (Oscillation of wavelets) Let ( j;k )j2ZZ;k2ZZd be a N -regular wavelet basis of L2 (IR) (with associated dual wavelet R l and associated scaling functions ', ' ). Then, for 0 l N , we have x (x) dx = 0. In particular, there exists a compactly supported function so that = dN . dxN
R Proof: Let L be the smallest integer so that xL (x) dx 6= 0 and let us assume that L N . Let x0 2 IR so that '(L) (x0 ) 6= 0,R and for j ' dx = 0, let Rq the biggest integer smaller than 2j x0 ; we have 0 = j;q R k + y j= 2 j (1 = 2 L ) 2 (y) '( 2j ) dy. But we easily check that limj!1 2 j;q ' dx R L (L) ' ( x 0) = ( x (x) dx) L! ; hence, we get a contradiction. Thus, L > N .
Remark: Since the number L in the proof above is necessarily nite (otherwise, we would get = 0), this proves that ' cannot be inde nitely smooth. We begin by giving a stronger version of the vaguelettes lemma (extended to the case of multivariate functions) :
Lemma 9.2: (The vaguelettes lemma) A) Let 0 < Æ 1 and let (gk )k2ZZd be a sequence of functions such that jgk (x)j (1+jx1j)d+Æ and jgk (x) gk (y)j Pjx yjÆ ( (1+jx1j)d+Æ +P(1+jy1j)d+Æ ). Then, we have, for 2 fÆ=8; Æ=8g, k( ) ( k2ZZd k gk )k2 C ( k2ZZd jk j2 )1=2 . B) Let Æ > 0 and let (gj;k )j2ZZ;k2ZZd be a sequence of functions so that jgj;k (x)j 2jd=2 (1+j2j x1 kj)d+Æ , and
jgj;k (x) gj;k (y)j 2j(d=2+Æ) jx yjÆ ( (1 + j2j x1 kj)d+Æ + (1 + j2j y1 kj)d+Æ ): Then, we have
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k P;j2ZZ;k2ZZd j;k gj;k k2 C (Pj2ZZ;k2ZZd jj;k j2 )1=2 .
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Real harmonic analysis (C) With the same hypotheses, we have
P
2 ;j 2ZZ;k2ZZd jhf jgj;k ij
C kf k22 .
Proof: For proving (A), we want to prove that for 2 fÆ=8; Æ=8g, the matrix M = (mk;l = h( ) gk j( ) gl i)k;l2ZZd is bounded on l2 (ZZd ). We write mk;l = h( )3 gk j( ) gl i = h( ) gk j( )3 gl i. Thus, if we can prove that for 0 < 1 ; 2 < Æ=2 the matrix M~ = (h( ) 1 gk j( ) 2 gl i)k;l2ZZd is bounded on l1 (ZZd ), we would obtain that M and t M are bounded on l1, hence that M is bounded on l1 and l1, hence on l2 . We rst prove that k( ) 2 gl k1 C 2 . Let K 2 be the kernel of the d convolution operator ( ) 2 ; K 2 (x) = jxjcd 22 2 . K 2 2 L d 2 2 ;1 and kgl kL 2 d2 ;1 C ; hence, k( ) 2 gl k1 C . Now, for jx lj 1, we write R ( ) 2 gl (x) = I1 + I2 + I3 = jy ljjx lj=2(K 2 (x y) K 2 (x l))gl (y) dy + R R jy ljjx lj=2 K 2 (x Rl)gl (y) dy + jy ljjx lj=2 K 2 (x y)gl (y) dy. We have (choosing 2 2 < < Æ) jRI1 j C jx ljd1 2 2+ (1+jjyy lljj)d+Æ dy; this gives R jI1 j dx C . We have I2 = K 2 (x R l) jy lj>jx lj=2 gl (y) dy; hence, we have jI2 j C jx lj1d 2 2 jx 1ljÆ and thus jI2 j dx C . We have jI3 j R C jx yj3jy lj jx yj1d 2 2 (1+jy 1 lj)d+Æ dy and we get easily by the Fubini theorem R R 2 2 P that jI3 j dx C (1+jyjy ljlj)d+Æ dy C 0 . Thus, k( ) 2 ( l2ZZd l gl )k1 C 2 k(l )k1 . P Moreover, we have k k2ZZd k gk kB1 Æ;1 C k(k )k1 : the estimation of P P g ( x ) g (y) (for jx y j 1) is straightfork k k2ZZd k gk (x) and of k2ZZd k jx P y jÆ Æ 2 1 ;1 L1 and ward; hence, for 1 < Æ=2, ( ) 1 ( k2ZZd k gk ) 2 B1 P
k( ) 1 ( k2ZZd k gk )k1 C 1 k(k )k1 . This proves that jhM~ (l )j(k )ij C k(l )k1 k(k )k1 , and (A) is proved. (B) is then easy : we have by (A) that
k( and
X
)Æ=8(
k2ZZd
X
j;k gj;k )k2 C 2jÆ=4 (
k2ZZd
jj;k j2 )1=2
X X ) Æ=8 ( j;k gj;k )k2 C 2 jÆ=4 ( jj;k j2 )1=2 k2ZZd k2ZZd hence, in conclusion, we nd that
k(
X
jh
k2ZZd
j;k gj;k j
X
X X l;k gl;k )ij C 2 jj ljÆ=4 ( jj;k j2 )1=2 jl;k j2 )1=2 k2ZZd k2ZZd k2ZZd
(C) is a direct consequence of (B). We now begin the analysis of Besov spaces with the description of the easy case of quadratic Sobolev spaces.
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Theorem 9.2: (Sobolev spaces) Let ( ;j;k )12d 1;j2ZZ;k2ZZd be a N -regular wavelet basis (with associated dual wavelet and associated scaling functions ', ' ). Then for 0 < s < N , the family ('(x k))k2ZZd [( ;j;k )12d 1;j2IN;k2ZZd is a Riesz basis of H s (IRd); more precisely, there exists two positive constants 0 < As Bs < 1 so that for all f 2 H s we have As kf kH s
X
k2ZZd
jhf j'0;k ij2 +
X X
X
ij2 1=2 Bs kf kH s 4js jhf j ;j;k
j 2IN k2ZZd 12d 1
P P i j;k; and we de ne j = Proof: We write fj = k2ZZd 12d 1 hf j ;j;k P P 2 1=2 . We have to prove that for some constants k2ZZd 12d 1 jhf jP;j;k ij P P Cs , Ds , we have : Cs k j0 fj kH s ( j 4js j2 )1=2 Ds k j0 fj kH s . Using the vaguelettes lemma, we obviously nd that for = 0 or = N we have kfj kH C 2j j . This gives 2js max(kfj k2 ; 2 jN kfj kH N ) C 2js j ; hence, kf k[L2;H N ]s=N;2 C k2js j kl2 . Since [L2 ; H N ]s=N;2 = H s , this gives P P Cs k j0 fj kH s ( j 4js j2 )1=2 . Conversely, we have (using Lemma 9.1) that for 1 2d 1 the (sep@ N , where l depends on and arable) wavelet may be written as = @x N l 2 L2comp. Using integration by parts and the vaguelettes lemma then gives P @ N f k ). Now, if f 2 H s , we may write that j C min(kf k2; 2 jN 1lN k @x N 2 l P f = p0 Fp with (2ps max(kFp k2 ; 2 pN kFp kH N )) 2 l2 (IN). Then, we nd that X
2js j C (
X 2(j p)(s N ) 2ps 2 pN kFp kH N + 2(j p)s 2ps kFp k2 ) p<j pj P
and this is enough to get that ( j2IN 4js j2 )1=2 C kf kH s . We use a simple application of this characterization to prove the compactness of the inclusion H0s ( ) H0 ( ) for bounded and s > 0.
De nition 9.2: (Sobolev space on a domain) For a bounded open subset of IRn and s 0 the Sobolev space H0s ( ) is the space of functions f 2 H s (IRn ) such that f = 0 almost everywhere outside from . It is a closed subspace of H s (IRn ). Proposition 9.1: (Compact inclusion) For a bounded open subset of IRn and s > 0 the inclusion operator s H0 ( ) ! H0 ( ) is a compact operator. Proof: Let fn 2 H0s ( ), n 2 IN, be a bounded sequence in H s (IRd ). Then a subsequence gp = fnp , p 2 IN, is weakly convergent to some g 2 H s (IRd ):
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Real harmonic analysis R
R
for all ' 2 D, limn!1 gp ' dx = g' dx. We must prove that the convergence is strong in H (IRd ) for < s. We use the wavelet decomposition P Id wavelet P = Pj + qj Qq on a N -regular P P basis (N > s), where Pj f = h f j ' i ' and Q f = d d j;k q j;k k2ZZ 12 1 k2ZZd hf j ;q;k i ;q;k . Theorem 9.2 P gives that, for j 0, k qj Qq (gp g)kH C 2j( s) supp kgp kH s . Moreover, if Kj is the ( nite) set P of indexes such that Supp 'j;k \ 6= ;, then we have kPj (gp g)kH C 2j k2Kj jhgp gj'j;k ij, hence limp!1 kPj (gp g)kH = 0. Another easy case to deal with is the analysis of Holder spaces:
Theorem 9.3: (Holder spaces) Let ( ;j;k )12d 1;j2ZZ;k2ZZd be a N -regular wavelet basis (with associated dual wavelet and associated scaling functions ', ' ). Then for 0 < < N , there exists two positive constants 0
Proof: The convergence of the wavelet series is obvious. If M is the maximum of the diameters of the supports of the functions ', ' , and , then the behavior of the wavelet series of f is determined in a neighborhood of any compact subset K of IRd by the values of f on the set fx 2 IRd = d(x; K ) 2M g. ;1 for > 0 are locally It is then suÆcient to notice that the elements of B1 square integrable and that the wavelet series of a square-integrable function converges to this function in L2 norm. We now give two proofs of Theorem 9.3. The rst one works only for ;1 is equal to the Holder space C de ned by: if 2= IN. In that case, B1 = m + , m 2 IN et 0 < < 1, then f 2 C if and only the rst derivatives of f , up to order m, are continuous and bounded and the mth derivatives of f are Holderian of exponent ; C is normed by
kf kC =
X
j jm
@ k @x f k1 +
@
@
j f (x) @x f (y)j : sup @x jx yj j j=m x=6 y X
2 C , we use the Taylor expansion of f (x):
If f X
(x
j j<m
@ f( k ) Z 1 k @x k k k k (1 t)m 1 2j ) + dm f ( j +t(x j )):(x j ; : : : ; x j ) dt j 2 ! 2 2 2 2 (m 1)! 0
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then we use the oscillation of the wavelets (Lemma 9.1) to get
h
X
(x j j<m
and
@ f( k ) k @x 2j i=0 ) j ;j;k j 2 !
Z 1 m 1 h dm f ( 2kj ):(x 2kj ; : : : ; x 2kj ) (1(m t) 1)! dtj
0
;j;k i = 0
1 1 ij C kf kC j to get jhf j ;j;k 2 2jd=2 k k1 ; we easily obtain jhf j'0;k ij kf k1k' k1 and thus control the size of the coeÆcients of the wavelet expansions of f . Conversely, if we control thePsize of the coeÆcients of the wavelet expansion of f , we de ne f 1 = k2ZZd hf j'0;k i '0;k and, for j 0, fj = P P N k2ZZd 12d 1 hf j ;j;k i j;k; . f 1 is of class C and all its derivatives are bounded ; thus, f 1 belongs to C . Similarly, fj is of class C N and P its derivatives of order k (k N ) are bounded by C 2j(k ) . This gives that j0 fj is a function of class C m and that its derivatives up to order m are bounded. If j j = m, we control j @x@ fj (x) @x@ fj (y)j by min(2k @x@ fj k1; kr~ ( @x@ fj )k1 jx @ f (x) @ f (y )j yj) C 2 j min(1; 2j jx yj); summing over j , we get j @x j j @x C jx yj . We thus have proved Theorem 9.3 for 2= IN. When 2 IN , the space ;1 is a Zygmund class C , and the proof is more delicate. We then prefer B1 to use the characterization of the Besov space as a real interpolation space; we ;1 = [L1 ; W N;1 ] have shown in Chapter 3 that B1 =N;1 . If we control the size of the coeÆcients in the wavelet expansion of f , we again introduce f 1 = P P P h f j ' i ' and, for j 0, f = 0;k j 0;k k2ZZd k2ZZd 12d 1 hf j ;j;k i j;k; . N; 1 ; 1 j Then, f 1 2 W P B1 while we have kfj k1 C 2 and kfj kW N;1 C 2j(N ) which gives j0 fj 2 [L1 ; W N;1] =N;1 . Conversely, if we assume that f 2 [L1 ; W N;1] =N;1 , then f 2 L1 so that we control the coeÆcients hf j'0;k i ; moreover, we may decompose for all j 0 f into f = gj + hj where kgj k1 C 2 j and khj kW N;1 C 2j(N ) ; using the Taylor expansion of ij hj up to order N and the oscillation of the wavelet , we get jhf j ;j;k jN 0 jd= 2 j C k k1 (kgj k1 + 2 khj kW N;1 ) C 2 2 .
An example of application of the wavelet analysis of Holder spaces, the characterization of (Morrey)-Campanato spaces follows:
De nition 9.3: (Campanato spaces) d a) For 1 < p q 1, the Morrey space M p;q (IR of locally R ) is the space p d ( p=q 1) p dx < 1, L functions so that sup0
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Real harmonic analysis
Similarly, for 1 q 1, the Morrey space M 1;q (IRd )Ris the space of locally nite measures d so that sup0
sup sup inf R p jf I X1 ;:::;Xd ];deg P
P jp dx < 1
normed by
kf k ;N = kf kLpuloc + p
Z
jf P jpdx)1=p sup sup inf R ( P 2 C[ I X ;:::;X ] ; deg P
d Similarly, the Campanato space ;N 1 (IR ) is the space of uniformly locally nite measures d so that
sup sup
inf
I X1 ;:::;Xd ];deg P
R
Z
jx x0 j
dj j < 1
normed by
kdk ;N = kdkMuloc + 1
.
1
Z
sup sup inf R dj j 0
Remark: We are not interested in < 0 since, in that case, the space ;N p would coincide with 0p;N = Lpuloc (p > 1) [uniformly locally Lp functions] or 1 (p = 1). 01;N = Muloc Proposition 9.2: Let p 2 [1; +1], > 0 and N 2 IN. If > d=p and d=p 2= IN, then, for N < d=p, ;N = CI and, for N > d=p, p ;1 for = d=p. ;N = B p 1 Proof: Let us choose M > max(N; ) (where = d=p); we then use a basis of M -regular wavelets. Let f 2 ;N p . The coeÆcients hf j'0;k i are p 1 q bounded since f 2 Luloc or f 2 Muloc and ' 2 L (1=p + 1=q = 1) or ' 2 C0 . Now, for any polynomial P with degree less than N , the oscillation i = hf P j i; hence, for j 0 we have jhf j ij of gives hf j ;j;k ;j;k ;j;k ;1 C 2 j 2j(d=p d=2) k kq . Thus, we have proved that ;N p B1 . Conversely, ;1 and = m + (m 2 IN and 0 < < 1), then if Px is the Taylor if f 2 B1 0 expansion of f at x up to order m , we have j f ( x ) P ( x ) j C jx x0 j ; 0 x 0 R ( p 1 =p + d=p ;N ;1 for hence, jx x0jR jf (x) Px0 (x)j dx) CR . Thus, p = B1
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N m + 1. If N m, we introduce the di erence operators i; de ned by i; g(x) = g(x + ei ) g(x) where (e1 ; : : : ; ed) is the canonical basis of IRd . Then, we have for jx x0 j ji1 ; : : : iN ;f (x) @x@i1 : : : @x@iN f (x0 )N j Cmin(N +1; ) while for any P with degree less than N we have i1 ; : : : iN ; P = R 0; this gives ( jx x0j ji1 ; : : : iN ;f jp dx)1=p C ; hence, we nd that j @x@i1 : : : @x@iN f (x0 )j Cmin(1; N ) ; thus, f is a polynomial with degree less than N ; its uniform integrability gives moreover that f is constant. Theorems 9.2 and 9.3 are easily extended to the case of Besov spaces:
Theorem 9.4: (Besov spaces) Let ( ;j;k )12d 1;j2ZZ;k2ZZd be a N -regular wavelet basis (with associated dual wavelet and associated scaling functions ', ' ). Then for 0 < s < N , 1 p < 1, 1 q < 1, the family ('(x k))k2ZZd [ ( ;j;k )12d 1;j2IN;k2ZZd is a Riesz basis of Bps;q (IRd); more precisely, there exists two positive constants 0 < As;p;q Bs;p;q < 1 so that for all f 2 Bps;q we have As;p;q kf kBps;q Ns;p;q (f ) Bs;p;q kf kBps;q with
Ns;p;q (f ) =
X
k2ZZd
jhf j'0;k ijp
1=p
+
X
j 2IN
d d )q X X p ( f k2ZZd12d 1
2j(s+ 2
jh j
p q q1 ;j;k ij ) p
Remark: The equivalence of norms is also true in the case p = 1 or q = 1, but in that case we have only weak convergence of the wavelet series. P P i j;k; and we de ne j = Proof: We write fj = k2ZZd 12d 1 hf j ;j;k P P ijp 1=p . We must prove that for some constants f j ;j;k k2ZZd 12d 1 jhP P P C , D, we have: C k j0 fj kBps;q ( j 2jsq jq )1=q Dk j0 fj kBps;q . We check easily that for = 0 or = N we have kfj kW ;p C 2j j . This gives, using the J -method of interpolation, that kf kBps;q C k2js j klq . Conversely, we write again, for 1 2d 1, the (separable) wavelet p p 1 . Using inte as = @ NN where l depends on and 2 Lcomp @xl gration by parts and a one-scale P vaguelettes lemma for Lp then gives that @ N f k ). Now, if f 2 B s;q , we j ( d=p d= 2) jN j C 2 min(kf kp; 2 p p 1lN k @xN l P may write f = l0 Fl with (2ls max(kFl kp; 2 lN kFl kW N;p ) 2 lq (IN). Then, we conclude that
2j(s+d=2 d=p) j
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C(
X
X 2(j l)(s N ) 2ls 2 lN kFl kW N;p + 2(j l)s 2ls kFl kp) l<j lj
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Real harmonic analysis
3. Singular integrals Wavelets are a useful tool for the study of singular integral operators.
De nition 9.4: (Singular integral operators) A singular integral operator is a continuous linear operator T from D(IRd ) 0 to D (IRd ) so that there exists a continuous function K de ned on IRd IRd (where is the diagonal set x = y), which satis es: i) 9C0 > 0 8x 8y jK (x; y)j jxC0yjd Æ ii) 9Æ > 0 9C1 > 0 8x; y; z; jz j < 12 jx yj ) jK (x+z; y) K (x; y)j C1 jx jzyjjd+Æ Æ iii) 9Æ > 0 9C1 > 0 8x; y; z; jz j < 12 jx yj ) jK (x; y+z ) K (x; y)j C1 jx jzyjjd+Æ RR iv) 8f; g 2 D(IRd ); Suppf \ Suppg = ; ) hT f jgi = K (x; y)f (y)g(x) dx dy A Calderon{Zygmund operator is a singular integral operator T , which is bounded on L2(IRd) (T 2 L(L2 ; L2 )). When Æ = 1, we de ne the Calderon{ Zygmund norm of T as the sum kT kCZO = kT kL(L2;L2) + kjx yjd K (x; y)k1 + kjx yjd+1 r~ x K (x; y)k1 + kjx yjd+1 r~ y K (x; y)k1 kT kL(L2;L2) . We now introduce two useful notions associated to singular integral operators: the weak boundedness property and the distribution T (1). De nition 9.5: (Weak boundedness property) A continuous linear operator T from D(IRd) to D0 (IRd ) satis es the weak boundedness property (what we shall write T 2 W BP ) if there exist a constant C and a number N such that for all ; 2 D(IRd ) with support in B (0; 1), all x0 2 IRd and all R > 0, writing x0 ;R (x) = ( x Rx0 ) and x0 ;R (x) = ( x Rx0 ), we have
jhT ( x ;R )j 0
X
x0 ;R ij C (
j jN
X
Rj j k@ ( x0 ;R )k2 ) (
j jN
Rj jk@ ( x0 ;R )k2 ):
De nition 9.6: (The distribution T (1)) If T is a singular integral operator, we may de ne T (1) 2 D0 (IRd )=CI by choosing ' 2R D(IRd ) equal to 1 in a neighborhood of 0 and computing for 2 D(IRd) with dx = 0 hT (1)j i as hT (1)j i = limR!1 hT ('( Rx ))j i. We may see easily that, if ! 2 D(IRd ) is equal to 1 in a neighborhood of Supp and x0 2 Supp , then
hT (1)j i = hT (!)j i +
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Z Z
(K (x; y) K (x0 ; y))(1 !(y)) ;j;k (x) dx dy:
Wavelets and functional spaces
89
We shall now prove the T (1) theorem of David and Journe [DAVJ 84] in the case T (1) = T (1) = 0 (the general T (1) theorem is proved in Chapter 10):
Theorem 9.4: (The T (1) theorem) Let T be a singular integral operator. If T then T is bounded from L2 to L2 .
2 W BP
and T (1) = T (1) = 0,
N C N Proof: Since T 2 W BP , the bracket hT f jgi may be extended to Ccomp comp N . for some N . We thus introduce an orthonormal wavelet basis with ' 2 Ccomp N N , the operator Pj T Pj is well de ned and we Since Pj maps Ccomp to Ccomp N have, for all f; g 2 Ccomp , limj!+1 hT Pj f jPj gi = hT f jgi. Moreover, we have limj! 1 hT Pj f jPj gi = 0: let K be the ( nite) set of indexes k 2 ZZd so that 0 2 Supp '(x k); then, P for f; g with compact support, P we have for j close enough to 1 that Pj f = k2K hf j'j;k i'j;k and Pj g = k2K hgj'j;k i'j;k ; thus, using the weak boundedness property to get that supj2ZZ supk;k0 2K jhT 'j;k j'j;k0 ij < 1, we nd that jhT Pj f jPj gij C supk2K jhf j'j;k ij supk2K jhgj'j;k ij C 0 2j kf k1kgk1. Thus, T may be approximated by PJ T PJ P J T P J with J going to +1. Writing Pj+1 T Pj+1 Pj T Pj = Pj T Qj + Qj T Pj + Qj T Qj , we nd that, for f; g with compact support,
hT f jgi = Nlim !1
X
jjjN
hT Qj f jPj gi +
X
jjjN
hT Qj f jQj gi +
X
jjjN
hT Pj f jQj gi:
Thus, the theorem will be proved when we have shown that the operators P
P
f 7! A;N f = P jjjN P k2ZZd hf j ;j;k iPj T ;j;k f 7! B;N f = P jjjN P k2ZZd hf j ;j;k iQj T ;j;k f 7! C;N f = jjjN k2ZZd hf j ;j;k iPj T ;j;k are bounded in L2 norm with a bound that does not depend onP N . This is easily accomplished. Indeed, let ;j;k = Pj T ;j;k . We have ;j;k = l2ZZd ;j;k;l 'j;l with j ;j;k;l j C (1+jk 1 lj)d+Æ : for k close to l, we use the weak boundedness property; for k and l far from each other, we write ;j;k;l = hT ;j;k j'j;l i = RR (K (x; y) K (x; k=2j )) ;j;k (y)'j;l (x) dx dy. The estimates on ;j;k;l then give the estimates j;j;k (x)j C 2jd=2 1+j2j x1 kjd+Æ and
j;j;k (x) ;j;k (y)j C 2j(d=2+Æ) jx yjÆ ( (1 + j2j x1 kj)d+Æ + (1 + j2j y1 kj)d+Æ ) R
P
Moreover, we have ;j;k dx = 0: indeed, we know that k2ZZd '(x k) = 1; P we choose L large enough so that jljL '(2j x l) = 1 is equal to 1 on a neighborhood of the support of ;j;k . Then we have hT (1)j ;j;k i = A + B with P A = hT ( jljL 2 jd=2 'j;l )j ;j;k i RR P : B = (K (x; y) K (x; k=2j ))( jlj>L 2 jd=2 'j;l (x)) (y) dy dx
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Real harmonic analysis
We have A =
B = Tlim !1
P
jljL 2
Z Z
jd=2 ;j;k;l .
Similarly, we have
(K (x; y) K (x; k=2j ))( P
X
L<jlj
2 jd=2 'j;l (x)) (y) dy dx R
which gives B = limT !1 L<jlj
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Chapter 10 The space BMO
In this chapter, we present the space BMO and three basic results of harmonic analysis: the celebrated duality theorem of Fe erman and Stein [FEFS 72], the T (1) theorem of David and Journe [DAVJ 84] (see also Meyer [MEY 97]) and Calderon's commutator theorem [MEY 97].
1. Carleson measures and the duality between H1 and BMO
We give several equivalent characterizations of BMO. For one characterization, we need the Littlewood{Paley decomposition, i.e., the operators Sj and j de ned by F (Sj f ) = '(=2j )F f and F (j b) = (=2j )F b, where ' 2 D(IRd ) is such that j j 12 ) '( ) = 1 and j j 1 ) '( ) = 0 and where is de ned as ( ) = '(=2) '( ). Before stating the theorem, we prove a useful result on Carleson measures:
Proposition 10.1: (Carleson measures) Let j (x) be a sequence of measurable functions on IRd de ning a Carleson measure on ZZ IRd: Z 1 X sup j j (x)j2 dx < 1 d x0 2IRd ;R>0 R 2j R>1 jx x0 j
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Real harmonic analysis
for k 0. Thus, we may deduce the general case from the case ' = 1B(0;1) . R P We thus try to estimate IRd j2ZZ (jmB(x;2 j ) jf j)2 j j j2 dx. We introduce, for the Hardy{Littlewood maximal function Mf of f , k = fx = Mf (x) > 2Pk g, k the set of maximal dyadic open cubes contained in k (so that k = k k+1 Q2 k Q [ Nk with jNk j = 0) and k = f(j; x) = 2 < jmB (x;2 j )j 2 g. When (j; x) 2 k , then we clearly have x 2 k ; moreover, the ball B (x; p2 j ) j is wholly contained in k ; it contains a dyadic cube Q0 of size 2 = d; if Q = 2klQQ + 2l1Q (0; 1)d is the maximal dyadic cube of containing Q0 , then p p clearly x 2 B ( 2klQQ ; 2 d2 lQ ) and 2j+lQ d. Thus, we have
Z
X
IRdj 2ZZ
C
(jmB(x;2 X
k2ZZ
j)
22k+2
jf j)2 j j j2 dx
X
Q2 k
jQj = C
X
k2ZZ
X
k2ZZ
22k+2
X Z
Q2
X j 2 kQ p lQ p k B ( 2lQ ;2 d2 )2j+lQ d Z
j j dx
22k+2 j k j C 0
jMf (x)j2 dx C 00 kf k22
The classical characterizations of BMO follows:
Theorem 10.1: (Characterizations of BMO) Let b 2 D0 (IRd ). Then, the following assertions are equivalent: (A1) b 2 (HR1 )0 : more precisely, there exists a constant C so that for all ! 2 D(IRd) with ! dx = 0, we have jhbj!ij C k!kH1R. (A2) b is locally integrable and we have: supB2B jB1 j B jb mB bj dx < 1 where R B is the collection of all balls B (x; r), x 2 IRd ; r > 0, and mB b = jB1 j B b(x) dx. R (A3) For q 2 [1; 1), b is locally Lq and: supB2B jB1 j B jb mB bjq dx < 1. R (A4) b 2 S 0 (IRd ) and for all 2 S (IRd) so that dx = 0, de ning t (x) = 1 ( x ), the measure jb (x)j2 dt dx is a Carleson measure on (0; 1) IRd : t d t t t 1 sup d R R>0;x0 2IRd (A5) b 2 S 0 (IRd ) and
Z Z
0
jb
t (x)j
2 dt dx < 1
t
Z Z
1 @ t 2 sup j e bj dt dx < 1 d 0
0;1 . This is the reProof: We begin with some comments on the space B_ 1 0 ; 1 _ alization of the homogeneous space B1 . Indeed, if we consider a sequence
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BMO
93
of bounded functions (fj )j2ZZ 2 l1 (ZZ; L1) so that f^ is supported P in f 2 P j d 1 j j IR = 2 2 j j 2 2 g; then it is easy to check that j<0 fj (x) fj (0)+ j0 fj is convergent in S 0 (IRd): for j < 0 we have jfj (x) fj (0)j C jxj2j while for 0;1 j 0, we have fj = Fj with kFj k1 C 4 j . Thus, we may de ne B_ 1 as a Banach space of tempered distributions modulo the constants through the 0;1 , that ( b) double condition, for b to belong to B_ 1 (ZZ; L1 ) and that j j 2ZZ 2 l1P Z Z there exist a sequence of constants ( j ) 2 CI so that b = j2ZZ j b + j in S 0 (IRd ).PEquivalently, b 2 S 0 (IRd ), (j b)j2ZZ 2 l1 (ZZ; L1 ) and for l = 1; : : : ; d, @l b = j2ZZ @l j b in S 0 (IRd ): indeed, if (j b)j2ZZ 2 l1 (ZZ; L1 ) and if = P P ^ ^ j<0 j b(x) j b(0) + j 0 j b, we nd that b is supported in f0g, hence is a polynomial P ; but a polynomial has a homogeneous Littlewood{ P Paley decomposition equal to 0, so that the assumption @l b = j2ZZ @l j b gives r~ P = 0; hence, b is constant. In particular, we see that BMO (de ned as the dual of H1 ) is a subspace 0 of B_ 1;1 : j b(x) = hb(y)j2jd (2j (x y)i where = F 1 ; hence, we nd that kj bk1 kbkBMO k kH1 ; moreover, @l Sj b = hb(y)j2j(d+1) l (2j (xP y)i, where l = @l F 1 '; this gives k@l Sj bk1 2j kbkBMO k lkH1 and @l b = j2ZZ @l j b. We now prove the theorem : (A1) ) (A3): for q > 1, use the atomic decomposition with Lp atoms, where 1=p + 1=q = 1; for q = 1, use the result for q > 1 and the Holder inequality. (A3) ) (A2) is obvious (Holder inequality). (A2) ) (A1): use the atomic decomposition with bounded atoms. (A3) ) (A4): We rst assume that is compactly supported, with support in B (0; 1). Then for jx x0 j R and 0 < t < R, the valuesR of b t (x) depends only on the values of b on the ball B (x0 ; 2R) and, since dx = 0, we may change b into b mB(x0 ;2R b. Moreover, for f 2 L2 , we have Z Z
jf
t
dt (x)j2 dx = t
with C = maxjj=1
Z Z
0
R +1
0
jb
1 (2)d
j ^(t )j2
dt t
Z Z
C k k2W ; . This gives
2 dt dx C
t (x)j
t
jf^( )j2 j ^(t )j2 dtt d C kf k22 1 1
Z
B (x0 ;2R)
jb mB(x ;2R bj2 dx CRd : 0
R This proves (A4) for with support in B (0; 1). For 2 S 0 with dx = 0, Pd we write = j=1 @j !j with !j 2 S . !j may be decomposed as a sum of !j = P P n n(d+2) k!j;n k 2;1 < 1, W n2IN !j;n , with !j;n supported in B (0; 2 ) and n2IN 2 and this is enough to arrive at an estimate. d @ t p p1 p (A4)dt) (2Ad 5): write @xj e b = t b t for some 2 S (IR ) and, for t = , t = . (A5) ) (A6): We rst prove that @i b 2 S00 for i = 1; : : : ; d. Indeed, for j 2 ZZ and 0 < t < 4 j , we may write Sj @i b = e tSj et@i b. The operator
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Real harmonic analysis
e tSj is a convolution operator with 2jd F4j t (2j x), where the functions F = F 1 (ejdjj2 '( )) (0 < < 1) are a bounded set in S (IRd ). Thus, j2jd F4j t (2j x)j C (1+j22j xj)d+1 . Summing for t 2 (0; 4 j ), we get
jSj @i
b(x)j 4j
Z 4 jZ
IRd
0
2jd jet @i b(y)j dy dt (1 + j2j (x y)j)d+1
which gives
jSj @i
b(x)j C 2j
Z 4 jZ
1 jd d+1 2 (1 + j k j ) d k2ZZ X
0
jet @i b(y)j2 dy dt 1=2
p jy x 2kj j 2jd
and thus, kSj @i bk1 C 2j . Thus, @i b 2 S00 and b 2 B_ 11;1 . We now prove that the family (j b) de nes a Carleson measure on ZZ IRd . P d For j 2 ZZ and 4 j 1 < t < 4 j , we may write j b = i=1 e t @i j et @i b. The operator e t 1 j is a convolution operator with 2j(d 2) F4j t (2j x), where 2 the functions F = F 1 (ejj j(j2) ) (1=4 < < 1) are a bounded set in S (IRd ). P P We then decompose F into F '(x) + k0 F (=2k ) = g 1; + k0 gk; , with ' supported in B (0; 1) and supported in fx = 1=2 jxj 2g. We have, for j j 2 and L 2 IN, j@ gk; j CL 2 kL (1+jx1j)d+1 . This gives for every L 2 IN: k2jd(@i gk; )(2j x) f k22 CL 2 2kL kf k22 We thus write j b = 4j+1
Z 4 j
d X
4 (j+1) i=1
and get
4(j+2)
X
k 1
2k
Z
j
X
k 1
2k(1 2L)
d X
Z 4 j
2j(d
X
k 1
2k(1 2L)
1) (@
j x) (et @
i gk;4j t )(2
Z
4 (j+1) jx x0 jR+2k+1 j
X Z
CL
(@i gk;4j t )(2j x) (et @i b) dt
jj b(x)j2 dx
4 (j+1) jx x0 jR i=1
2j R1
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k 1
jx x0jR
Z 4 j
X
1)
Z
CL Finally:
2j(d
jx x0 jR 0
jet@i bj2 dt dx:
jj b(x)j2 dx
Z R2 Z
jx x0jR(1+2k+1 )
i b)j
jet @i bj2 dt dx
2
dt dx
BMO
CCL Rd
X
k 1
95 2k(d+1 2L)
which gives the result for 2L > d + 1. The Fourier spectrum of Sj 2 f j b is contained f = 2j 2 R in the2annulus P j +2 2 2 j j 2 g, thus we have : k(f; b)k2 C j2ZZ jSj 2 f j jj bj dx. We may then apply Proposition 10.1 to get the boundedness of (:; b) on L2 . 0;1 and f 2 L2 then the (A6) ) (A1): WePeasily check that if b 2 B_ 1 paraproduct (b; f ) = j2ZZ Sj 1 f j b is well de ned in S 0 (IRd ) (it belongs to B1d=2;1 ). The Schwartz kernel theorem for continuous linear operators from D to D0 allowsP us to speak from the distribution kernel K (x; y) of (b; :). We have K (x; y) = j2ZZ kj (x; y) with kj (x; y) = 2jd (2j (x y))j b(y) with ~ x kj (x; y)j + 2 S (IRd ). This gives jkj (x; y)j C 2jd min(1; (2j jx 1yj)d+1 and jr jr~ y kj (x; y)j C 2j(d+1) min(1; (2j jx 1yj)d+2 ), and nally we get that, outside of the diagonal set , the distribution K is a locally Lipschitzian function so that jK (x; y)j C jx 1yjd and jr~ x K (x; y)j + jr~ y K (x; y)j C jx y1jd+1 . If (b; :) is a bounded operator on L2 , then it is a Calderon{Zygmund operator and we have seen in Chapter 6 that by duality, it maps LR1 to BMO. Thus, there exists 2 RBMO so that, for all 2 D(IRd ) with dx = 0, R t T dx = h j i. But t T dx may easily be computed: we have, for ! 2 D t equal P to R1 in a neighborhood of the support P of R and for y0 2 Supp , T = ! j2ZZ kj (Ry; x) (y) dy + (1 !(x)) j2ZZ (kj (y; x) kj (y0 ; x)) (y) dy. P Since j2ZZ j(kj (y; x) kj (y0 ; x)) (y)j dy is bounded by jx yC0jd+1 far from R RR P the support of , we obtain t T dx = Pj2ZZ kj (y; x) (y) dy dx = R P 0 I Thus, b 2 j 2ZZ j b(y ) (y ) dy . This gives that = j 2ZZ j b in D =C. BMO.
2. The T (1) theorem David and Journe have given the following L2 boundedness criterion for singular integral operators [DAVJ 84]:
Theorem 10.2: (The T (1) theorem) Let T be a singular integral operator. Then the following assertions are equivalent : (A1) T is bounded on L2 : there exists a constant C so that for all 2 D(IRd) T ( ) 2 L2 and kT ( )k2 C k k2 . (A2) T (1) 2 BMO, T (1) 2 BMO and T 2 W BP . Proof: (A1) ) (A2) is obvious: if T is a singular integral operator bounded on L2 , then T and T are Calderon{Zygmund operators and we know that a Calderon{Zygmund operator maps L1 to BMO.
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Real harmonic analysis
(A2) ) (A1) is a direct consequence of Theorem 10.1, since we may modify T by T (:; T (1)) (:; T (1)) to get a singular integral operator T~ so that T~ 2 W BP , T~(1) = T~ (1) = 0. We have proved in Chapter 9 that such a singular integral operator is bounded on L2 . We may now collect some results on singular integral operators:
Proposition 10.2: a) Let !j;k and j;k , j 2 ZZ, k 2 ZZd , be two sequences of functions on IRd so that for some positive constants C and Æ: 8 <
j! (x) : j;k
j!j;k (x)j C 2jd=2 1+j2j x1 kjd Æ !j;k (y)j C 2j(d=2+RÆ) jx yjÆ ( (1+j2j x1 kj)d +
Æ
+ (1+j2j y1 kj)d+Æ )
Æ
+ (1+j2j y1 kj)d+Æ )
+
!j;k dx = 0
and (
jj;k (x)
jj;k (x)j C 2jd=2 1+j2j x1 kjd Æ j;k (y)j C 2j(d=2+Æ) jx yjÆ ( (1+j2j x1 kj)d +
+
P
Then the operator T (f ) = j2ZZ;k2ZZd hf j!j;k ij;k is a singular integral operator, T 2 W BP and T (1) = 0. b) When b 2 BMO and when ( ;j;k )12d 1;j2ZZ;k2ZZd is an orthonormal basis of L2(IRd) with compactly supported wavelets and compactly supported scaling function ' (with P ' continuously P P di erentiable), then the operator (b; :) de ned by (b; f ) = j2ZZ k2ZZd 12d 1 2jd=2 hf j'j;k i hbj ;j;k i ;j;k is a Calderon{Zygmund operator T so that T (1) = b and T (1) = 0. c) When T is a Calderon{Zygmund operator, then it may be decomposed into T = T0 + (T (1); :) + (T (1); :) with
T0(f ) =
X X
X
hf j
j 2ZZ k2ZZd 12d 1
;j;k i!;j;k
where ;j;k are the wavelets described in b) and where the functions !;j;k = T0 ( ;j;k ) satisfy for some positive constants C and Æ: 8 < :
j!;j;k (x)
j!;j;k (x)j C 2jd=2 1+j2j x1 kjd Æ !;j;k (y)j C 2j(d=R2+Æ) jx yjÆ ( (1+j2j x1 kj)d +
!;j;k dx = 0
Æ
+
+ (1+j2j y1 kj)d+Æ )
d) When T and S are two Calderon{Zygmund operators so that T (1) = 0 and S (1) = 0, then T Æ S is a Calderon{Zygmund operator. e) When T is a Calderon{Zygmund operator so that T (1) = 0, then T is bounded from the Hardy space H1 to H1 .
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BMO
97
f) When T is a Calderon{Zygmund operator so that T (1) = 0, then T is bounded from BMO to BMO.
Remark: (b; :) is a slight modi cation , proposed by Y. Meyer [MEY 92], of the paraproduct operator (b; :) of Bony [BON 81] (described in Section 1). P
Proof: a) We rst estimate the sum k2ZZd ! j;k (y)j;k (x) = kj (x; y): jkj (x; y)j C 2jd (2j (x y)) with (x) = 1+jx1jd+Æ with (x) = 1+jx1jd+Æ ; for jhj max(2 j ; jx yj=2) : { jkj (x; y) kj (x + h; y)j C 2jd (2j jhj)Æ=2 (2j (x y)) { jkj (x; y) kj (x; y + h)j C 2jd (2j jhj)Æ=2 (2j (x y)). This P gives P P { j2ZZ jkj (x; y)j C ( 2j jx yj1 2jd + 2j jx yj>1 2 jÆ jx yj d Æ ) C 0 jx yj d { for jhP j jx yj=2 : P { j2ZZ jkj (x; y) kj (x + h; y)j j2ZZ C 2jd (2j jhj)Æ=2 (2j (x y)) 2 jx y j d Æ=2 C 0 jhjÆ=P { j2ZZ jkj (x; y) kj (x; y + h)j C 0 jhjÆ=2 jx yj d Æ=2. Moreover, if x0 2 IRd, r0 > 0 and if f and g are supported in B (x0 ; r0 ), we have: RR Pj kj (Rx;Ry)f (y)g(x) dx dyj C 2jd kf k1kgk1, hence we get the estimate kj (x; y)f (y)g(x) dx dyj Cr0 d kf k1kgk1 C 0 kf k2kgk2; 2j r0 1Rj Rsince !j;k dxy = 0, we may write !j;k = Pdl=1 @l !j;k;l where !j;k;l (x) = C !j;k (x y)( jyjld jxxjld ) dy, we nd j!j;k;l (x)j 2j(d 1)(1+ j2j xj) d+1 Æ and P j (d 1) (1+ j2j (x y )j) d+1 Æ ; we then choose so k2ZZd j!j;k;l (y )j;k (x)j C 2 1 d that d+Æ 1 < < d 1 (which is always possible, since d(d + Æ 1) (d 1) = (d 1)2 + dÆ > 0) and we get
j
Z Z
kj (x; y)f (y)g(x) dx dyj C 2j(d
1 d= ) kr ~ fk
1
kgk1;
RR P ~ f k kgk1 and hence, 2j r0 >1 j kj (x; y)f (y)g(x) dx dyj Cr0 d+1+d kr 1 this allows us to conclude that T is well de ned and satis es the weak boundedness property. Now, we prove T (1) = 0. Indeed, let g 2 D and let ' 2 D. We want to R prove that, if g dx = 0, then limR!1 hT ('(x=R))jgi = 0. We have proved that
j
Z Z
kj (x; y)'(y=R)g(x) dx dyj C (2j R)(d
1 d= ) kr ~ 'k
1
kgk1;
with d 1 d= < 0, and thus we have no problem with R the terms indexed by jR R 0. For the sum over j > 0, we use the condition R R g dx = 0: we nd that j(P kj (x; yR) kj (y; y))g(x)j dy dx C 2jÆ while kj (x; y)g (x) dy dx = 0, thus j<0 kj (x; y)g(x) dx converges in L1(dy) to a function whose integral is equal to 0.
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Real harmonic analysis
b) is now obvious. We rst notice that 2jd=2 jhbj ;j;k ij C kbkBMO . Hence, we may deduce from a) that T = (b; :) is a singular integral operator, T 2 R W BP and T (1) = 0.R Moreover, TP(1) =Pb: indeed, if f 2 D with f dx = 0, P we have hf jT (1)i = T f dx = j2ZZ k2ZZd 12d 1 hf j ;j;k i h ;j;k jbi. If f is supported by B (x0 ; r0 ), the series limited to j > j0 uses the value of b only on the ball B (x0 ; r0 + C 2R j0 ) ; since b is locally square integrable, we may P rewrite the truncated series as f (b mB(x0;r0 ) b) dx k2ZZd hf j'j0 ;k i h'j0 ;k jb mB(x0 ;r0 ) bi. When j0 goes to 1, there exist an index J and a xed nite set of indices K so that, for j0 < J and k 2= K , hf j'j0 ;k i = 0, while, for k 2 K , jhf j'j0 ;k ij C 2j0 (d=2+1) and jh'j0 ;k jb mB(x0 ;r0) bij C jj0 j2 j0 d=2 (see Stein [STE 93] for the last estimate). In order to prove c), we just need to estimate the size of the functions T0 ( ;j;k ) = !;j;k . We easily estimate the size of !;j;k (x) when Rjx k=2j j C 2 j : we write, using the kernel K0 of T0 , that !;j;k (x) = (K0 (x; y) K0 (x; k=2j )) ;j;k (y) dy; hence, j!;j;k (x)j C 2 jÆ jx k=2j j d Æ k ;j;k k1 jd=2 j2j x k j d Æ , while for j2j hj 1 we have j!;j;k (x) !;j;k (x + h)j C R2 jK0 (x; y) K0(x + h; y)jj ;j;k (y)j dy C 2jd=2 (2j jhj)Æ j2j x kj d Æ . We now consider the case P jx k=R2j j C 2 j . We have already proved (through Theorem 9.4) that T0f = l2ZZ Kl (x; y)f (y) dy, where for some positive constants C and Æ,
jKl (x; y)j C 2ld 1+j2l (x1 y)jd Æ jKl (x; y) Kl (x + z; y)j C 2l(d+Æ=2)jz jÆ=2( (1+j2l (x1 y)j)d jKl (x; y) Kl(x; y + z )j C 2l(dR+Æ=2) jz jÆ=2( (1+j2l (x1 y)j)d +
Kl (x; y) dy = 0
R
1
Æ + (1+j2l (x+z y)j)d+Æ ) 1 +Æ + (1+j2l (x z y )j)d+Æ ) +
R
j We R then write Kl (x; y ) ;j;k (y ) dy = (Kl (x; y ) K R l (x; k=2 )) ;j;k (y ) dy = Kl (x; y)( ;j;k (y) ( x )) dy and we obtain j K ( x; y ) ;j;k ;j;k l R (y ) dy j 2jd=2 2 jj ljÆ=2 . Hence, j!;j;k (x)j C 2jd=2 . Similarly, we nd j (Kl (x; y) Kl (x + h; y)) ;j;k (y) dyj 2jd=2 min(2lÆ=2 jhjÆ=2 ; 2(j l)Æ=2 ) and thus j!;j;k (x) !;j;k (x + h)j C (2j jhj)Æ=4 . d) is a direct consequence of c). Indeed, write T = (T (1); :) + T0 and S = (S (1); :) + S0 andPwritePthe kernels of T and S as the series (conP 0 (IRd IRd )): verging in D j 2ZZ k2ZZd 12d 1 T ( ;j;k )(x) ;j;k (y ) and P P P S ( ;j;k )(y) ;j;k (x); we obtain that the kernel of j 2ZZ k2ZZd 1P 2d 1P P T Æ S is given by Pj2ZZ Pk2ZZd P12d 1 T ( ;j;k )(x)S ( ;j;k )(y) and thus may be written as j2ZZ k2ZZd 12d 1 ;j;k (x)!;j;k (y), where 8 < ;j;k (x)
= 2jd=2 h ;j;k jT (1)i'j;k (x) + T0 ( ;j;k )(x) and : !;j;k (y) = 2jd=2 h ;j;k jS (1)i'j;k (x) + S0 ( ;j;k )(y) Now, we apply the same computations as for point a) to conclude that we have good estimates for the kernel of T Æ S .
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BMO
99
e) may be viewed as a consequence of d) since H1 is characterized through the Riesz transforms as H1 = ff 2 L1 = For j = 1; : : : ; d; Rj f 2 L1g (Fefferman and Stein [FEFS 72]). Another way is to check directly that the image of an atom is bounded in H1 : we have seen in Chapter 6 that the image of an atom a supported by B (0; 0) through a Calderon-Zygmund operator T is locally square-integrable and is Rbounded near in nity with a size 0(jxj d ; if T (1) = 0, we have, moreover, T (a) dx = 0 and this is enough to prove that T (a) 2 H1 . f) is a consequence of e) (through the duality between H1 and BMO). An important example of Calderon{Zygmund operator is the so-called Calderon commutator:
Theorem 10.3: (The Calderon commutator) ~ A 2 (L1 )d ) and be a Let A be a Lipschitzian function on IRd (i.e., r d smooth function on IR nf0g homogeneous with degree 1. Let MA be the pointwise multiplication operator by A (MA f = Af ) and T the Fourier multiplication operator of symbol (F (T f ) = f^). Then the commutator between T and MA, [T ; MA ] = T MA MAP T , is a Calderon{Zygmund operator and ~ Ak1 j j
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100
Real harmonic analysis Thus, we have d x x X lim h'( )jT (A ) AT ( )i = lim h'( )j (@i A)Ti ( )i R!1 R R!1 R i=1
which gives T (1) =
Pd
(@i A) 2 BMO. Similarly, we have T (1) 2 BMO.
i=1 Ti
3. The local Hardy space h1 and the local space bmo The Hardy space H1 and the space BMO are homogeneous spaces in the sense that their norms are homogeneous under dilations (for all positive , kf (x)kH = d kf kH and kf (x)kBMO = kf kBMO ); but BMO is not embedded into D0 , its elements being de ned only modulo constants. These 1
1
spaces admit a local counterpart, where dilations are not isometries but which are Banach spaces of distributions: the local Hardy space h1 and the local space bmo:
De nition 10.4: (Local atomic Hardy space) d 1 The local (atomic) Hardy space h1 (IR P ) is de ned as: f 2 H if and only if f may be decomposed into a series f = j2IN j ajR where Supp aj is contained d in P a ball B (xj ; Rj ), Rj 1, j aj j 1 (Rj ) , aj dx = 0 if Rj < 1 and jj j < 1. The function aj is called an atom P carried by the ball B (xj ; Rj ). P It is normed by kf kh1 = inf f j jj j = f = j j aj g where the minimum runs over all the atomic decompositions of f . The local space bmo is de ned as bmo = (h1 ) . Theorem 10.4: (Characterizations of bmo) Let b 2 D0 (IRd ). Then, the following assertions are equivalent: (A1) b 2 (h1 )0 : there exists a constant C such that for all ! 2 D(IRd ) we have jhbj!ij C k!kh1 . R (A2) b is a uniformly locally integrable function (supx0 2IRd jx x0j<1 jbj dx < R 1) and satis es: supjBj<1 jB1 j B jb mB bj dx < 1. (A3) For q R2 [1; 1), we have that b is uniformly locally Lq and satis es: supjBj<1 jB1 j B jb mB bjq dx < 1. (A4) Rb is a uniformly locally integrable function and for all 2 S (IRd ) such that dx = 0, de ning t (x) = t1d ( xt ), the measure jb t (x)j2 dtt dx is a local Carleson measure on (0; 1) IRd : Z Z 1 dt 8T > 0 sup j b t (x)j2 dx < 1 d R t 0
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1 d 0
Z Z
j @x@ etbj2 dt dx < 1 p j 0
BMO 0;1 and the operator f (A6) b 2 B1 d 2 L (IR ).
7! (b; f ) = Pj2IN Sj 1 f j b is bounded on
Proof: This is essentially the same proof as for Theorem 10.1.
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101
Part 2:
A general framework for shift-invariant estimates for the Navier{Stokes equations
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Chapter 11 Weak solutions equations
for
the
Navier{Stokes
In this chapter, we shall present the integral formulation for the Navier{ Stokes equations and discuss its equivalence with the di erential formulation. The results presented in this chapter are a slight extension of results contained in the paper of Furioli, Lemarie-Rieusset and Terraneo [FURLT 00].
1. The Leray projection operator and the Oseen kernel In this book, we consider the Navier{Stokes equations in d dimensions in the special setting of a viscous, homogeneous, incompressible uid which lls the entire space and is not submitted to external forces. Thus, the equations we consider are the system:
(11:1)
@t ~u = ~u
r~ :(~u u~) r~ p
r~ :~u = 0
which is a condensed writing for (11:10)
P
d @ :(u u ) @ p For 1 k d; @t uP k = uk k l=1 l l k d @u =0 l l l=1
The unknown quantities are the velocity ~u(x; t) of the uid element at time t and position x and the pressure p, whose action is to maintain the divergence of ~u to be 0 (this divergence free condition expresses the incompressibility of the
uid). Taking the divergence of (11.1), we obtain: (11:2)
~ r ~ :(~u: ~u) = p = r
d X d X k=1 l=1
Thus, we formally get the equations (11:3)
~ :(~u u~) @t ~u = ~u IPr r~ :~u = 0 105
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@k @l (uk ul )
106
Shift-invariant estimates
where IP is de ned as IPf~ = f~
(11:4)
r~ 1 (r~ :f~)
We shall study the Cauchy problem for equation (11.3) (looking for a solution on (0; T ) IRd with initial value ~u0 ), and transform (11.3) into the integral equation (11:5)
~u = et u~0
R t (t s) e IP ~ :(~u
0
r~ :u~0 = 0
r u~) ds
Our rst task is then to give a meaning to the operators IP (the Leray projection operator) and et IP (a convolution operator with the Oseen kernel). On many occasions, we use a quite direct de nition of IP. Let us recall that the Riesz transforms Rj are de ned by Rj = p@j , i.e. for f 2 L2 by F (Rj f ) = ijjj f^( ). Then IP is easily de ned on (L2 )d as IP = Id+R R where R P is the vector of the Riesz transformations: (IPf~)j = fj + 1kd Rj Rk fk . Since Rj Rk is a Calderon{Zygmund operator, IP may be de ned on many Banach spaces. ~ :). The However, we never quite use IP, but more often the operator IP(r use of one di erentiation allows one to extend the de nition of the operator to a larger class of Banach spaces. To give precise assertions, we use an auxiliary space:
De nition 11.1: (The space W L1 ) The space W L1 is de ned as the Banach space of Lebesgue measurable P functions ' on IRd so that the norm k'kW L1 = k2ZZd supx k2[0;1]d j'(x)j is nite. Lemma 11.1: (The Leray projection operator) For 1 k; l; m d, the operator 1 @k @l @m is a convolution operator with a distribution Tk;l;m which may be decomposed into Tk;l;m = k;l;m + @k @l m where k;l;m 2 W L1 and m 2 L1comp. ~ : is de ned on the space (L1uloc (IRd ))dd Remark: This lemma proves that IPr 1 where Luloc is the space of uniformly locally integrable functions on IRd , since ~ is L1comp L1uloc L1uloc and W L1 L1uloc L1 . In the next section, IPr de ned in a slightly more general setting. Proof: Let ! 2 D be equal to 1 in a neighborhood of 0. Let Tm be the distribution F ( 1) ( im ). The distribution Tk;l;m is equal to k;l;m + @k @l m
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The Oseen kernel
107
where k;l;m = @k @l (1 !)Tm and m = !Tm. We have j m (x)j C jxj d+1 , thus m 2 L1comp. We have j k;l;m (x)j C jxj d 1 , thus k;l;m 2 W L1 . We now turn to the Oseen kernel. The main property we use throughout ~ : is a matrix of convolution operators with this book is that the operator etIPr bounded integrable kernels.
Proposition 11.1: (The Oseen kernel) For 1 j; k d and t > 0, the operator Oj;k;t = 1 @j @k et is a convolution operator Oj;k;t f = Kj;k;t f , where the kernel Kj;k;t (x) satis es Kj;k;t (x) = x 1 td=2 Kj;k ( pt ) for a smooth function Kj;k such that for all 2 INd (1 + jxj)d+j j @ Kj;k 2 L1 (IRd ) 2 Proof: We have Kj;k = F ( 1) jjj2k e jj , thus, for all 2 INd, @ Kj;k 2 L1 (IRd ). For jxj 1, P we use the Littlewood{Paley decomposition and write Kj;k = (Id S0 )Kj;k + l<0 l Kj;k . We have (Id S0 )Kj;k 2 S , and we have l 2 l Kj;k = 2ld !j;k;l (2l x) with !^j;k;l ( ) = ( ) jjj2k e j2 j . The functions !j;k;l , l < 0, are a bounded set in S and, thus, we may write for every N 2 IN and uniformly in l that (1 + 2l jxj)N 2l(d+j j)j@ l Kj;k (x)j CN . This gives (for N > d + )
j@ S0 Kj;k (x)j C
X
2l jxj1
2l(d+j j) +
X
2l jxj>1
2l(d+j j N jxj N
C jxj
d j j :
This kernel was introduced by Oseen [OSE 27] in IR3 and used by Fabes, Riviere and Jones for describing strong solutions in Lp (IRd ) [FABJR 72].
2. Elimination of the pressure We focus on the invariance of equation (11.1) under spatial translations and dilations, in that we consider the problem on the whole space IRd. We begin by de ning what we call a \weak solution" for the Navier{Stokes equations:
De nition 11.2: (Weak solutions) A weak solution of the Navier{Stokes equations on (0; T ) IRd is a distribution vector eld ~u(t; x) in (D0 ((0; T ) IRd)d so that: a) ~u is locally square integrable on (0; T ) IRd ~ :~u = 0 b) r ~ :(~u u~) r ~p c) 9p 2 D0 ((0; T ) IRd) @t ~u = ~u r
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108
Shift-invariant estimates
The following invariance of the set of solutions is used throughout the book: i) shift-invariance: if ~u(t; x) is a weak solution of the Navier{Stokes equations on (0; T ) IRd, then ~u(t; x x0 ) is a weak solution on (0; T ) IRd ii) dilation invariance: for > 0, 1 ~u( t2 ; x ) is a solution on (0; 2 T ) IRd iii) delay invariance: if ~u(t; x) is a weak solution of the Navier{Stokes equations on (0; T ) IRd and if t0 2 (0; T ) then ~u(t + t0 ; x) is a weak solution of the Navier{Stokes equations on (0; T t0 ) IRd . In order to use the space invariance, we introduce a more restrictive class of solutions:
De nition 11.3: (Uniformly locally square integrable weak solutions) A weak solution of the Navier{Stokes equations on (0; T ) IRd is said to be uniformly if for all ' 2 D((0; T ) IRd ) we have R R locally square-integrable 2 supx0 2IRd j'(x x0 ; t)~u(t; x)j dx dt < 1. Equivalently, ~u is uniformly locally square-integrable if and only if for all R t0 < t1 2 (0; T ), the function Ut0 ;t1 (x) = ( tt01 j~u(t; x)j2 dt)1=2 belongs to the Morrey space L2uloc . We introduce some useful notations about uniform local integrability:
De nition 11.4: (Uniform local integrability) For 1 p 1, the Morrey space of uniformly locally Lp functions on IRd is the Banach space Lpuloc of Lebesgue measurable functions f on IRd so that R the norm kf kp;uloc = supx0 2IRd jx x0j<1 jf (x)jp dx 1=p is nite. For t0 < t1 , 1 p; q 1 the space Lpuloc;xLqt ((t0 ; t1 ) IRd) is the Banach space of Lebesgue measurable functions f on (t0 ; t1 ) IRd such that the norm R R supx0 2IRd jx x0j<1 f tt01 jf (t; x)jq dtgp=q dx 1=p is nite. We now explain the utility of introducing uniform weak solutions. In order to make sense of equation (11.1), we only need to assume that ~u is locally square-integrable on (0; T ) IRd . But we need an extra assumption on ~u to ~ :(~u ~u)). We use Lemma 11.1 to give assign meaning to (11.3) (namely, to IPr ~ sense to IPr:(~u ~u) with the help of the following obvious lemma:
Lemma 11.2: If ' belongs to W L1 (IRd ), then the (spatial) convolution R operator with ': f (t; x) 7! IRd f (t; y) '(x y) dy = f '(t; x) is well de ned and operates boundedly from L1uloc;xL1t ((t0 ; t1 ) IRd ) to L1uloc;xL1t ((t0 ; t1 ) IRd ). The same conclusion holds if ' belongs to L1comp(IRd ) (i.e., if ' is compactly supported and belongs to L1 ). We may now give a precise description of the relationships between equations (11.1) and (11.3):
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The Oseen kernel
109
Theorem 11.1: (Elimination of the pressure) i) If ~u is uniformly locally square-integrable on (0; T ) IRd (in the sense of ~ :(~u ~u) is well de ned in (D0 ((0; T ) IRd ))d , and there De nition 3), then IPr ~ :(~u ~u) = r ~ :(~u ~u)+ r ~ P. exists a distribution P 2 D0 ((0; T ) IRd ) so that IPr Thus, if ~u is a solution for (11.3), then it is also a solution for (11.1). ii) Conversely, if ~u is a uniform weak solution for (11.1), and if ~u vanishes at in nity in the sense that for all t0 < t1 2 (0; T ) we have Z
Z
1 t1 j~uj2 dx dt = 0; lim sup d R!1 x0 2IRd R t0 jx x0 j
Proof: We begin by proving that a solution ~u for (11.1) in D0 , which is uniformly locally square-integrable on (0; T ) IRd, is solution for (11.1) in a more precise space of distributions, namely in the space T 0 of spatially tempered distributions (de ned in Chapter 5). Recall that the space T 0 is the space of distributions ! on (0; T ) IRd so that for all [a; b] (0; T ) there exist C 0 and N 2 IN such that for all ' 2 D((0; T ) IRd with Supp' [a; b] IRd, we have the inequality jh!j'ij C Pj jN Pj jN PpN kx @x@ @t@ pp f (t; x)k1 . T 0 is the dual space of the space T ((0; T ) IRd) of smooth functions compactly supported in time and rapidly decaying in space. T is the inductive limit of the Frechet spaces Tn = ff 2 T = Suppf [1=n; T 1=n] IRd g, where the space Tn is equipped with the semi-norms: @ @p sup jx p f (t; x)j; 2 INd; 2 INd; p 2 IN: @x @t 1=n
Since the components of ~u are in L2uloc;xL2t ((t0 ; t1 ) IRd ) for all 0 < t0 < t1 < T , we easily nd that ~u belongs to (T 0 )d ; hence, so do @t~u and ~u. Moreover, the components of ~u ~u are in L1uloc;xL1t ((t0 ; t1 ) IRd ) for all 0 < ~ :~u ~u belongs to (T 0 )d . t0 < t1 < T ; thus ~u ~u belongs to (T 0 )dd and r ~ p belongs Since all the other terms in (11.1) belong to (T 0 )d , we nd that r 0 d to (T ) . We shall now de ne the Leray projection operator IP. On T 0 , we may use the Littlewood{Paley decomposition with respect to the space variable x. This wil be very useful in de ning IP. High frequencies do not present a problem, since IP(Id S0 ) operates boundedly on T 0 . In order to deal with low frequencies, we use the following properties: for f 2 L1uloc (IRd ) and g 2 W L1 , fg 2 L1 and kfgk1 C kf kL1uloc kgkW L1 ;
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Shift-invariant estimates
more precisely, W L1 is the dual space of E1 , the closure of D in L1uloc and,
thus, W L1 is a shift-invariant Banach space of distributions; we thus may conclude that for f 2 L1 (IRd ) and g 2 W L1 , f g 2 W L1 and kf gkW L1 C kf k1kgkW L1 ; If '(x) belongs to W L1 (IRd ) and f (t; x)R belongs to L1uloc;xL1t ((t0 ; t1 ) IRd ), then the (spatial) convolution f '(t; x) = IRd f (t; y) '(x y) dy is well de ned d 1 1 and belongs to L1 x Lt ((t0 ; t1 ) IR ) and kf 'kL1 x L1t C kf kL1uloc;x L1t k'kW L . Pd Now, let us consider a vector eld ~g with gi = k=1 @k fi;k with fi;k 2 \0
kSj ~gkL1x Lt ((t ;t )IRd ) C 2j 1
0
1
X
kfi;k kLuloc;xLt ((t ;t )IRd ) kkW L1 1
i;k
1
0
1
X
k
k@k k1 ;
P we nd that we have S0~g = j<0 j ~gPin T 0 . Since IPj operates on (T 0 )d , we shall try to de ne IPS0~g as the sum j<0 j IP~g. Since we have (j IP~g)i = P jd j k 2 i;k (2 x) gk with i;k 2 S , we nd that
kj IP~gkL1x Lt ((t ;t )IRd ) C 2j 1
0
1
X
kfi;k kLuloc;xLt ((t ;t )IRd ) 1
i;k
1
0
1
P and thus the series j<0 j IP~g is convergent in T 0 ((0; T ) IRd ). In particular, when ~u belongs to \0
t0
jqj (t; x)j dt C 2j jxjk~ukLuloc;x Lt ((t ;t )IRd ) ; 2
2
0
1
P ~ :(~u
and this is enough to grant that j<0 qj is convergent in T 0 . Thus, IPr P P ~ :(~u ~u) + r ~ p with p = j0 pj + j<0 qj and ~u is a solution of (11.1). ~u) = r We now prove point ii) of Theorem 11.1. We start from the property Z
Z
1 t1 j~uj2 dx dt = 0; lim sup d R!1 x0 2IRd R t0 jx x0 j
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The Oseen kernel
111
1 R t1 R and we estimate Sj ~u by kSj ~ukL1 uj2 dx dt 1=2, x L1t C supx0 2IRd Rd t0 jx x0 j
@t~u = limj! 1 (Id Sj )@t ~u = limj! 1 IP(Id Sj )@t~u ~ :(~u ~u) r ~ p) = limj! 1 IP(Id Sj )(~u r ~ ~p = ~u IPr:(~u ~u) limj! 1 IP(Id Sj )r ~ p = 0. Of Thus, the proof of point ii) is reduced to proving that IP(Id Sj )r 0 ~ course, IP(Id Sj )r is equal to 0 on T , but we do not know whether p belongs ~ pj~!i = hr ~ pjIP(Id Sj )~!i. We to T 0 . If ~! 2 (T )d , we have hIP(Id Sj )r d 0 ~ p 2 (T 0 )d thus have to prove that, when p 2 D ((0; T ) IR ) is such that r d ~ ~ ~ ~ ~ and 2 (T ) is such that r: = 0, then hrpj i = 0. We are going to prove that there exists a matrix = ('k;l )1k;ld 2 T dd with 'k;l = 'l;k so that ~=r ~ :. Then, it will be enough to approximate by a sequence n 2 Ddd (with t n = n) to get: ~ ~ hr~ pj ~ i = nlim !1hrpjr:n i =
~ :(r ~ :n )i = 0: lim hpjr
n!1
The construction of is simple. We rst notice that R a function f in T may be written as f = @1 g with g 2 T if and only if f (t; x1 ; : : : ; xd ) dx1 = 0 for all t 2 (0; T ) and all (x2 ; : : : ; xd ) 2 IRd 1: indeed, if is a univariate cuto function equal to 1 on a neighbourhood of 0, we de ne g as F g(t; ) = R i(1 ) 01 @@1 F f (t; 1 ; 2 ; : : : ; d ) d + 1 i(11 ) F f (t; ). We thus write ~ = ~ + ~ , where 0
1
0 1 1 0 R B C B 2C B 2 (t; y1 ; x2 : : : ; xd ) dy1 ! (x1 ) C C ~ = B B C=@ A ::: @:::A R ( t; y ; x : : : ; x ) dy ! ( x ) d 1 2 d 1 1 d R ~ :~ = 0; hence, r ~ : ~ = 0. with !(x1 ) dx1 = 1, and ~ = ~ ~ . R We have r We know that for 2 k d we have k (t; y1 ; x2 ; : : : ; xd ) dy1 = 0; hence, ~ : ~ = 0 to get @1 1 = therePexists '1;k so that k = @P 1 '1;k . Now, we use r @1 k2 @k '1;k , hence 1 = . Then, we deal with ~ : we have k2 @k '1;k0 1
2 C d 1 ))d 1 ~ = !(x1 )~ with 1 = 0, ~Æ(t; x2 ; : : : ; xd ) = B @ : : : A 2 (T ((0; T ) IR
d ~ ~ and r:Æ = 0; we thus may iterate the construction and nally get the matrix .
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Shift-invariant estimates
3. Di erential formulation and integral formulation for the Navier{Stokes equations We now turn to the equivalence between equations (11.3) and (11.5):
Theorem 11.2: (The equivalence theorem) Let ~u 2 \t1
~ :(~u ~u) @t ~u = ~u IPr ~r:~u = 0
(A2) ~u is a solution of the integral Navier{Stokes equations
t 9~u0 2 (S 0 (IRd ))d ~u = e u~0
R t (t s) e IP ~ :(~u
0
r~ :u~0 = 0
r u~) ds
We begin with an easy and useful lemma:
Lemma 11.3: R Let T 2 (0; 1) and w 2 L1uloc;xL1t ((0; T ) IRd ). Then, 0t e(t s) w(s; :)ds de nes a function W (t; x) 2 L1uloc;xL1t ((0; T ) IRd ). Moreover: i) @t W = W + w ii) for all ' 2 D(IRd ) with Supp ' [ 1; 1]d, we have that t 7! ' W (t; :) is continuous from [0; T ] to L1(IRd) and satis es sup0
0
je(t s) jds dx
IRd
Z tZ
0
jw(t; x)j ds dx C kwkL1uloc;x L1t ((0;T )IRd ) : jx x0 j100pd p
Now, p we estimate the contribution of . When jx x0 j d and jx yj 100 d, we may easily check that the kernel K (t s; x y) of e(t s) satis es
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The Oseen kernel 113 pt s R R 0 K (t s; x y) C jx0 yjd+1 . This gives 0t x0 x2[ 1;1]d je(t s) jds dx pRR p )j C t 0t jx0 yj99pd jxjw0 (s;y yjd+1 ds dyR C tkw(x; t)kL1uloc;x L1t ((0;T )IRd ) . We obviously see that t 7! 0t e(t s) ds is continuous from (0; T ) to R L1 (B (x0 ; 1)) and that limt!0 k 0t e(t s) dskL1 (B(x0 ;1)) = 0. The estimate R limt!0 k 0t e(t s) dskL1(B(x0 ;1)) = 0 is easy as well, since we have limT !0 kw(x; t)kL1uloc;x L1t ((0;T )IRd ) = 0. For the continuity in L1-norm of t 7! R A(t) = 0t e(t s) ds, it is enough to notice that jje (e Id)jj1 C ; thus, writing for 0 < t1 < t2 < T = (t1 + t2)=2 and p= jt1 t2p j=2, we have kA(t1 ) R + R A(t2 )kL1 (dx) C p k (s)kL1 (dx) ds + p p 0 k (s)kL1(dx) ds , hence limt2 !t1 kA(t1 ) A(t2 )kL1 (dx) = 0. The simplest way to compute @t W is to notice that for T 2 L1uloc;xL1t we may easily check that @ (RetT ) = et (@t T + T ) (just use the partial Fourier transform) and that @t ( 0t T (s) ds) = T ; then, we write, for the Littlewood{ R Paley approximation of W , that Sj W = et 0t (Sj e s )w ds; hence, Sj @t W = @t Sj W = Sj W + et (Sj e t)w = Sj (W + w); letting j go to +1 gives then the required result.
Proof of Theorem 11.2 : Using Lemma 11.3, we nd that, for ~u0 2 (S 0 )d R t ~ :(~u ~u) ds is and ~u 2 (L1uloc;x L1t )d , we nd that F~ (~u) = et~u0 0 e(t s) IPr ~ :(~u ~u). Thus, if ~u = F~ (~u), then ~u is a a solution for @t F~ (~u) = F~ (~u) IPr solution for the Navier{Stokes equations. ~ :(~u ~u), Conversely, if ~u 2 (L1uloc;x L1t ((0; T ) IRd ))d and @t~u = ~u IPr then, choosing ! 2 D(IR) equal to 1 on [0; T=4] and to 0 on [T=2; +1) and de ning ~v = !(t)(Id ) 2~u, we nd that @t~v 2 (L1uloc;xL1t ((0; T ) IRd ))d ; R hence, ~v(t) = tT @t~v(s) ds 2 (C ([0; T ]; L1uloc))d ; thus we nd that the mapping t 7! (Id ) 2 ~u is continuous from [0; T ) to (L1uloc )d and that t 7! ~u is weakly continuous from [0; T ) to (S 0 )d . Thus, ~uR0 = limt!0 ~u is well de ned. ~ :(~u ~u) ds; we have @t (~u We again introduce F~ (~u) = et~u0 0t e(t s)IPr F~ (~u)) = (~u F~ (~u)) and limt!0 ~u F~ (~u) = 0. Using the Littlewood{Paley ap proximation Sj , we write: @t (e t Sj )(~u F~ (~u)) = 0 and limt!0 (e tSj )(~u F~ (~u)) = 0; hence, (e t Sj )(~u F~ (~u)) = 0; we get Sj (~u F~ (~u)) = 0 and nally ~u = F~ (~u)). Remark: We proved that t 7! ~u is continuous from [0; T ) to ((Id )2 L1uloc )d . It means that the distribution ~u, which is a locally integrable vector function (0; T ) IRd and hence may be modi ed arbitrarily on subsets of measure 0, may be de ned by a function such that the mapping t 7! ~u is continuous. In this book, we always assume that ~u denotes this representative.
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Chapter 12 Divergence-free vector wavelets
In this chapter, we consider divergence-free vector wavelets as a tool for approximating divergence free vector elds vanishing at in nity by compactly supported divergence-free vector elds.
1. A short survey in divergence-free vector wavelets Wavelets have been advocated by many authors as a tool for investigating the structure of turbulent solutions of the Navier-Stokes equations (Farge, Kevlahan, Perrier and Schneider [FARKPS 97], Frick and Zimin [FRIZ 93]). According to Farge, applications of the wavelet transform to the theory of turbulence should be developed in three directions: analysis, ltering, and numerical modeling. The wavelet analysis relies on the localisation of wavelets both in space and frequency; thus, wavelets may be a useful tool for relating the dynamics of coherent structures in physical space to the redistribution of energy among the Fourier modes. Adaptive nonlinear ltering of the wavelet coeÆcients appears to help in isolating dynamically active structures supported by thin sets and exploring their interaction. Then a wavelet-based numerical modeling could be developed at a reasonable computational cost. Special attention has been given to divergence-free vector wavelet bases, ~ :f~ = 0g which are derived through i.e., bases for H = ff~ 2 (L2 (IRd ))d = r dyadic dilations and translations from a nite number of basic functions: the basis is of the form (2jd=2 ~ (2j x k))1 (d 1)(2d 1);j2ZZ;k2ZZd . Battle and Lemarie-Rieusset noticed independently that in the case of d = 2, the construction of an Hilbertian basis for H is very easy: if R1 , R2 are the Riesz transforms p@1 , p@2 and if ( ;j;k )13;j2ZZ;k2ZZd is a Hilbertian basis of L2(IR2 ), then we have a basis ( ~;j;k )13;j2ZZ;k2ZZd of H with ~ = (R2 ; R1 ). We may choose ~ with good regularity and localization properties: If we start from a Meyer{Lemarie wavelet basis ( 2 S (IR2 ) and ^ ( ) = 0 on a neighbourhood of 0 [LEMM 86]), then ~ belongs to the Schwartz class (S (IR2 ))2 . p If we start from a Battle basis = where the wavelets and their rst derivatives have exponential decay at in nity (Hibertian basis for 115
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Shift-invariant estimates RP
the homogeneous Sobolev scalar product j @j u@j v dx [BAT 87]), then the divergence-free vector wavelets ~ have exponential decay. However, there does not exist a Hilbertian wavelet basis for H with compactly supported wavelets ~ (Lemarie-Rieusset [LEM 94]). Moreover, the projection operator on the low-frequency vector wavelets is ~ j f~ as Q ~ j f~ = P Pk2ZZd hf~j ~;j;k i ~;j;k , we always ill-localized: if we de ne Q P ~ j f~; the kernel of IP P~0 is rapidly ~ j f~ = P~0 f~ + Pj0 Q have IPf~ = j2ZZ Q decaying at in nity (provided that the vector wavelets ~ have rapid decay) while the kernel of IP is decaying with order O(jx yj d). Battle and Federbush have constructed Hilbertian bases of divergence-free vector wavelets in higher dimensions [BATF 95]. The construction is much more intricate than in dimension d = 2. Federbush [FED 93] used those bases to construct mild solutions for the Navier{Stokes equations on IR3 through a Galerkin approach. (See also the book by Cannone [CAN 95]). This approach was criticized in the book by Meyer [MEY 99] who claimed that Cannone's approach by Littlewood{Paley estimates [CAN 95] and even a direct approach using spatial estimates as in the papers of Kato on Ld solutions [KAT 84] or on solutions in the Morrey{Campanato spaces [KAT 92] could give the same results (or even more precise results) at lesser expense. Meyer also proved that the interaction of two vector wavelets was not well localized in frequency and thus required many wavelet coeÆcients to be restored. Lemarie-Rieusset proposed another construction [LEM 92] that is described in the next section. The idea was to get rid of orthogonality to allow a direct and local representation of the approximation on the low frequencies. We start from a bi-orthogonal multiresolution analysis V~j of the whole space (L2 (IRd )d so that the projection operators P~j on Vj are local. Moreover, we construct ~ j \ H , where V~j so that P~j (H ) H . Then, we de ne a wavelet basis of W 2 d ~ ~ ~ ~ Wj = (Pj+1 Pj )((L ) ). Thus, the approximation of f 2 H by P~j f~ may be described both as an external approximation (using a well-localized basis (2j=2 '~ l (2j x k))1ld;k2ZZd of V~j ) or as an internal approximation (using the well-localized divergence-free vector wavelets ( ~;l;k )1(d 1)(2d 1);l<j;k2ZZd ). Such representations have been used by Urban [URB 95] to numerically approximate the Stokes problem.
2. Bi-orthogonal bases We now introduce the divergence-free vector basis of compactly supported wavelets constructed in [LEM 92]. Let '; ' be compactly supported univariate bi-orthogonal scaling functions so that, for instance, ' and ' are C 2 . Let '~ and '~ be the scaling functions de ned by: d '(x) = '~(x) '~(x 1) dx
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Divergence-free wavelets
'~ (x) =
Z x+1
x
117
' (t) dt
If Vj ; Vj ; V~j and V~j are the multiresolution analyses generated by '; ' ; '~ and '~ , if Pj is the projection operator onto Vj in the direction of (Vj )? and if P~j is the projection operator onto V~j in direction of (V~j )? , then we have:
d d (P f ) = P~j ( f ) dx j dx where the formula is valid for all locally integrable f . We then de ne the vector projection operator P~j = (Pj[1] ; Pj[2]; : : : ; Pj[d]) with Pj[k] = PjfÆk;1 g P~jfÆk;2 g : : : P~jfÆk;d g with Pjf0g = P~j and Pjf1g = Pj so that for all f 2 (L1loc )d we have:
r~ :P~j (f~) = P~j P~j : : : P~j (r~ :f~): Thus, a divergence-free vector eld is projected onto a vector eld which is still divergence free. The main property of this approximation process for vector elds is that the term (P~j+1 P~j )f~ may be locally reorganized into divergence free \atoms" when f~ is itself a divergence free vector eld. In the \classical" wavelet theory, we would decompose (P~j+1 P~j )f~ into (P~j+1
P~j )f~ =
X
X
X
1ld 1 2d 1 k2ZZd
hfl j
;l;j;k i ~ ;l;j;k
(2j x k) and ~ ;l;j;k = 2dj=2 ~ ;l (2j x k) and where where ;l;j;k = 2dj=2 ;l d are derived from the d(2 1) wavelets ~ ;l and their d(2d 1) dual wavelets ;l the dual wavelets ; (associated to '; ' ) and ~; ~ (associated to '; ~ '~ ) through the following formulas where we write (~el )1ld for the canonical basis P of IRd (so that f~ = l fl ~el ):
' [l;n]
= ' for l = n and = 0; = '~ for l 6= n and = 0 = = for l = n and = 1; = ~ for l 6= n and = 1
= = ' for l = n and = 0; = '~ for l 6= n and = 0 '[l;n ] = for l = n and = 1; = ~ for l 6= n and = 1 d ~ ;l = ' [l;11] ' [l;22] : : : ' [l;d ] ~el
= ' 1 ' 2 : : : ' d [l;1] [l;2] [l;d]
;l
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for = for =
d X
n=1 d X
n=1
n 2n
n 2n
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Shift-invariant estimates
In the theory of divergence-free vector wavelets, we decompose (P~j+1 P~j )f~ in 2d 1 groups (following the values of ):
~ ;j f~ = Q
X
X
1ld k2ZZ
3
hfl j
;l;j;k i ~ ;l;j;k
~ :Q~ ;j f~ = Q~ ;j (r ~ :f~). Finally, we notice which satisfy a formula of the kind r that, since 6= 0, at least one of the associated n 's is equal to 1; we call d to n the smallest index so that n = 1 and we use the fact that ~ = dx ~ ;l = get that for l 6= n we have ~ ;l = @n ;l ~el where ;l is de ned as 1 2 n d k k [l;1] [l;2] : : : [l;d] with [l;k] = '[l;k] for k 6= n and [l;n] = . We then de ne ~ ;l = @n ;l ~el @l ;l ~en and ~ ;l;j;k = 2dj=2 ~ ;l (2j x k). We may write: X X i ~ ;l;j;k + Rn (f~) ~en ~ ;j f~ = Q hfl j ;l;j;k d l= 6 n k2ZZ ~ :f~ = 0, we get that @n Rn (f~) = 0 with Rn (f~) vanishing When f~ 2 (E1 )d and r at in nity, and thus Rn (f~) = 0. = We may now de ne three series of operators on (L2 (IRd )d , de ning ~ ;l jd=2 ~ (2j x k ), ~ ~ ~ ;l (x)~el , ~ ;l = IP ;l and ;l;j;k (x) = 2 ;l ;l;j;k (x) = jd= 2 j 2 ~ ;l (2 x k): 8 ~ ~ > < Qj f > :
R~ j f~ S~j f~
= = =
P
P
P
d kP 2ZZd hf~j ~ ;l;j;k i ~ ;l;j;k P 1ld P1 2 1 ~ ~ 1 2d 1 1ld;l6=n( ) k2ZZd hf j ;l;j;k i ~ ;l;j;k P P P ~ 1 2d 1 1ld;l6=n( ) k2ZZd hf j~ ;l;j;k i ~ ;l;j;k
Then, we have i) The family ( ~ ;l;j;k )1 2d 1;1ld;j2ZZ;k2ZZd is a bi-orthogonal wavelet basis )1 2d 1;1ld;j2ZZ;k2ZZd : of (L2 (IRd ))d with dual basis the wavelets ( ~ ;l;j;k P have ~ j = Id(L2 )d . The wavelets ~ ;l and ~ ;l Q~ j = P~j+1 P~j and j2ZZ Q compact supports and so do the associated scaling functions ' ;l~el and ' ;l~el . ii) (~ ;l;j;k )1 2d 1;1ld;l=6 n( );j2ZZ;k2ZZd is a bi-orthogonal wavelet basis of )1 2d 1;1ld;l=6 n( );j2ZZ;k2ZZd: S~j f~ = H with dual basis the wavelets (~ ;l;j;k P Q~ j IPf~ and j2ZZ S~j = IP. The wavelets ~ ;l have compact supports but the have only polynomial decay at in nity: for some associated dual wavelets ~ ;l (x)j = O(jxj N ). N 2 IN, j~ ;l ~j = iii) The use of R~ j is intermediate between i) and ii): on H , we have R~ jP= Q ~Sj , but on H ? the three operators take di erent values. The operator j2ZZ R~ j is a Calderon{Zygmund operator with range H and it is a bounded oblique projection operator from (L2 )d onto H . The main features of R~ j is that the have compact supports and that we may use the related wavelets ~ ;l and ~ ;l
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Divergence-free wavelets
119
~ j )j2ZZ for compactly supported scaling functions associated with the analysis (Q P ~ ~ expressing the low-frequency approximate j<0 Rj f of a divergence-free vector eld f~. We shall now prove a useful lemma on approximation of divergence-free vector elds by square-integrable divergence-free vector elds:
Proposition 12.1: (The approximation lemma) Let p 2 [1; 1) and let Ep be the space ofR locally Lp functions which vanish at in nity (de ned by kf kEp = supx2IRd ( jx yj<1 jf (y)jp dy)1=p < 1 and R ~ :~u = 0. Then limx!1 jx yj<1 jf (y)jp dy = 0). Let ~u 2 (L2 + Ep )d such that r ~ :~v = r ~ :w~ = 0 so that for all > 0, there exists ~v 2 (L2 )d ; w~ 2 (Ep )d ; r ~u = ~v + w~ et kw~ kEp . For p = 1, the same result holds when de ning E1 as E1 = C0 . Proof: Since D is dense in Ep , we may decompose each component ui of ~u in Vi + Wi where the Ep norm of Wi is small. The diÆculty lies in the requirement that ~v and w~ be divergence-free. It is not enough to project V~ and ~ on divergence-free vector elds with the help of the projection operator IP, W ~ :~u)) is not bounded on (Ep )d . Instead of ~ ( 1 r because IP (de ned by IP~u = ~u r IP, we use our oblique projection operators R~ j onto the divergence-free vector wavelet basis. We rst consider the projection operators P~j on (Ep )d . We have, for any ~v 2 (Ep )d , limj!+1 kP~j ~v ~vkEp = 0 and limj! 1 kP~j ~vkEp = 0: indeed, the operators P~j are equicontinuous on (Ep )d , while for ~v 2 (D)d we obviously have that limj!+1 kP~j ~v ~vk1 = 0 and limj! 1 kP~j ~vk1 = 0. When ~v is divergence free, we nd that for N large enough ~zN = P~ N ~v + ~v P~N +1~v is divergence free and has a norm in (Ep )d less than =2. Thus, we just have to split P~N +1~v P~ N ~v =
X
X
X
X
N j N 1 2d 1 l6=n( ) k2ZZd
h~vj
;l;j;k i ~ ;l;j;k :
P i ~ ;l;j;k . When We consider the operators A~ j; ;l;K (~v ) = k2ZZd;jkj>K h~vj ;l;j;k j is xed, the operators A~ j; ;l;K are equicontinuous on (Ep )d and we have for every ~v 2 (Ep )d limK !+1 kA~ j; ;l;K ~vkEp = 0. Since we consider a nite numberPof j , we P may nd for every ~v a K 2 IN such that Z~ N;K = P d N j N 1 2d 1 l= 6 n( ) A~ j; ;l;K (~v) has a norm in (Ep ) smaller than =2. We then de ne w~ = ~zN + Z~ N;K and ~v = ~v w~ . The proof gives a more precise result: we may choose ~v to have a compact support.
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Shift-invariant estimates
3. The div-curl theorem The following theorem proved by Coifman, Lions, Meyer and Semmes [COILMS 92] will be very useful :
Theorem 12.1: (The div-curl theorem) ~ :~u = 0 and let ~v = r ~ V with V 2 H_ 1 (IRd ). Let ~u 2 (L2 (IRd ))d with r 1 Then ~u:~v belongs to the Hardy space H and k~u:~vkH1 C k~uk2k~vk2 where the constant C does depend only on d. Proof: We adapt the proof given by Dobyinsky [DOB 92]. We use the multiresolution analysis P~j described in Section 2 associated with the projection operd Æ P = P~ Æ d , and we use as well the projection ators Pj and P~j , which satisfy dx j j dx ~ j = P~j+1 P~j and Qj = Pj+1 Pj . operator Pj = P~j : : : P~j . We de ne Q P P ~ j ~u:r ~ Ql V = A(~u; ~v) + B (~u; ~v) + C (~u; ~v) We then write ~u:~v = j2ZZ l2ZZ Q P ~ j ~u:r ~ Pj V , C (~u; ~v) = ~ j ~u:r ~ Qj V , B (~u; ~v) = Pj2ZZ Q with A(~u; ~v) = j2ZZ Q P ~ ~ j 2ZZ Pj ~u:rQj V . ~ j by R~ j (which is harmless on divergence-free vector elds). We replace Q P P We write Qj V = 12d 1 k2ZZd hV j2jd=2 (2j x k)i2jd=2 (2j x k); for each wavelet , we may write, for some index l() and some compactly ~ Qj V = T~j ~v with supported function , = @l() so that we get r ~Tj f~ = P12d 1 Pk2ZZd hfl() j2jd=2 (2j x k)i2jd=2 (r ~ )(2j x k). We thus P P replace A, B and C by A~(~u; ~v) = j2ZZ R~ j ~u:T~j ~v, B~ (~u; ~v) = j2ZZ R~ j ~u:P~j~v, P C~ (~u; ~v) = j2ZZ P~j ~u:T~j ~v. We rst check that B~ and C~ map (L2 )d (L2 )d to H1 : we use the duality of H 1 and CMO (the closure of C0 in BMO) (see Coifman and Weiss [COIW 77]Pand Bourdaud [BOU 01]) and try to prove that the operators B(f;~ g) = j2ZZ R~ j (gP~j f~) and C (f;~ g) = Pj2ZZ T~j(gP~j f~) map (L2 )d BMO to (L2 )d . First we notice that R~ j Æ P~j = 0 and T~j Æ P~j = 0, so that we may add any constant to g and nd the same result (recall that BMO is de ned modulo the constants). Moreover, we nd that B(:; g) and C (:; g) are matrices of singular integrals operators (Bi;j (:; g))1i;jd and (Ci;j (:; g))1i;jd : expand P~j , P~j using scaling functions and R~ j and T~j with help of wavelets. There are only a few terms that interact, because of the localization of the supports. To estimate the size of the kernels of B and C and the size of their derivatives, it must be shown that sup jh2jd=2~ ;m(2j x k1 )jg2jd=2 0;l (2j x k2 )~el ij < 1 d j 2 ZZ; k0 2 ZZ k1 2 ZZd ; 1 l d : 1 2d 1; m 6= n( ) 8 <
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Divergence-free wavelets
121
and sup jh2jd=2 r~ (2j x k1 )jg2jd=2 0;l (2j x k2 )~el ij < 1: d j 2 ZZ; k0 2 ZZ k1 2 ZZd ; 1 l d : 1 2d 1 8 <
Proposition 10.2 allows us to conclude that Bi;j (:; g) is a singular integral operator with Bi;j (:; g) 2 W BP and Bi;j (:; g) (1) = 0 and similarly, that Ci;j (:; g) is a singular integral operator with Ci;j (:; g) 2 W BP and Ci;j (:; g) (1) = 0. MoreP over, B(:; g)(1~el) = ( j2ZZ R~ j )g~el and we may again apply Proposition 10.2 P to conclude that the matrix of Calderon{Zygmund operators j2ZZ R~ j maps BMOd to BMOd ; this gives that for all i; j in f1; : : : ; dg we have Bi;j (:; g)(1) 2 BMO; hence, that Bi;j (:; g) is bounded on L2. ThePsame proof works for Ci;j , using the matrix of Calderon{Zygmund operators j2ZZ T~j. Estimating Ij;k0 ;k1 ;l; ;m = h2jd=2~ ;m (2j x k1 )jg2jd=2 0;l (2j x k2 )~el i and ~ (2j x k1 )jg2jd=2 0;l (2j x k2 )~el i is easy. Indeed, on Jj;k0 ;k1 ;l; = h2jd=2 r let M be large enough so that the supports of the scaling functions 0;l , and 0;l are contained in B (0; M ) for every l 2 f1; : : : ; dg; then, since we have h~ ;m (2j x k1 )j 0;l (2j x k2 )~el i = 0 and hr~ (2j x k1 )j 0;l (2j x k2 )~el i = 0, we may write (writing Bj;k2 for the ball Bj;k2 = B ( k22j ; M 2j )):
Ij;k0 ;k1 ;l; ;m = and
Jj;k0 ;k1 ;l; =
Z
Bj;k2 Z
Bj;k2
2jd (g(x) mBj;k2 g) 0;l (2j x k2 )~ ;m (2j x k1 ):~el dx
2jd (g(x) mBj;k2 g) 0;l (2j x k2 )@l (2j x k1 ) dx
which gives the required estimates jIj;k0 ;k1 ;l; ;mj C kgkBMO k 0;l k1 k~ ;mk1 and jJj;k0 ;k1 ;l;j C kgkBMO k 0;l k1 k@l kP 1. We still have to deal with A~(~u; ~v) = j2ZZ R~ j ~u:T~j ~v. But this remainder ~ j ~u and T~j ~v in their wavelet term obviously belongs to H1 , since expanding R bases yields directly an atomic decomposition: we have indeed
R~ j ~u = with
P
X
X
1 2d 1 1md;m6=n( ) k2ZZd
1 2d 1
P
1md;m6=n( )
T~j ~v =
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X
X
P
X
12d 1 k2ZZd
j;k; ;m 2jd=2 (~ ;m )(2j x k)
k2ZZd jj;k; ;m j
2
C k~uk22 and
~ )(2j x k) j;k; 2jd=2 (r
122
Shift-invariant estimates P
P
with 12d 1 k2ZZd jj;k; j2 C k~vk22 ; now, we recall that the pointwise ~ (2j x k2 ) are identically equal to O for jk1 k2 j products ~ ;m (2j x k1 ):r K , where K is a constant depending only on the supports of and of ~ ;m and we obtain that
A~(~u; ~v) =
X
(j;k1 ;k2 ; ;m;)2I
~ )(2j x k2 ) j;k1 ; ;mj;k2 ; 2jd~ ;m(2j x k1 ):(r
with I = f(j; k1 ; k2 ; ; m; ) = j 2 ZZ; 1 2d 1; 1 m d; m 6= n( ); k1 2 ZP Zd ; 1 2d 1; k2 2 ZZd ; jk2 k1 j < K g. We may now conclude, since jd j ~ j (j;k1 ;k2 ; ;m;)2I jj;k1 ; ;m j;k2 ; j < 1 and 2 ~ ;m (2 x k1 ):(r )(2 x k2 ) 1 is a bounded atom in H .
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Chapter 13 The molli ed Navier{Stokes equations In this chapter, we present a classical tool of Leray [LER 34] to prove the existence of weak solutions: in order to be able to use the the Picard contraction algorithm, we rst mollify the equations, replacing theRnonlinearity ~u ~u into the smoother term (~u !) ~u with ! 2 D(IRd ), ! 0, IRd ! dx = 1 and ! = 1d !( x ); then we try to get uniform estimates on the solutions with respect to the perturbation and to use a limiting process to go back to the initial equations. 1. The molli ed equations
In order to introduce R the molli ed equations, we choose a function ! 2 D(IRd) with ! 0 and IRd ! dx = 1; then the molli ed equations are given for > 0 by
(13:1)
8 u < @t ~ :
= ~u r~ :((~u !) ~u) r~ p r~ :~u = 0 ~u(0; :) = ~u0
where ! = 1d !( x ). For ~u0 2 (B1N;1)d for some N > 0, we seek a solution ~u 2 (L1t B1N;1)d. We replace the term r~ :((~u !) ~u) + r~ p by the term IPr~ :((~u !) ~u). We have no problem de ning IP in that case, since (~u !) ~u 2 (L1t B1N;1)dd and since IPr~ : maps (B1N;1)dd to (B1N 1;1 )d. Thus, we shall solve the equation 8 ~ :((~u ! ) ~u) u = ~u IPr < @t ~ (13:2) r~ :~u = 0 : ~u(0; :) = ~u0 which is a special case of (13.1). Equation (13.2) is equivalent to the xed-point problem ~u = et~u0 B(~u; ~u) where the bilinear operator B is given by B (~u; ~v) =
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Z t
0
~ :((~u ! ) ~v) ds: e(t s)IPr
123
124 Shift-invariant estimates It is very easy to construct a local solution to (13.2): Theorem 13.1: (Solution of the molli ed equations) ~ :~u0 = 0. Then there exists i) Let N > 0 and ~u0 2 (B1N;1 )d with r a maximal T > 0 so that the equations (13.2) have a solution ~u with ~u 2 \T
Let E be a Banach space and let B be a bounded bilinear transform from E E to E : kB (e; f )kE CB kekE kf kE : Then, if 0 < Æ < 4C1B and if e0 2 E is such that ke0 kE Æ , the equation e = e0 B (e; e) has a solution with kekE 2Æ. This solution is unique in the ball B (0; 2Æ ). Moreover, the solution continuously depends on e0 : if kf0 kE Æ , f = f0 B (f; f ) and kf kE 2Æ, then ke f kE 1 41CB Æ ke0 f0 kE .
We construct e by iteration. We start from e0 and we de ne inductively a sequence en by en+1 = e0 B(en; en). By induction, we show that kenkE 2Æ: indeed, we have ken+1kE ke0kE + CB kenk2E Æ + 4CB Æ2 2Æ. Moreover, we have ken+1 en kE = kB (en en 1; en )+B (en 1 ; en en 1kE (4CB Æ)ken en 1kE ; hence, ken+1 enkE (4CB Æ)nke1 e0kE . Since 4CB Æ < 1, we nd that en converges to a limit e, which is the required solution. The uniqueness of e in B (0; 2Æ) is then obvious. The continuous dependence on e0 is easy as well: just write e f = e0 f0 B(e f; e) B(f; e f ), which gives ke f kE ke0 f0 kE + CB kekE ke f kE + CB kf kE ke f kE ke0 f0 kE +4CB Æke f kE . Proof:
We may now prove Theorem 13.1: Proof of Theorem 13.1: For T > 0, let ET = (L1 ((0; T ); B1N;1(IRd)))d . If ~u0 belongs to (B1N;1)d, then we have for all T > 0 ket~u0kET k~u0kB1N;1 . Moreover, B is continuous from ET ET to ET : rst, we have2Nfor1 2 B1N;1 N +1 ; 1 and > 0 ! 2 B1 and k !kB1N ;1 C (1 + )k kB1N;1 ; N +1 ; 1 N; 1 moreover, for 2 B1 and 2 B1 , we have 2 B1N;1 and k kB1N;1 C k kB1N ;1 k kB1N;1 ; nally, the operator e(t s) IPr~ : is given +1
+1
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The molli ed equations 125 by convolutions with functions whose L1 norm is controlled in (t s) 1=2 (see Chapter 11); thus, we get p kB(~u; ~v)kET C T (1 + 2N 1 )k~ukET k~vkET where the constant C does not depend on T nor on . Thus, Theorem 13.2 gives that we of a solution ~u in ET of ~u = et~u0 B(~u; ~u) as p have existence 2 N 1 soon as 4C T (1 + )k~u0kB1N;1 < 1, thus for T small enough. A solution in ET of (2) satis es that @t~u 2 (L1((0; T ); B1N 2;1))d; hence, that ~u 2 (C ([0; T ]; B1N 2;1))d . Thus, uniqueness of solutions in ET is obvious: the contraction principle proves that two solutions of (13.2) in ET must coincide on (0; T0) where C pT0(1 + 2N 1)(k~ukET + k~vkET ) < 1. Now, if T1 is the supremum of the times t so that ~u = ~v on [0; t), we have ~u(T1) = ~v(T1) in (B1N 2;1)d by continuity; hence, in (B1N;1)d by *-weak continuity; if T1 < T , we may notice that ~u(t + T1) and ~v(t + T1) are solutions in ET T of a similar Cauchy problem (replacing the initial value ~u0 by ~u(T1)); thus, we would nd that ~u = ~v on [T1; T1 + T2) for small enough T2, and this is in contradiction with the de nition of T1 : thus T1 = T and we have uniqueness in ET . Because of this uniqueness, we nd that there is a maximal interval [0; T ) on which ~u is de ned. If T < 1, we must have for t < T that k~u(t)kB1N;1 1 4C pT t(1+ N ) , hence limt!T k~u(t)kB1N;1 = +1. We now prove the regularity of ~u. More precisely, we prove by induction that ~u 2 \0
2
1
1
2
~u(t) = e(t
Z t
T0 )~ u
(T0)
T0
~ :((~u ! ) ~u) ds; e(t s)IPr
we have ke(t T )~u(T0)kB1N k= ;1 C sup((t T0) k=4 ; 1)k~u(T0)kB1N;1 ; moreover, if ~u 2 (B1N +k=2;1 )d for some k 2 IN, then (~u !) ~u 2 (B1N +k=2;1 )dd; thus, we nd Z t k IPr~ :((~u ! ) ~u) dskB1N k = ;1 0 Z t C (1 + N 1 )k~uk2L1((T ;T );B1N k= ;1) sup( (t 1s)3=4 ; 1) ds: T Thus far, we have proved the spatial regularity of ~u. In particular, we have k;1 )d for all k 2 IN; a similar estimate then ~u 2 \0
+
2
+( +1) 2
0
1
+
2
0
2
+1
+1
0
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2
126
Shift-invariant estimates FT = f~u 2 ET = ~u 2 (C ((0; T ); C0(IRd )))d and sup0
2
2
2
2
1
2
1
1
2
2
2
2
1
0
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The molli ed equations 127 by the Picard contraction principle in (C ([T; T1]; C0(IRd))d. We nd that, for a constant C which does not depend on T nor , thepproblem may be solved on [T; T1] (with supT tT k~vk1 2) as soon as C T1 T < 1. We shall then choose in order to get T1 > T . Now, we write ~u = ~v + w~ where w~ is solution of 8 ~ :((~v !) w~ + (w~ ! ) ~v + (w~ !) w~ ) ~ = w~ IPr < @t w r~ :w~ = 0 : ~ w~ (T; :) = W This may be solved by the Picard contraction principle in (C ([T; T2]; L2(IRd))d: we nd that, for a constant C , which does not depend on T nor , the problem may be solved on [T; T2] as soon as C ( + kW~ k2)pT2 T < 1. This is the same solution as ~u w~ , because of uniqueness of solutions in (L1 ((T; T2); B1N;1))d 2 N; 1 (we may assume N > d=2, hence L B1 ). Now, we shall prove the energy estimate to get that this solution w~ never explodes in L2 norm, hence may be continued up to T , and even beyond T ; this will prove that ~u may be continued beyond T , din contradiction with the de nition dof T . Indeed, if w~ 2 (C ([T; T ]; L2(IR ))d , we de ne p 2 C ([T; T ]; L2(IR ) by p = 1 P P @j @k (wk (w3j ! ) + wk (vj ! ) + vk (wj ! ));3 since w~ = ~u ~v j k is smooth on (0; T3) IRd, we have ~ w~ j2 = jw~ j2 r ~ :Z~ + 2r ~ w: @t jw~ j2 + 2jr ~ (w~ ! ) ~v where Z~ P = jw~ j2 (w~ !+~v!)+2(~v:w~ ) w~ !+2p w~ and where r~ w~ :(w~ !) ~v = P ~ w~ j2 and Z~ are 1jd 1kd (wj !) vk @j wk ; we now notice that jw~ j2 , jr integrable on [0; T3] IRd: we just have to check it for jr~ w~ j2; we just write 1
r~ w~ = r~ e(t T )W~ +
Z t T
r~ e(t s) p
~ pr :M ds
IP
with M = (~v !) R Rw~+1+ (w~ !) ~v + (w~ !) w~ ; r~ e(t T )W~ is square-integrable: T jr~ e(t T )W~ j2 dt = kW~ k22; moreover M belongs to (C ([T; T3]; L2))dd, hence is square-integrable on (T; T3) IRd; the maximal regularity theorem for the heat kernel (Chapter 7) then gives that R t (t s) e pr~ IP pr~ :M ds is square-integrable. We now choose 2 T D((IRd ) equal to 1 in a neighborhood of 0 and integrate this equality against (t)(x=R) for 2 D((0; T3 )) and R > 0, and then let R go to in nity; we get kw~ (t)k22 + 2
Z Z t T
jr~ w~ j2 dx ds = kW~ k22 + 2
which gives (using the inequality 2ab a2 + b2) kw~ (t)k22 kW~ k22 +
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Z Z t T
Z Z t T
r~ w: ~ (w~ ! ) ~v ds
j(w~ ! ) ~vj2 ds kW~ k22 + C2
Z Z t T
j(w~ (s)j2 ds
128 Shift-invariant estimates which gives kw~ (t)k2 kW~ k2e C (t T )=2 . Thus, we get T = +1. 2
2. The limiting process
In this section, we discuss how to recover a solution ~u for the Navier{Stokes equations: 8 ~ :((~u ! ) ~u) u = ~u IPr < @t ~ ~ (13:3) r:~u = 0 : ~u(0; :) = ~u0 from the solutions ~u of the molli ed equations, 8 ~ :((u~ ! ) u~ ) < @t u~ = u~ IPr (13:4) r~ :u~ = 0 : ~u(0; :) = ~u0 Following the discussion in Chapter 11, we search for a solution ~u in the space \0
ff 2 L2loc ((0; T )IRd )=
X
k2ZZd
kf kL ((0;T )k+[0;1]d) < 1g l1(ZZ; L2((0; T )[0; 1]d)) 2
we may extract from those solutions ~u (through a diagonal process) a subsequence ~uk (with limk!1 k = 0), which is *-weakly convergent to some function ~u in the space \0
2
2
0
then there exists ~u \0
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The molli ed equations
129
ii) Assume that we have the following uniform estimation of spatial regularity: for all 2 D((0; T0 ) IRd );
sup k(Id )=2 ( ~u)kL ((0;T )IRd) < 1; 2
0<<
0
0
then ~u is solution of the Navier{Stokes equations (13.3).
We have already proved the existence of a *-weak limit ~u for some subsequence ~uk . Thus, we have just to prove point ii). We rst estimate the local regularity of ~u (uniformly with respect to ). We have, for 2 D((0; T0 ) IRd ), @t ( ~u ) = (@t )~u + ~u IPr~ :((~u !) ~ud). Since we have uniform estimates in L2uloc;xL2t norm on ~u on (0; T ) IR (where Supp (0; T ) IRd) and since the kernel of IPr~ : is a sum of a matrix of convolutions with kernels in W L1 and of a matrix of convolutions with kernels that are a sum ofPdouble derivatives ofPfunctions in L1comp (Lemma 11.1), we nd that @t( ~u) = j;kd @j @k~vj;k; ; + jd @j ~vj; ; + ~v ;, where the functions ~vj;k; ; , ~vj; ; and ~v ; are (uniformly with respect to ) in (L1comp ((0; T0) IRd ))d and are supported in a compact neighborhood V of Supp independent of . Thus, (Id ) 2 @t( ~u) 2 (L1(IR IRd))d uniformly with respect to . Let us call F~( ; ) the Fourier transform (in t and x) of ~u 2 (L2(IR IRd))d. We provedR Rthat (1+jj jj ) jF~( ; )j C ( ) with C ( ) a constant independent of , hence (1 + 2 )1=4 (1 + j j2 ) (d+9)=2 jF~ ( ; )j2 d d D( ) with D( ) a constant independent of .. Moreover, the hypothesis in point ii) states that RR (1 + jj2) jF~( ; )j2 d d E ( ) with E ( ) a constant independent of .
This gives for > 0 small enough (so that 4 < 1 and (2(1d+19) 4 ) ) that RR (1 + 2 + jj2) jF~ ( ; )j2 d d F ( ) with F ( ) a constant independent of . This proves that the family ( ~u)0<< is bounded in (H (IR d IRd))d; hence, the subsequence ~uk is weakly2convergent to ~u in (H (IR IR ))d and thus strongly convergent to ~u in (L (IR IRd))d since all the functions are supported in the compact set Supp (Proposition 9.1). If we look more closely at the proof of Lemma 11.1, we easily see (replacing in the proof !(x) by !(x=R) and letting R go to = 1) that the kernel of IPr~ : is a sum of a matrix of convolutions with kernels in W L1 with arbitrarily small 1 norms in W L and of a matrix of convolutions with kernels that are a sum of double derivatives of functions in L1comp. Now, if we look at IPr~ :((~uk !k )
~uk ), the convolution with the W L1 kernel gives a result in (L1 (IR IRd ))d with a small norm (uniformly in ) and the convolution with the compactly supported kernel converges to the convolution with ~u ~u, since we have local strong convergence of (~uk !k ) ~uk to ~u ~u in L1 norm. This proves that IPr~ :((~uk !k ) ~uk ) converges to IPr~ :(~u ~u) in (D0 ((0; T0) IRd))d . Proof:
2 2
0
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130
Shift-invariant estimates
3. Mild solutions
In this section, we consider the case of ~u0 so that the Picard contraction principle works directly for the Navier{Stokes equations and not only for the molli ed equations. Theorem 13.4: (Convergence to mild solutions) Let ET L2uloc;x L2t ((0; T ) IRd ) be such that : R ~ :(~u ~v) ds is i) the bilinear transform B de ned by B (~u; ~v ) = 0t e(t s) IPr d bounded on ET : kB (~u; ~v ))kET C0 k~ukET k~v kET ; d ii) R for all f 2 ET and all 2 D(IR ), the function f de ned by f (t; x) = f (t; y) (x y) dy belongs to ET and kf kET kf kET k k1 ; iii) there exists a > 0 so that for all ~u, ~v in ETd and all 2 D(0; T ) IRd ), k(Id )=2 ( B (~u; ~v))kL ((0;T )IRd ) C ( )k~ukET k~vkET ; iv) ET is the dual space of a Banach space in which D((0; T ) IRd ) is densely 2
included and the norm k kET is the dual norm. Then: ~ :~u0 = 0 and ket~u0 kET < 4C1 , there exists a weak a) For all ~u0 satisfying r 0 d solution ~u 2 ET ofRthe Navier{Stokes equations associated with the initial value ~ :(~u ~u) ds. ~u0 : ~u = et u~0 0t e(t s) IPr b) Moreover, for every > 0, the molli ed Navier{Stokes equations have a R ~ :((u~ ! u~ ) ds. solution ~u de ned on (0; T ) IRd u~ = et u~0 0t e(t s) IPr The solutions ~u belong to (ET )d uniformly with respect to and converge *weakly to ~u. c) If, moreover, D((0; T ) IRd ) is continuously and densely included in ET , then the solutions ~u converge strongly to ~u in ET .
Point a) is a direct application of the Picard contraction principle (Theorem 13.2). To prove point b) we rst note that the Picard contraction principle works for nding a solution for the molli ed equations in ET , since R k 0t e(t s)IPr~ :((~u !) ~v) dskET C0 k~ukET k~vkET with the same constant C0 as for B. Moreover, choosing 2 D((0; 1)) withR (1) = 1, and writing et~u0 = Rt R 0 t t ( s=t) s s s that p0 @s ((s=t)e 1 ~ut0 ) ds = 01 2t e ~u0 ds + d 0 d(s=t)e ~u0 ds, we nd 3;1 )d t(Id ) e ~u0 2 (Lt Luloc;x((0; T ) IR )) ; hence, ~u0 2 (BLuloc it is easy to see that the bilinear operator B maps (B13 d=2;1)d. Moreover, (L2uloc;xL2t ((0; T )IRd))d (L2uloc;xL2t ((0; T )IRd))d to (L1((0; T ); B1d 1;1))d. Thus, the solution ~u computed in ETd is the same one as the solution computed in (L1((0; T ); B1max(d+1;3+d=2);1))d . We know (Theorem 13.3) that given any sequence ~un with n ! 0, we may nd a subsequence ~unk , which is convergent in (D0 ((0; T ) IRd)d to a solution ~v of the Navier{Stokes equations. But,tsince ET is a dual space and since k~ukET 2ket~u0kET , we obtain k~vkET 2ket~u0kET ; hence, ~u = ~v by uniqueness of solutions in the ball f~g = k~gkET 2ke ~u0kET . Thus, point b) is proved. Proof:
2
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The molli ed equations 131 To prove point c), we write ~u ~u = B (~u; ~u ~u) + B ((~u ~u) !; ~u) + B (~u (~u !); ~u ) which gives t k~u ~ukET 1 2C40Ckeket~u0~ukEkT k~u ~u !kET : 0 0 ET The operators f 7! f ! are equicontinuous on ET and when 2 D((0; T )IRd) the functions ! converge to in D (hence in ET ) as goes to O. Thus, we have lim!0 k~u ~u !kET = 0. Thus, we have proved that the limiting process in Theorem 13.3 and the direct application of the Picard contraction principle lead to the same solution.
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Part 3:
Classical existence results for the Navier{Stokes equations
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Chapter 14 The Leray solutions for the Navier{Stokes equations In this chapter, we shall present the classical proof of the Leray theorem on existence of square integrable weak solutions: Theorem 14.1: (Leray's theorem [LER 34]) ~ :~u0 = 0, there exists a weak solution For all ~u0 2 (L2 (IRd ))d so that r 1 2 d 2 _ ~u 2 L ((0; 1); (L ) ) \ L ((0; 1); (H 1 )d ) for the Navier{Stokes equations on (0; 1) IRd so that limt!0+ k~u ~u0k2 = 0. Moreover, we may choose this solution ~u full lling the Leray energy inequality:
(14:1)
8t > 0 k~u(t; :)k22 + 2
Z tZ
0 IR
d
jr~ ~uj2 dx ds k~u0k22
where d Z
X k~u(t; :)k22 = i=1
d
d
XX ~ ~u(s; x)j2 = ui (t; x) dx and jr j@i uj (s; x)j2 : i=1 j =1
Then, we introduce the results of Serrin [SER 62] giving criteria to ensure uniqueness for the Leray solutions. 1. The energy inequality
We rst recall that, when looking for a solution in L1((0; 1); (L2 (IRd)d), the Navier{Stokes equations may be equivalently formulated (see Chapter 11) as 8 ~u 2 L1 ((0; 1); (L2 (IRd )d ) > > < 0 9p 2 D ((0; 1) IRd ) @t~u = ~u r~ :(~u u~) r~ p (14:2) > r~ :~u = 0 > : limt!0 ~u = ~u0 135 © 2002 by CRC Press LLC
136 or as
Classical results 2 L1 ((0; 1); (L2 (IRd )d ) ~ :(~u u~) @t~u = ~u IPr r~ :~u = 0
8 ~u > > <
(14:3)
> > :
or as
8 <
(14:4)
:
~u =
limt!0 ~u = ~u0
~u 2 L1R((0; 1); (L2 (IRd )d ) t ~ :(~u u~) ds 0 0 e(t s) IPr ~r:u~0 = 0
etu~
We now turn to the proof of Theorem 14.1. As a matter of fact, we have already proved Theorem 14.1. Indeed, the method introduced by Leray in [LER 34] for solving the Navier{Stokes equations is the molli cation discussed in Chapter 13. Theorem 13.1 proves that, when ~u0 2 (L2)d with r~ :~u0 = 0, the molli ed equations 8 ~ :((~u !) ~u) r ~p u = ~u r < @t ~ ~r:~u = 0 (14:5) : ~u(0; :) = ~u0 have a solution on ~u on (0; +1) IRd; this theorem proved that we have for all R R ~ ~uj2 dx ds = t0 0 and all t t0 the energy equality k~u(t; :)k22 + 2 tt IRd jr 2 k~u(t0 ; :)k2 . Thus, the functions ~u are (uniformly with respect to ) in L1 ((0; 1); (L2 )d ) \ L2 ((0; T ); (H_ 1)d ) (where H_ 1 is the homogeneous Sobolev space H_ 1 = ff 2 S00 = r~ f 2 (L2)dg with norm kf kH_ = kr~ f k2). Thus, we may apply Theorem 013.3 and conclude that there exists a subsequence ~uk , which converges in (D ((0; 1) IRd))d to a solution ~u. If (t) is a function in D((0; +1)), we use the weak convergence of ~uk in (L2((0; +1) IRd)d (since we have convergence in (D0 )d and uniform control in ~ ~uk L2 norm on (0; T ) IRd for every nite T ) and the weak convergence of r d dd 2 0 d d in (L ((0; +1) IR ) (convergence in (D ) and uniform control in L2 norm) to get that RR R R R j RR(t)j2 j~u(t)j2 dt dx + 2 j (tR)j2 ( 0t jRr~ R ~u(s)j2 ds dx)dt lim inf k !0 j (t)j2 j~uk(t)j2 dt dxR+2 j (t)j2 ( 0t jr~ ~uk(s)j2 ds dx) dt k~u0 k22 j (t)j2 dt: R We choose 2 D(IR) with jj2 dt = 1 and we take for t0 > 0 and > 0 (t) = p1 ( t t ); we then obtain that 0
1
0
Z 1 j( t lim sup !0
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t0 2 )j k~u(t)k2L2(dx) dt + 2
Z t0 Z
0
jr~ ~u(s)j2 ds dx k~u0k22 :
Leray's theory 137 If t0 is a Lebesgue point for the measurable function t 7! k~ukL (dx), then the limit in the left-hand side of the above inequality is equal to k~u(t0 )k2L (dx). Thus, inequality (14.1) is valid for almost every t > 0. Moreover, since t 7! ~u is bounded in L2 -norm and continuous in some Besov norm (namely, in B1max(2+d=2;d+1);1 norm), we nd that this is continuous from [0; 1) to (L2 )d endowed with the weak topology. If t > 0 and if tn is a sequence of times2 so that (14.1) is valid for tnRand so that tn converges to t, we nd that k~u(t)k2 R tn R tR ~ 2 2 lim inf tn!t k~u(tn )k2 and 0 jr ~u(s)j ds dx = limtn!t 0 jr~ ~u(s)j2 ds dx. Thus, (14.1) is valid everywhere. The continuity of ~u in L2(dx) norm at t = 0 is then obvious: since ~u(t) is weakly convergent to ~u0, we just have to check that limt!0 k~u(t)k22 = k~u0k22. But the weak convergence gives that k~u0k22 lim inf t!0 k~u(t)k22 while inequality (14.1) gives that lim supt!0 k~u(t)k22 k~u0k22. We may now de ne the Leray solutions for the Navier{Stokes equations: De nition 14.1: (Leray solution) ~ :~u0 = 0. A Leray solution on (0; T ) for (A) Let ~u0 2 (L2 (IRd ))d so that r 2
2
+
+
+
the Navier{Stokes equations with initial value ~u0 is a weak solution ~u for
9p 2 D0 ((0; T ) IRd ) @t~u = ~u r~ :(~u u~) r~ p r~ :~u = 0 so that ~u 2 L1 ((0; T ); (L2 )d ) \R LR2 ((0; T ); (H_ 1 )d ), limt!0 k~u ~u0 k2 = 0 and, ~ ~uj2 dx ds k~u0 k22. for all t 2 (0; T ), k~u(t; :)k22 + 2 0t IRd jr +
(B) A restricted Leray solution on (0; T ) is a solution provided by the limiting process applied to the solutions of the molli ed equations.
The last two sections of this chapter are devoted to classical criteria for uniqueness of the Leray solutions. We shall say that we have uniqueness in the Leray class on (0; T ) if there is only one Leray solution on (0; T ) associated with ~u0; we have uniqueness in the restricted Leray class if there is only one restricted Leray solution on (0; T ) associated with ~u0. Proofs of uniqueness in the Leray class are based on the energy inequality ; proofs of uniqueness in the restricted Leray class are based on direct estimates for the mild solutions of the equations (Theorem 13.4). Before turning our attention to uniqueness issues, we recall a regularity result for (restricted) Leray solutions when d 4: Proposition 14.1: (Strong energy inequality) When d 4, inequality (14.1) may be strengthened into: (14:6)
Z tZ
for almost all t0 > 0; 8t >t0 k~u(t; :)k22+2
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t0
IR
jr~ ~uj2 dx ds k~u(t0 ; :)k22
d
138
Classical results
This strong energy inequality is ful lled by all the restricted Leray solutions.
Let us consider a restricted Leray solution. We follow the proof of the Leray energy inequality in Theorem 14.1. For a xed t0 > 0 and for t1 > t0, we write (t) = p1 ( t t ) with R jj2 dt = 1 and Proof:
1
RR
R
R R
j RR (t)j2 j~u(t)j2 dt dx + 2 j (t)Rj2 ( tt jrR~ R ~u(s)j2ds dx) dt lim inf k !0 j (t)j2 j~uk(t)j2 dt dx +2 j (t)j2 ( tt jr~ ~uk(s)j2 ds dx)dt lim inf k !0 k~uk (t0 )k22: 0
0
Thus, we obtain
k~u(t1 )k2L (dx) + 2 2
Z t1 Z t0
jr~ ~u(s)j2 ds dx lim kinf!0 k~uk (t0 )k22 :
for any Lebesgue point t1 > t0 of the measurable function t 7! k~ukL (dx); hence, for every t1 > t0 (due to the continuity in the weak L2 topology of t 7! ~u). Thus, we shall prove Proposition 14.1 by establishing that there is a subsequence l of the sequence k such that ~ul (t0 ) converges to ~u(t0) strongly in (L2(dx))d for almost every t0 > 0. Recall that we proved (at least for a subsequence of (~uk )) that we have strong convergence in (L2loc(IR IRd))d . (See the proof of Theorem 13.3.) Now, we are going to prove that we have more precisely, for every 0 < T0 < T1 < 1, limk !0 RTT k~uk ~uk22 dt = 0. This is suÆcient to reach a conclusion, since we shall then be able to extract a subsequence so that liml!0 k~ul ~uk22 = 0 almost everywhere. In order to estimate RTT k~uk ~uk22 dt, we choose a cuto function ' 2 D(IRd) so that 0 ' 1 and '(x) =R 1 for jxj 1 and de ne for R > 0 'R (x) = '(x=R). We know that liml!0 TT k'R (x) (~uk ~u)k22 dt = 0, so that 2
1
0
1
0
1
0
lim supl!0 RTT k~uk ~uk22 dt (T1 T0 ) lim supR!1 lim supl !0 supT
0
0
1
0
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1
Leray's theory 139 as Z~R; = (1 'R )2 (j~uj2 u~ ! + 2p ~u). Since Z~R; is integrable on every strip (S ; S ) IRd with 0 < S1 < S2 < R+R1t , we nd the equality k~uR;(t)k22 + RRt 1 2 2 R R0 jr~ ~uR;j2 dx ds = k~uR;(0)k22 + 0 j~uj2(~u !):r~ (1 'R )2 dx ds + 2 0t p~u:r~ (1 'R )2 dx ds. Now, we use the fact that ~u 2 d(L2((0; +1); H_ 1))d and that (through the Sobolev embeddings) H_ 1(IRd) L d (d 3) or BMO (d = 2). Since we have for all T > 0 ~u 2 (L1((02d; T ); L2)d (L2((0; T ); L2)d, we nd that ~u 2 (L2((0; T ); L4)d provided that d 2 4, i.e. d 4. This gives j~uj2 (~u ! ) 2 (L1 ((0; T ) IRd)d and p~u 2 (L1 ((0; T ) IRd )d ; moreover, their norms in L1 are controlled uniformly with respect to , hence we nd that for every T > 0 there exists a positive constant CT (independent from ) such that for 0 < t < T we have k~uR;(t)k22 k(1 'R)~u0k22 + CT R1 . Thus, the proof is complete. 2
2
2. Energy equality
We begin with a celebrated theorem of Serrin [SER 62] on cases where we have equality, not inequality as in (14.1): Proposition 14.2: (Energy equality) Let T 2 (0; +1] and let ~u 2 L1 ((0; T ); (L2 (IRd )d ), be a solution for the Navier{Stokes equations:
9p 2 D0 ((0; 1) IRd ) @t~u = ~u r~ :(~u u~) r~ p r~ :~u = 0
Assume that: i) ~u 2 L2 ((0; T ); (H_ 1 (IRd )d ) ii) For some q 2 [d; +1], ~u 2 Lp ((0; T ); (Lq (IRd )d ), with 1p Then, ~u 2 C ([0; T ); (L2 (IRd )d ) and the energy equality
k~u(t; :)k22 + 2
= 12
d
2q .
Z tZ
jr~ ~uj2 dx ds = k~u( ; :)k22 IRd holds for all , t so that 0 < t < T . The same result holds when, for q = +1, we replace the assumption ~u 2 L2 ((0; T ); (L1(IRd )d ) by the weaker assumption ~u 2 L2 ((0; T ); (BMO(IRd)d ).
We even prove a more general result: Proposition 14.3: Let T 2 (0; +1] and let ~u1 and ~u2 be two solutions for the Navier{Stokes equations:
for i = 1; 2
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9pi 2 D0 ((0; 1) IRd ) @t u~i = u~i r~ :(u~i u~i ) r~ pi r~ :~ui = 0
140
Classical results
Assume that: i) for i = 1; 2, ~ui 2 L1 ((0; T ); (L2 (IRd )d ), ii) for i = 1; 2, ~ui 2 L2 ((0; T ); (H_ 1 (IRd )d ) iii) For some q 2 [d; +1], ~u1 2 Lp ((0; T ); (Lq (IRd )d ), with p1 = 21 2dq . R Then, t 7! ~u1 (t; x):~u2 (t; x) dx is continuous on [0; T ) and we have the equality Z
Z tZ
~u1 (t; x):~u2 (t; x) dx + 2 Z tZ
~ )~u2 dx ds ~u1 :(~u1 :r
Z tZ
IRd
r~ ~u1:r~ ~u2 dx ds = Z
~ )~u2 dx ds + ~u1 ( ; x):~u2 ( ; x) dx ~u1 :(~u2 :r
IRd IRd for all , t so that 0 < t < T .
The same result holds when, for q = +1, we replace the assumption ~u1 2 L2 ((0; T ); (L1(IRd )d ) by the weaker assumption ~u1 2 L2 ((0; T ); (BMO(IRd)d ).
We begin the proof with an useful lemma: Lemma 14.1: Let T 2 (0; +1] and j 2 f1; : : : ; dg. R t (t s) then 0 e @j f ds 2 Cb ([0; T ); L2(IRd )). Proof:
Let g(t) = R0t e(t
s) @
jf
hg(t)j'iL ;L = 2
and thus, jhg(t)j'iL ;L j 2
2
2
If f
ds and let ' 2 L2(IRd). Z t
0
2 L2((0; T ) IRd ),
We write
hf (s)je(t s) @j 'iL ;L ds 2
2
R R j f j2 ds dx 1=2 0t je(t s)@j 'j2 ds dx 1=2 0 R R R R 0T jf j2 ds dx 1=2 0+1 jes @j 'j2 ds dx 1=2 ; R tR
we then use the Plancherel equality to get Z +1 Z Z +1 Z je sjj j '^( )j2 d ds 21 k'k22 : jes @j 'j2 ds dx = (21 )d 0 0 2 Thus, the mapping f 7! g is bounded from L2t L2x to L1 t Lx ; when f belongs to d 2 D((0; T ) IR ), we check easily that g 2 C ([0; T ]; L ); since D((0; T ) IRd) is d 2 dense in L ((0; T ) IR , we may conclude that the mapping f 7! g is bounded from L2t L2x to Cb([0; T ]; L2). Proof of Proposition 14.3: We use a smoothing R R function (t; x) = (t) (x) 2 D(IRd+1 ), where is supported in [ 1; 1], with dx dt = 1, and de ne, for > 0, (t; x) = d1 ( t ; x ). Then, ~ui is a smooth function on (; T ) IRd 2
+1
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Leray's theory 141 andwe may write @t ( ~u1):( ~u2) = @t ( ~u1) :( ~u2)+( ~u1):@t ( ~u2 ) = ( @t~u1):( ~u2)+( ~u1 ):( @t~u2). We then write for i; j 2 f1; 2g: ( @t~ui):( ~uj ) = r~ : ( [r~ ~ui ]):( ~uj ) ( [r~ ~ui ]):( [r~ ~uj ]) r~ : ( [~ui ~ui ]):( ~uj ) + ( [~ui ~ui ]):( [r~ ~uj ]) r~ : ( pi )( ~uj ) We then integrate this equality against a test function (t)'(x=R) with ' equal to 1 on a neighbourhood of 0 and wed let R go to 1. The terms which are written1r~ :F2~ dwith F2~ 2 (L1((0; T ) IR ))d will have a null contribution. Since ~ui 2 (L L ) \ (L H_ 1 )d (L4 H_ 1=2 )d (L4 Lq )d with 1=q = 1=2 1=(2d), we nd that pi 2 L2Ld=(d 1), pi 2 L2Ld=(d 1) and ~uj 2 (L2Ld)d, so that (R pi )( ~uj ) 2 (L1((0; T ) IRR d))d. Hence, we get the equality in DR 0 (; T ): @t ( ~u1 ):( ~u2) dx = 2 ( [r~ ~u1]):( [r~ ~u2]) dx + ( [~u1
R ~ ~u2 ]) dx + ( [~u2 ~u2]):( [r ~ ~u1 ]) dx. We may rewrite the last ~u1 ]):( [r R ~ summand in ( [~u2 ~u2]):( [r ~u1]) dx = R ( [r~ :(~u2 ~u2)]):( ~u1) dx. The next step is devoted to proving that ~u1 ~u1 2 (L2((0; T )IRd))dd.2 We rst deal 2with the case d > 2 and q < 1. (Notice that, when d = 2, H_ 1(IR ) BMO(IR ) so that we do not need the assumption iii) on ~u1 ). We write (for r 2 d > 2) L2t H_ 1 \ L1 2) and 2= = d=2 d=r ; t Lx Lt Lx for 2 r 2d=(d p we then write ~u1 2 (L ((0; T ); Lq ))d for some (p; q) with d q < 1 and 1=p = 1=2 d=(2q) (assumption iii)), while ~u1 2 (L((0; T ); Lr ))d where we choose 1=r = 1=2 1=q so that 1=r +1=q = 1=2 and 1= +1=p =d d=4 d=2(1=2 1=q)+ 1=2 d=(2q) = 1=2 ; this gives ~u1 ~u1 2 (L2((0; T ) IR ))dd. We now discuss the case ~u1 2 (L2((0; T ); BMO))d. We know that the complex interpolation p space [L2; BMO]1=2 is equal to L4, hence that kf k4 C kf k2kf kBMO . This gives the inclusion L1 ((0; T ); L2) \ L2((0; T ); BMO) L4 ((0; T ) IRd ). Hence, ~u1 ~u1 2 (L2 ((0; T ) IRd ))dd . We now let go to 0. When ! 0 and f 2 L2 ((0; T ) IRd ), f (which d may be de ned on (0; T ) IR by rst de ning d f by the convolutiond on (; T ) and by extending it by 0 on (0; d) IR and on (T ; T ) IR ) is strongly convergent to f in RL2((0; T ) IR ). We therefore have theR following R 0 convergences in D (0; T R): @t ( ~u1):( ~u2)R dx ! @t ~u1:~u2 dx, ( [r~
~ ~u2 ]) dx ! r ~ ~u1 :r ~ ~u2 dx and ( [~u1 ~u1 ]):( [r ~ ~u2]) dx ! ~uR 1 ]):( [r R ~ ~ ~u1 ~u1 :r ~u2 dx = ~u1 :(~u1 :r)~u2 dx. Thus, we just have to study the convergence of the last summand R ( [r~ :(~u2 ~u2)]):( ~u1) dx. We begin by rewriting [r~ :(~u2 ~u2)] as [(~u2:r~ )~u2]. Indeed, we easily check that, for ~v 2 (L1((0; T ); L2)d with r~ :~v = 0 and w~ 2 (L2((0; T ); H 1)d, we have r~ :f~v w~ g = (~v:r~ ):w~ in (D0 ((0; T ) IRd))d : just smoothen one more time ~v into ~v and w~ into w~ and let go to 0. Now, if we have ~u1 2 (Lp((0; T ); Lq ))d with 1=p = 1=2 d=(2q) and q < 1, we shall write ~u2 2 (L((0; T ); Lr ))d where we choose 1=r = 1=2 1=q so that 1=r +
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142 Classical results 1=q = 1=2 and 1= + 1=p = 1=2. Since r~ ~u2 2 (L2((0; T ) IRd))dd, this gives that (~u2:r~ ):~q=u2(q 21) (Lp=d (p 1) ((0; T ); Lq=(q 1)))d. When ! 0 and p= ( p 1) f 2 L ((0; T ); L (IR )), then f is strongly convergent to f in Lp=(p 1)((0; T ); Lq=(q 1)) whilep when g q2 Lp((0; T ); Lq (IRd)), then g is strongly Rconvergent to g in L ((0; T ); L ). This gives the convergence (in R D0 ((0; T ))) ( [r~ :(~u2 ~u2 )]):( ~u1 ) dx ! ~u1:(~u2 :r~ )~u2 dx. The proof when ~u1 2 (L2((0; T ); BMO))d is quite similar. We use the div-curl theorem in Chapter 12: we get that, for ~u2 2 (L1((0; T ); L2)d \ (L2((0; T1); H 1)d with r~ :~u2 = 0, we have (~u2:r~ ):~u2 22 (L2((0; T1 ); Hd1(IRd)))d where H is the Hardy space. When ! 0 and df 2 L ((0; T ); H (IR )), f is strongly convergent to f in L2((0; T ); H1(IR )), while when g belongs to d 2 L ((0; T ); BMO(IR )), g is *-weakly convergent toR g in L2 ((0; T ); BMO). Thus, we obtain again the convergence in D0((0; T )) of ( [r~ :(~u2 ~u2)]):( R ~ )~u2 dx. ~u1 ) dx to ~u1:(~u2 :r We have thus obtainedR the following equality in D0 (0; T ): @t R ~u1:~u2 dx = R R 2 r~ ~u1:r~ ~u2 dx + ~u1:(~u1:r~ )~u2 dx ~u1:(~u2:r~ )~u2 dx. Using Lemma 14.1, we get that the map t 7! ~u1 is continuous from [0; T ] to R(tL2(dx))d (since ~u1 ~u1 2 (L2((0; T ) IRd))dd and ~u1 = et~u1(0) IP 0 e(t s)r~ :(~u1 ~u1) ds). Since t 7! ~u2 is weakly continuous from [0; T ] to (L2(dx))d , we nd that t 7! R ~u1:~u2 dx is continuous. Thus, we may integrate our equality and obtain the equality in Proposition 14.3. Remarks: i) The condition ~u 2 (Lpt Lq )d is a limitation on the initial data ~u0 . Indeed, if ~u 2 (Lpt Lq )d with d < q < 1 and 1=p = 1=2 d=(2q) is a solution of the Navier{Stokes equations, then in Chapter 20, we see that we must have ~u0 2 (Bqd=q 1;p)d. We see conversely that when ~u0 2 (Bqd=q 1;p )d with d < q < 1 and 1=p = 1=2 rd=(2q) then there exist a positive T and a mild solution ~u 2 (Lp ((0; T ); Lq (IR )))d for the Navier{Stokes equations with initial value ~u0; when we have ~u0 2 (Bqd=q 1;p \ L2)d, this mild solution is a Leray solution, hence hypotheses i) and ii) for ~u1 are always full lled. ii) Similarly, the condition ~u 2 (Ld1t dLd)d implies obviously (by weak continuity of t 7! ~u(t; :d))dthat ~u0 2 (L ) . Conversely, we see in Chapter 15 that when ~u0 2 (L ) , then there exist a positive T and a mild solution ~u 2 (C ([0; T ]; Ld(IRr )))d of the Navier{Stokes equations with initial value ~u0 ; when we have ~u0 2 (Ld \ L2)d, this mild solution is a Leray solution. iii) The replacement of hypothesis ~u 2 (L2t L1)d by ~u 2 (L2t BMO)d was recently discussed in a similar context by Kozono and Taniuchi [KOZT 00]. 3. Uniqueness theorems Theorem 14.2: (Serrin's uniqueness theorem) ~ :~u0 = 0. Assume that there exists a solution ~u Let ~u0 2 (L2 (IRd ))d with r for the Navier{Stokes equations on (0; T ) IRd (for some T 2 (0; +1]) with initial value ~u0 so that:
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Leray's theory
143
i) ~u 2 L1 ((0; T ); (L2 (IRd )d ); ii) ~u 2 L2 ((0; T ); (H_ 1 (IRd )d ); iii) For some q 2 (d; +1], ~u 2 Lp ((0; T ); (Lq (IRd )d ), with p1 = 12 2dq . Then, ~u is the unique Leray solution associated with ~u0 on (0; T ). d The same result holds when the assumption ~u 2 (L2t (L1 x )) is replaced by
~u 2 (L2t (BMO))d
The same result is also true for q is smaller than a constant Æd .
= d provided that the norm k~ukL1(Ld)
The proof is easy, due to Proposition 14.3. If ~v is another Leray solution, we write thatR Rk~u(t; :) ~v(t; :)k22 = k~u(t; :)k22 +R kR~v(t; :)k22 2 R ~u(t; :):~v(t; :) dx Rk~uR0k22 2 0t IRd jr~ ~uj2 dx ds + k~u0k22 2 0t IRd jr~ R ~vRj2 dx ds 2k~u0k22 + R R 4 0tR IRRd r~ ~u:r~ ~v dx ds 2 0t IRRd~uR:(~u:r~ )~v dx ds +2 0t IRd~u:(~v:r~ )~v dx ds = 2 0t IRd jr~ (~u ~v)j2 dx ds R2 R0t IRd~u: (~u ~v):r~ ~v dx ds. Moreover, we have the antisymmetry property 0t IRd~u: (~u ~v):r~ ~u dx ds = 0. Now, when 2=r _ r d 2 2 _1 d d q < 1, we de ne r = d=q. We have ~v ~u 2 (L1 t Lx \ Lt H ) (Lt H ) 2 =r 1 r 1 (since kf kH_ r = (2)d= kjjr f^k2 kf k2 kf krH_ ) ; hence ~u ~v 2 (Lt Lx )d with 1= = 1=2 r=d, r~ (~u ~v) 2 (L2L2)d and ~u 2 (Lp Lq )d with 1= +1=2+1=q = 1 and r=2 + 1=2 + 1=p = 1; this gives for every 0 < t < T : Proof:
2
1
R R
j t IRd~u: (~u ~v):r~ (~v ~u) dx dsj Rt R R Cr ( k~ukpq ds)1=p ( t k~v ~uk2H_ ds)1=2 ( t k~v ~ukHr_ r ds) r R R R CRr0 ( t k~ukpq ds)1=p ( 0tRIRRd jr~ (~v ~u)j2 ds dx) r sup0<s
1
2
1+ 2
If ~u = ~v on [0; ] and if t > Ris such that 1+2 r Cr0 (R t k~ukpq ds)1=p < 1, we obtain sup0<s
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144
Classical results Since the case d = 2 is well understood, we shall mainly be interested in the case d 3, discussed in the chapters that follow. In Theorem 14.2, the case q = d remains open. This is dealt with by the uniqueness theorem of Sohr and Von Wahl [WAH 85] : Theorem 14.4: (Von Wahl's uniqueness theorem) ~ :~u0 = 0. Assume that there exist a solution ~u of Let ~u0 2 (L2 (IRd ))d with r the Navier{Stokes equations on (0; T ) IRd (for some T 2 (0; +1]) with initial value ~u0 so that : i) ~u 2 L1 ((0; T ); (L2 (IRd )d ); ii) ~u 2 L2 ((0; T ); (H_ 1 (IRd )d ); iii) ~u 2 C ([0; T ); (Ld(IRd )d ). Then, ~u is the unique Leray solution associated to ~u0 on (0; T ). Proof: The proof is very easy. If T0 < T , then for each > 0 we may split ~u on [0; T0] in ~u = ~ + ~ with ~ 2 (L1 ((0; T0) IRd )d and k ~ kL1Ld < : indeed, by uniform continuity of t 2 [0; T0] 7! ~u(t; :) 2 Ld, we may nd P N so that k~u 0k
2
C
1
0
The Gronwall lemma gives then that ~u = ~v.
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2
Chapter 15 The Kato theory of mild solutions for the Navier{Stokes equations In this chapter, we present the classical results of Kato and Fujita on the existence of mild solutions [FUJK 64] [KAT 84]. The idea is to construct the solution ~u for the Navier{Stokes equations as a solution for the integral equation (15:1)
~u =
Z t
et u~
0
0
~ :(~u ~u) ds e(t s) IPr
and, thus, to search for a xed point of the transform (15:2)
~v 7! et u~0
Z t
0
~ :(~v ~v ) ds = et u~0 B (~v ; ~v) e(t s) IPr
This is the so-called Picard method, which had been already used by Oseen in the beginning of the 20th century to establish the existence of a classical solution for a regular initial value [OSE 27]. 1. Picard's contraction principle
A simple approach to problem (15.2) is trying to nd a Banach space ET of functions de ned on (0; T ) IRd so that the bilinear transform B (~u; ~v) =
Z t
0
~ :(~u ~v) ds e(t s)IPr
is bounded from ETd ETd to ETd . Then, we may consider the space ET S 0 de ned by f 2 ET i f 2 S 0 and (etf )0
@t~u = ~u
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IPr~ :(~u ~u) and r~ :~u = 0 145
146
Classical results
then the associated initial value ~u0 belongs to ETd . b) Conversely, there exists a positive constant C so that for all ~u0 2 ETd satis~ :~u0 = 0 and ket~u0kET < C there exists a weak solution ~u 2 ETd of the fying r Navier{Stokes equations associated with the initial value ~u0 :
~u =
etu~
0
Z t
0
~ :(~u ~u) ds: e(t s) IPr
The requirement that ET L2uloc;xL2t ((0; T ) IRd) grants that the operator B is well-de ned: indeed, we saw in Chapter 11 that B is well-de ned on (L2uloc;xL2t ((0; T ) IRd))d (L2uloc;xL2t ((0; T ) IRd))d (with values in (S 0 )d) and that the equationd \ @t~u = ~u IPr~ :(~u ~u) and r~ :~u = 0 " for ~u 2 (L2uloc;xL2t ((0; T ) IR ))d is equivalent to the existence of some ~u0 2 (S 0 )d with R t (t s) t r~ :~u0 = 0 so that ~u = e ~u0 0 e IPr~ :(~u ~u) ds. Proof: If ~u 2 (ET )d and B maps (ET )d (ET )d to (ET )d and if ~u = et~u0 B (~u; ~u), then et~u0 belongs to (ET )d and thus ~u0 belongs to (ET )d . Conversely, if kB(~v; w~ )kET C0k~vkET kw~ kET , then if ket~u0kET Æ < 4C1 0 we nd that the map ~v 7! et~u0 B(~v; ~v) is contractive on(0)the ball f~v = k~vkET 2Æg. Hence, the sequence de ned by the recursion ~u = et~u0, ~u(n+1) = et~u0 B (~u(n) ; ~u(n)) converges to a xed point ~u of the map; hence, we have a solution ~u in (ET )d.
We then de ne Kato's mild solutions as solutions obtained through the iterative process ~u(0) = et~u0, ~u(n+1) = et~u0 B(~u(n) ; ~u(n)). We call a space ET , on which we may apply the Picard contraction principle, an admissible path space for the Navier{Stokes equations, and the associated space ET an adapted value space . At times, we consider a slight generalization of the pairing between admissibletpath spaces and adapted value spaces, because we may use a weaker norm on e ~u0 than the norm in (ET )d to grant existence of a solution in ET : Theorem 15.1 bis:
(Variation on the Picard contraction principle)
Let ET FT L2uloc;x L2t ((0; T ) IRd ) (with continuous embeddings) be such that the bilinear transform B is bounded from FTd FTd to ETd . Then: a) Let ~u 2 FTd be a weak solution to the Navier{Stokes equations:
@t~u = ~u
IPr~ :(~u ~u) and r~ :~u = 0:
If, moreover, the initial value ~u0 belongs to ETd , then ~u belongs more precisely to ETd .
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Kato's theorem
147
b) Conversely, there exists a positive constant C so that for all ~u0 2 ETd satis~ :~u0 = 0 and ket~u0 kFT < C there exists a weak solution ~u 2 ETd to the fying r Navier{Stokes equations associated with the initial value ~u0 : Z t
~u = etu~0
0
~ :(~u ~u) ds: e(t s) IPr
The theorem is a direct consequence of Theorem 15.1 b). We may often need a useful regularity criterion for the solutions to the Navier{Stokes equations: Proposition 15.1: (Regularity criterion) Let ~u 2 (L1 ((0; T ) IRd ))d be a solution to the Navier{Stokes equations ~ :(~u u~) @t ~u = ~u IPr (15:3) ~r:~u = 0 Then ~u is C 1 on (0; T ) IRd . Proof: We follow the proof of Theorem 13.1, rst proving by induction that k=2;1 d ~u 2 \0
1
2
~u(t) = e(t
T0 ) ~ u
(T0)
Z t T0
~ :(~u ~u) ds; e(t s)IPr
we have ke(t T )~u(T0)kB1k= ;1 C sup((t T0) k=4 ; 1)k~u(T0)k1; moreover, if k=2;1 d ~u 2 (B1 ) for some k 2 IN, then ~u ~u 2 (B1k=2;1 )dd (while, for k = 0, we write that ~u ~u 2 (L1)dd (B10;1)dd; thus, we nd Z t Z t 2 ~ k IPr:(~u ~u) dskB1k = ;1 C k~ukL1((T ;T );B1k= ;1) sup( (t 1s)3=4 ; 1) ds 0 T (for k = 0, replace k~uk2L1((T ;T );B1k= ;1) by k~uk2L1((T ;T )IRd) ), which yields the result. Thus far, we have proved the spatial regularity of ~u. Moreover, similar estimates then khold ford @t~u since maps B1k;1 to B1k 2;1 and IPr~ : k; 1 d d 1 ; 1 maps (B1 ) p to (B1 ) . Then, we may conclude by pinduction that all p @ @ @ derivatives u satisfy similar estimates, since we have @tp ~u = @tp ~u @tp ~ IPr~ : @t@pp (~u ~u). This proves that ~u is smooth on (0; T ) IRd. 0
2
( +1) 2
0
0
2
2
2
2
0
0
2
+1
+1
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148
Classical results
2. Kato's mild solutions in H s , s d=2 1
Mild solutions were rst constructed by Kato and Fujita ins thed Sobolev spaces H s(IRd), s d=2 1 [FUJK 64]. The space ET is then H (IR ) and the space ET is the space C ([0; T ]; H d=2 1) if s > d=2 1 and C ([0; T ]; H d=2 1) \ L2 ((0; T ); H d=2) for s = d=2 1. A modern treatment for mild solutions in H s was given by Chemin [CHE 92]. Theorem 15.2: ~ :~u0 = 0, there (A) Let s > d=2 1. Then for all ~u0 2 (H s (IRd ))d so that r exist a positive T and a (unique) weak solution ~u 2 C ([0; T ); (H s (IRd ))d ) for the Navier{Stokes equations on (0; T ) IRd so that ~u(0; :) = ~u0 . This solution is then smooth on (0; T ) IRd . ~ :~u0 = 0, there exist a positive (B) For all ~u0 2 (H d=2 1 (IRd ))d so that r d= 2 1 (IRd ))d ) for the Navier{Stokes T and a weak solution ~u 2 C ([0; T ); (H d equations on (0; T ) IR so that ~u(0; :) = ~u0 . Moreover, this solution may be chosen so that for all T 2 (0; T ), we have ~u 2 L2 ((0; T ); (H d=2 (IRd ))d ). With this extra condition on the H d=2 norm, such a solution is unique and, moreover, it is smooth on (0; T ) IRd . R (C) There exists 0 > 0 such that if ( j jd 2 j~u^0 ( )j2 d )1=2 < 0 then the existence time T in point (A) or (B) for the smooth solution ~u is equal to
+1. (D) The smooth solutions ~u described in (A) and (B ) satisfy ~u(t; :) 2 (C0(IRd)d for all t < T . The maximal existence time T is nite if and only if limt!T R k~uk1 = +1. We may replace this latter condition by: T < 1 if and only if TT =2 k~u(t; :)k2BMO dt = +1.
Proof: (A) is easy. We want to prove that H s (IRd) is an adapted value space when s > d=2 1, with adapted path space ET = L1([0; T ]; H s(IRd)). We rst estimate the pointwise product of twos functions in H s. We have, due to the d=2;1 s;2 s Bernstein inequalities, H = B2 B1 ; thus (applying Theorem 4.1), we have that, whenever f 2 H s and g 2 H s , fg 2 H s (s > d=2) or fg 2 H 2s d=2 (d=4 < s < d=2); for s = d=2, we shall write that for any positive we have fg 2 H d=2 when f; g 2 H d=2 . Now, let f~ and ~g Rbelong to ETd , with ET = ~ g) = 0t e(t )IPr ~ :(f~ ~g) d . L1 ([0; T ]; H s(IRd )); let us estimate ~h = B (f;~ We write s = + with = s and = 0 for s > d=2, = 2s d=2 and = d=2 s for d=2 1 < sR< d=2 and = d=2 1=2 and = 1=2 for s = d=2; we then have k~h(t; :)kHs 0t ke(t )(Id )=R 2 IPr~ :(Id )=2 (f~ ~g)k2d 1 )=2 ) d . Cs sup0<
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Kato's theorem 149 \0
~u(0) = eit~u0 and ~u(k) = ~u(0)
4
B (~u(k) ; ~u(k) )
belongs to C ([0; T ]; (H s)d). The fact that ~u(0) 2 C ([0; T ]; (sH s)d) is obvious (since test functions are dense in the shift-invariant space H ) ; the fact that B maps C ([0; T ]; (H s)d ) C ([0; T ]; (H s)d ) to C ([0; T ]; (H s)d ) is easily checked: it is enough to prove it on a dense subspace, and, thus, we just have to check that B maps C ([0; T ]; (H s+2)d) C ([0; T ]; (H s+2)d) to C ([0; T ]; (H s)d) : indeed, ~ g) 2 (L1 ((0; T ); H s+2 ))d and for f~ and ~g in (L1 ((0; T ); H s+2))d, we have B(f;~ 1 s +1 d ~ ~ ~ IPr:(f ~g) 2 (L (0; T ); H )) , hence @t B(f;~g) 2 (L1(0; T ); H s))d. Moreover, we easily check that the solution ~u is C 1 on (0; T )IRd. Indeed, since H s B1s d=2;1, we may replace the space ET =d L1([0; T ]; H s(IRd)) by d 1 s ET = ff 2 L ([0; T ]; H (IR )) = t f 2 L1 ((0; T ) IR )g where we de ne as = 0 if s > d=2, = d=4 s=2 if d=2 1 < s < d=2 and = 1=4 if s = d=2. Since ~ g)(t)kL1((0;T )IRd ) C T 1=2 kt f~kL1((0;T )IRd ) kt~gkL1((0;T )IRd ) , kt B (f;~ where the constant C does not depend on T , we nd that the Picard contraction principle works in our new ET for some T which can be estimated by T min(1; s k~u0 kH s ; s0 k~u0kH s ). Hence, the solution ~u we constructed in \0
4
1
4 2
0
1
(
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1) 2
150 ~ g)(t=2) + et=2B (f;~
Rt
(t
t=2 e
Classical results s) IPr ~ :f~ ~g ds; hence,
~ g)(t)k1 C min(1; t) 1=2 kB (f;~ ~ g)(t=2)k d kB (f;~ H2
1
+C
Z t
kf~k1 k~gk1 ptds s t=2 p
where C does not depend on T . Moreover, we have limt!0 ket~u0k1 = 0 and we nd that the Picard iteration scheme will provide a solution in GT for some small enough positive T . Our estimation on T depends only on the size of ket~u0kLt Hxd = and sup0
(
1) 2
0
2
t
k~u(t)kH s k~u0kH s + C0
Z t
k~u( )k1 k~u( )kH s ptd 0
Now, we use the estimate k~u( )k1 C1 k~u(t)k k~u0k + C0 C1 Hs
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Hs
1=2 and we get
Z t
k~u( )kH s pt d p : 0
1
Kato's theorem If !(t) = sup
1
Z
0
p1 d p )!((1 )t) + C0 C1
151 1
Z
1
p1 d p !(t)
whichs gives, for a small enough , !(t) C ()!((1 )t). Thus, the norm of ~u in H remains bounded on every compact subset of [0; 1) and we know from (A) that the maximal existence time of the solution in H s is then T = +1. (As a matter of fact, we may even prove that the H s norm remains globally bounded on [0; 1): see the persistency theorem in Chapter 19). We now prove (D). We know from Proposition 15.1 that ~u(t) is Lipschitz for positive t (since r~ d ~u 2 (L1(IRd))dd); since ~u is square-integrable, we nd that ~u(t) 2 (C0(IR ))d for all positive t < T . Moreover, we check ~ g)k1 easily on (L1([0; T ) IRd))d : sup0
k~u(T =2)kH s + C0
Z t T =2
k~u( )k1 k~u( )kH s ptd
Now, we use the estimate k~u( )k1 C1 and we get R k~u(t)kH s k~u(T =2)kH s + C0 C1 Tt =2 k~u( )kH s pd t R C2 k~u(T =2)kH s + C3 Tt =2 k~u( )kH s d
with C2 = 1 + 2C0C1 T =2 and C3 = (C0 C1)2 R01 p1 dp . The Gronwall lemma shows then that the H s norm of ~u remains bounded. This is enough to get that ~u exists in H s beyond T if s > d=2 1, since we know that the existence time is related to the size of the norm. If s = d=2 1, we notice that ~u 2 (H d=2)d for 0 < t < T ; thus, we nd that the solution ~u exists in H d=2 beyond T , hence exists as well in H d=2 1 beyond T . Finally, we prove in the next section that, if T is nite, if ~u0 2 (Lp)d R T for some p 2 [d; +1) and if T =2 k~u(t; :)k2BMO dt < +1, then we have lim supt!T k~uk < 1. Since H s Ld for s d=2 1, this concludes the proof. p
3. Kato's mild solutions in Lp , p d
In 1984, following the ideas of Weissler [WEI 81], Kato proved the existence of mild solutions in Lp. Existence of solutions for initial values in Lp had also
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152 Classical results been studied by Fabes, Jones, and Riviere [FABJR 72] and by Giga [GIG 86] (see Chapter 20). Theorem 15.3: (Kato's theorem [KAT 84]) ~ :~u0 = 0, there (A) Let p 2 (d; 1). Then, for all ~u0 2 (Lp (IRd ))d so that r exist a positive T and a (unique) weak solution ~u 2 C ([0; T ); (Lp )d ) for the Navier{Stokes equations on (0; T ) IRd so that ~u(0; :) = ~u0 . This solution is then smooth on (0; T ) IRd . ~ :~u0 = 0, there exist a positive T (B) For all ~u0 2 (Ld (IRd ))d so that r d d and a weak solution ~u 2 C ([0; T ); (L ) ) for the Navier{Stokes equations on (0; T ) IRd so that ~u(0; :) =p~u0. This solution may be chosen p so that for all T 2 (0; T ) we have sup0
1 2+2
2
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1 2
2
Kato's theorem 153 and p): since C ([0; T ]; (Lp)d) is closed in ETd , it is enough to check that each iterate ~u(k) de ned by ~u(0) = eit~u0 and ~u(k) = ~u(0) B(~u(k) ; ~u(k)) belongs to C ([0; T ]; (Lp)d). The fact that ~u(0) 2 C ([0; T ]; (Lpp)d) is obvious (since test functions are dense in the shift-invariant space L ); the fact that B maps C ([0; T ]; (Lp)d ) C ([0; T ]; (Lp)d ) to C ([0; T ]; (Lp)d ) is easily checked: it is enough to prove it on a 2+dense subspace, and,2+thus, we just have to check that B d=p d d=p d maps C ([0; T ]; (Hp ) ) C ([0; T ]; (Hp ) ) to C ([0; T ]; (Lp)d): indeed, for ~ g) 2 (L1 ((0; T ); Hp2+d=p))d and f~ and ~g in (L1 ((0; T ); Hp2+d=p))d , we have B (f;~ 1+ d=p ~ g) 2 (L1 (0; T ); Lp))d . IPr~ :(f~ ~g) 2 (L1(0; T ); Hp ))d; hence, @tB(f;~ Moreover, we easily check that the solution ~u is C 1 on (0; T )IRd. Indeed, since Lp B1d=p;1, we may replace the space ET = L1([0; T ]; Lp(IRd)) by d ET = ff 2 L1([0; T ]; Lp(IRd )) = t p f 2 L1 ((0; T ) IRd )g. We nd that the Picard contraction principle works in our new ET for some T which can be estimated by T pk~u0kp . Hence, the solution ~u we have constructed in \0
1
4
0
1
For f~ and ~g in (FT )d, we have ~ g)k2d C kB (f;~
Z t
1
0 (t s)3=4
s1=4 kf~k2d s1=4 k~gk2d ds s1=2
~ g) 2 (FT )dd , while for f~ and ~g in (ET )d we have hence, B(f;~ ~ g)kd + kB (~g; f~)kd C kB (f;~
and ~ g)k1 + kB (~g; f~)k1 C kB (f;~
1
Z t
0 (t s)3=4
s1=4 kf~k2d k~gkd ds s1=4
s1=4 kf~k2d s1=2 k~gk1 ds s3=4 0 (t s)3=4
Z t
1
~ g) 2 (ET )dd and kB (f;~ ~ g)kET + kB (~g; f~)kET C kf~kFT k~gkET . hence, B(f;~ The estimate in FT proves that the Picard iteration scheme will provide a solution to the Navier{Stokes equations ~u 2 (FT )d provided that the norm of (et~u0)0
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154
Classical results ~u(n) converges to ~u in the ET norm. Moreover, for f~ and ~g in (ET )d , we nd easily ~ g) 2 (C ([0; T ]; Ld(IRd ))d : clearly, we have from the above estimates that B(f;~ ~ g)kd = 0; for 0 < t T and for t < min(T; 3t=2), we that limt!0 kB(f;~ write = (1 + )t and = (1 )t (with 0 < 1=2) and ~ g)() B (f;~ ~ g)(t) B (f;~
=
R t=2
(e( t) + Rt=2 (e( t) + t(e( Rt) R0
=
~ :f~ ~g ds Id)e(t s) IPr ( t s ) Id)e IPr~ :f~ ~g ds ( t s ) Id)e IPr~ :f~ ~g ds ( s) ~ ~ IPr:f ~g ds t e I1 + I2 + I3 + I4
For I1 and I2 , we write k(e( t) Id)f kd pC ( t)kf kd and get: { kI1kd C t t sup0<s
r
j tj sup kf~k sup ps k~gk d 1 t+ 0<s
0<s
Finally, we conclude that (B) is valid. So far, we have proved the inequality
~ g)kFT C kf~kFT k~gkFT where C is a positive constant that does not kB (f;~
depend on T . Thus, in order to prove (locally in time) existence of a solution, we just prove that for ~u0 2 (Ld)d, we have limT !0 sup0
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Kato's theorem 155 equations with initial value ~u(t0 ). The end of (C) is contained in assertion (D), which we prove now. We know from Proposition 15.1 that ~u(t) is Lipschitz for positive t (since ~r ~u 2 (L1 (IRd))dd ); since ~u is in Lp (with p < 1), we nd that ~u(t) 2 (C0(IRd))d for all positive t < T .1Moreover, we have seen in the proof of Theorem 15.1 that B is bounded on (L ([0; T )IRd))d and we thus get that, for every t < T , ~u(t + s) remains bounded for s 2 [0; T (t)] where T (t) = O(1=k~u(t)k21 ). Thus, we know that either lim supt!T k~u(t)k1 < +1 or limt!T k~uk1 = 1. In the rst case, we get that supT =2
Z t
d k ~u( )k1 k~u( )kp p t T =2
As in the case of the H s norm, we get the inequality k~u(t)kp C1 k~u(T =2)kp + C2
Z t
T =2
k~u( )kp d
and we may use the Gronwall lemma to prove thatp the Lp norm of ~u remains bounded. This is enough to get that ~u exists in L beyond T if p > d, since we know that the existence time is related to the size of the norm. If p = d, we notice that ~u 2 (L2d)d for 0 < t < T , thus we nd that the solution ~u exists in L2d beyond T , hence exists well in Ld beyond T . Indeed, we write R t as t ( t ~ =~u(s + T =2) ~u(s + T =2) ds = ~u(t + T =2) = e ~u(T =2) 0 e s)IPr et~u(T =2) w~ (t). Since ~u(T =2) 2 Ld, the term et~u(T =2) belongs to Ld for every positive t, while, since ~u 2 C ([T =2; T + ]; (L2d)d], we nd that w~ 2 C ([T =2; T + ]; (Ld)d ]. Finally, we consider the case where RTT=2 k~u(t; :)k2BMO dt < +1. We may consider only the case d < p < 1 (for p = d, we use the result for p = 2d just as in the preceding paragraph).In this case, we have only to prove that the Lp norm remains bounded Ron the whole interval [0; T ). Thus, we consider, for 0 < pt < T =2, w~ (t) = 0t e(t s)IPr~ :~u(s + T =2) ~u(s + T =2) ds. Let '~ 2 (L p )d . We have 1
Z
IRd
'~ (x):w~ (t; x) dx =
hence
Z tZ
0 IRd
(e(t
s)
IPr~ )'~ :~u(s + T =2) ~u(s + T =2) dx ds
R R p j IRd '~ (x):w~ (t; x) dxj C 0t ke(t s) '~ k p p k~u(s + T =2)k22p ds R R p C 0t ke(t s) '~ k2p p ds 1=2 0t k~u(s + T2 )k42p ds 1=2 1
Since p > 2, we have L p
p
1
B_ 0;p2 p
1
kw~ (t; x)kp C
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1
; hence, we get Z t
0
k~u(s + T2 )k42p ds 1=2
156 Classical results But we know that L2p = [Lp; BMO]1=2 (Chapter 6). Thus, we nd at last k~u(t + T =2)k2p C (k~u(T =2)k2p +
Z t
0
k~u(s + T2 )k2p k~u(s + T2 )k2BMO ds)
and we conclude with the Gronwall lemma. To conclude the proof, we notice that since ~u is extended beyond T , we have that ~u remains bounded in L1 norm on [T =2; T ] as stated in the proof of Theorem 15.1. There is another proof of existence of global solutions given by Calderon [CAL 93] and by Cannone [CAN 95]: Proposition 15.2: There exists 0 > 0 (which depends only on the dimension ~ :~u0 = 0 and k~u0 kd < 0 , there exists a d) such that if ~u0 2 (Ld(IRd))d with r weak solution ~u for the Navier{Stokes equations on (0; +1) IRd with initial value ~u(0; :) = ~u0 so that sup0
1
Now, UV 2 Ld=2 while jxj1d 2 L d d ;1, hence W = jxj1d UV 2 Ld. 1
1
1
The extra conditions on the solutions ~u 2 (C ([0; T ); Ld(IRd)))d are used to introduce good adapted path spaces for the proofs of existence of solutions in Theorem 15.2 and Proposition 15.2. We do not need them to prove uniqueness (Furioli, Lemarie-Rieusset and Terraneo [FURLT 00], Meyer [MEY 99], Monniaux [MON 99], Lions and Masmoudi [LIOM 98]); this point will be discussed in Chapter 27. Remark:
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Part 4:
New approaches to mild solutions
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Chapter 16 The mild solutions of Koch and Tataru : the space BMO 1 1. The space BMO 1
The space BMO 1 is de ned as the space of derivatives of functions in BMO. It plays a natural role in the search of global mild solutions for the Navier{Stokes equations. We have seen that the initial value problem for these equations was meaningful for solutions belonging to the space L2uloc;xL2t ((0; T ) IRd). When looking for solutions through the iterative scheme of Picard, we usually require the rst term of the iteration, et~u0, to belong to the space where solutions are searched for. Moreover, the search for global solutions is often derived from the proof of local existence provided that the space we work in satis es that its norm is invariant through the transforms ~u 7! ~u(2 t; x), > 0, (which operate on the set of solutions to the Navier{Stokes equations). BMO( 1) may be characterized precisely as the space of tempered distributions f so that the distribution F = et f satis es that the family F (2 t; x), > 0, is bounded in L2uloc;xL2t ((0; 1) IRd). De nition 16.1: (bmo( 1) and BMO( 1) ) We de ne bmo 1 as the space of tempered distributions f on IRd so that for R tR 1 all T 2 (0; 1) we have sup0
1.
2
0
0
Similarly, BMOR( R1) is the space of the distributions f sup0
We de ne the norms of bmo( kf kbmo
(
1)
(
1) and BMO( 1) by
Z tZ = 0
and kf kBMO
1)
1 = sup sup d= d t 2 0
jesf (x)j2 ds dx 1=2 p 0 jx x0j t
Z tZ
jesf (x)j2 ds dx 1=2 p 0 jx x0j t
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2 bmo( 1) such that
160 Mild solutions We are going to check that those spaces are shift-invariant Banach spaces of distributions. Lemma 16.1: (bmo( 1) and Besov spaces) There is a constant C so that for any f 2 S 0 (IRd ) and any t > 0, we have Z t=2 Z jes f (x)j2 ds dx 1=2 : ketf k C p1 sup 1 1
t x0 2IRd td=2 0
jx
p
x0 j t=2
In particular, bmo( 1) B11;1 and BMO 1 B_ 11;1
Proof: This isR very easy. It is enough to write that etf = e(t s)es f and thus etf = 2t 0t=2 e(t s)esf ds. Now, e is a convolution operator with a 2 j x j positive kernel W (x) = d=1 2 W ( px ), where W (x) = (4R1)d=2 e 4 . Thus, we use theR Cauchy{Schwarz inequality to get: jef (x0 )j = j W (x x0 )f (x) dxj W (x x0 )jf (x)j2 dx 1=2 , and using again the Cauchy{Schwarz inequality, we obtain: jetf (x0 )j 2t R0t=2RIRd Wt s (x x0)jesf (x)j2 ds dx1=2 . We then conclude by writing, for 0 < s < t=2, k 2 ZZd and xpxt 0 2 k + [0; 1]d that Wt s (x x0 ) C td=1 2 1+jk1jd+1 .
We may give the following useful characterization of BMO( 1): Proposition 16.1: (BMO( 1) as derivatives of BMO) i) A distribution f belongs to bmo( 1) P if and only if there exists d + 1 distributions f0 ; : : : ; fd 2 bmo with f = f0 + 1id @i fi . Moreover, the norm P
d of f in bmo( 1) is equivalent i=0 kfi kbmo for all P to the minimum of the sums decompositions f = f0 + 1id @i fi . ( 1) if and only if there exists ii) Similarly, a distribution f belongs to BMO P d distributions f1 ; : : : ; fd 2 BMO with f = 1id @i fi . Moreover, the norm P of f in BMO( 1) is equivalent to the minimum of the sums di=1 kfi kBMO for P all decompositions f = 1id @i fi .
We know that a distribution b belongs to BMO if and only if Z Z for j = 1; : : : ; d; sup 1 j @ et bj2 dt dx < 1;
Proof:
R>0;x0 2IRd
Rd
0
x0 j
@xj
thus, if f = P1idP@i fi with fi 2 BMO, we nd that f 2 BMO( 1) and that kfP kBMO 1id kfikBMO . A similar result holds for bmo( 1) and a sum 1id @i fi with fi 2 bmo. We easily check that bmo bmo( 1): indeed, we write for f 2 bmo f = S0f + Pj0 j f . We have S0f 2 L1; hence, ketS0 f k1 kS0 f k1 and this implies that S0 f 2 bmo( 1) . On the other hand, (
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1)
The Koch and Tataru theorem 161 we write j f = Pdi=1 @i @i j f and get kj f kbmo C 2 j kj f kbmo C 0 2 j kf kbmo. We now prove the converse implications. Since BMO( 1) B_ 11;1, we may write f = P1id @i fi with fi = Pj<0 fi;j fi;j (0) + Pj0 fi;j where 0;1 . We must check that fi 2 BMO. It means fi;j = j @i f . We have fi 2 B_ 1 (from Theorem 10.1) that we must check for f 2 BMO( 1) that Z Z 1 i @j t 2 for i; j = 1; : : : ; d; sup d Rd j @ e f j dt dx < 1 0
2
0
1)
0
Let ! be a xed function in D(IRd) with Supp ! B(0; 1) and R ! dx = 1; we de ne !R(x) = R1d !( Rx ). We then write @i@j etf = fR + gR with gR = !R @i@j et f (the convolution is with respect with the x variable in IRd ). We have, for xed t, kgRkR1R k!RkB_ ; k @i@j etf kB_ 1 ;1 C R1 kf kB_ 1 ;1 and this gives obviously that 0
2
1
1
1
0
0
0
0
0
2
0
0
2
0
(
R;x0 (t; x)=
ZZ
1)
!R (x z )(Ki;j (x y) Ki;j (z; y))(1 R;x0 (y))(et f )(t; y) dy dz
where Ki;j is the kernel of @i@j . We get, for jx j R;x (t; x)j C 0
Z
jx0
jx
x0 j
j R;x (t; x)j2 dx C 0 Rd 0
x0 j < R, that
R j(et f )(t; y)j dy yjd+1 yj10R jx0
hence Z
0
Z
jx0
R j(et f )(t; y)j2 dy yjd+1 yj10R jx0
and R R0
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0
(
1)
162 Mild solutions i.e., to prove sup1
0
0
0
1
1
0
2
1
Z 1Z
0 jx
pj
es
x0 j t
(Id S0 )f j2ds dx
and we easily check that R01Rjx
zj1
sup
Z 1Z
z2IRd
jes (Id S0 )f j2 ds dx 0 jx zj1
jes S0 f j2 ds dx C kf k2bmo
(
1)
2. Local and global existence of solutions
We may now state the theorem of Koch and Tataru: Theorem 16.1: (Koch and Tataru [KOCT 01]) a) [Admissible path space] If ~u is a solution of the Navier{Stokes equations on (0; T ) IRd ~ :~u ~u @t~u = ~u IPr ~r:~u = 0 so p that d d i) t ~u(t; x) 2 (L1 ((0 R ;RT ) IR ) 1 ii) supt
~ :~u ~u @t~u = ~u IPr ~r:~u = 0
such p that d d i) t ~u(t; x) 2 (L1 ((0 R ;RT ) IR ) 1 ii) supt
The proof relies on the following estimates:
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The Koch and Tataru theorem
163
Lemma 16.2: For (t; x) de ned on (0; 1) IRd , let A( ) be the quantity A( ) =
sup sup
x0 2IRd 0
t
d=2
Z tZ
0 jx
0
p j (s; x)j ds dx
xj< t
Then, the following inequality holds for (t; x) = Z 1Z
Z 1Z
j (s; x)j2 ds dx CA( ) d
0 IR Proof:
We write
p
et R0t ds:
0 IRd
j j ds dx
R 1R
2 0 IRd j R(tt;px)j dt dx = etp (s; :) dsj 0 pet (; :) diL (dx) dt = h t t 2Re 0R<<s
p
2
2
2
2
2
2
4
Rs
R sR x y 1 2s 0Pe j R(;s R:)j d = 0 IRd (2s)d= W ( p2s )j (; y)j dy d = ps(k+[0;1]d ) (2s1)d= W ( xp2ys )j (; y)j dy d k2ZZd 0 x P z 1 R sR p k2ZZd supz2k+[0;1]d W ( p2 ) (2s)d= 0 x s(k+[0;1]d ) j (; y )j dy d CA( ) 2
2
2
Lemma 16.3: For every T
2 (0; +1], the bilinear operator B de ned by
B (~u; ~v) = is continuous from by
ETd ETd
Z t
0
~ :(~u ~v) ds e(t s)IPr
to ETd , where
ET L2uloc;xL2t ((0; T ) IRd is de ned
sup0
0
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0
164
Mild solutions
with norm Z tZ p kf kET = sup tkf k1 + sup d sup t d=2
0
x0 2IR
0
0 jx
x0
jf (s; x)j2 ds dx 1=2 : p j< t
The estimate on the L1 norm is easy. We just split the integral de ning B in two domains 0 < s < t=2 and t=2 s < t. For t=2 s < t, we just write ke(t s)IPr~ :(~u ~v)k C pt1 s k~uk1~vk1 C sp1t s k~ukET k~vkET . For 0 < s < t=2, we write (using the estimate on the Oseen kernel from Proposition 11.1) that Z ( t s) ~ je IPr:(~u ~v)j C d (pt + jx1 yj)d+1 j~u(s; y)jj~v(s; y)j dy: IR This gives R X x y2pt(k+[0;1]d ) j~ u(s; y)jj~v(s; y)j dy ( t s) ~ : je IPr:(~u ~v)j C ( d +1) = 2 t (1 + jkj)d+1 k2ZZd Proof:
Since we have Z tZ
0
p
x y2 t(k+[0;1]d )
j~u(s; y)jj~v(s; y)j dy ds Ctd=2 k~ukET k~vkET ;
we obtain the desired estimate. We now turn to the estimate on the local L2 norm: we try to estimate for d 0 < t < T and x0 2 IR the quantity Z tZ
I=
0 jx
2 p jB (~u; ~v)j ds dx =
x0 j t
Z tZ
0 jx
pj
x0 j t
Z s
0
~ :~u ~v dj2 ds dx: e(s ) IPr
We use the function t;x (y) = 1B(x ;10pt)(y) and write B(~u; ~v) = B1 B2 with R 8 ~ :((1 t;x )~u ~v) d B1 = R0s e(s ) IPr > < s 1 ( s ) ~: 0 e p 1 (Id e )( t;x ~u ~v) d B2 = p IPr > : ~ :p es (R0s t;x ~u ~v d) B3 = p 1 IPr 0
0
0
0
B1
is easily estimated for s < t and jx jB1 j
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p
0
x0 j < t : RsR C 0 jy x0j10pt (ps +j1x yj)d+1 j~u(; y)jj~v (; y)j d R R C 0 0t jy x0j10pt jx0 1yjd+1 j~u(; y)jj~v (; y)j d C 00 p1t k~ukET k~vkET
B3
The Koch and Tataru theorem
165
RtR
which gives 0 jx x jpt jB1j2 ds dx Ctd=2k~uk2ET k~vk2ET . To estimate B2 and B3 , we write M (; y) = t;x (y)~u(; y) ~v(; y). We use the boundedness of the Riesz transforms on L2(IRd) (hence, on L2t L2x) to get rid of p 1 IPr~ and get 0
0
Z tZ
0 IR
jB2 j2 ds dx C
Z tZ
j
Z s
e(s )
p 1 (Id e )M dj2 ds dx
0 IR 0 Now, we use the maximal L2L2 regularity of the heat kernel (Theorem 7.3) and d
get
Z tZ
0 IR
d
jB2 j2 ds dx C d
Z tZ
0 IR
d
j p 1 (Id e )M j2 d dy:
Since the function 7! 1 e is bounded, we nd that p 1 (Id on L2(IRd) with operator norm O(p), which gives 2
Z tZ
jB2 j2 ds dx C
e) operates
Z tZ
jM j2 d dy CkM kL1((0;t)IRd ) kM kL ((0;t)IRd ) 0 IRd and we easily conclude, since we have kM kL1((0;t)IRd) C k~ukET k~vkET and kM kL ((0;t)IRd ) Ctd=2 k~ukET k~vkET . 1
0 IRd
1
Similarly, Z tZ
0 IRd
jB3 j2 ds dx C
Z tZ
0 IRd
j
p
es
(
Z s
M d)j2 ds dx
0
We make the change of variables s = t , = t, x = pt z, y = pt w and we nd Z tZ
0 IRd
jB3 j2 ds dx Ct2+d=2
Z
1Z
0 IRd
p
j
e
Z
(
0
N d)j2 d dz
with N (; w) = M (t; pt w). We then use Lemma 16.2 and get Z tZ
jB3 j2 ds dx Ct2+d=2 A(N )kN kL ((0;1)IRd : 0 IRd A(N ) Ct 1 k~ukET k~vkET and kN kL ((0;1)IRd Ct 1 k~ukET k~vkET , 1
Since get the required estimate.
1
we
Point a) is easy. By assumption, ~u belongs to ETd . Through Lemma 16.3, we nd that B(~u; ~u) belongs to ETd . Since ~u is a dsolution of the Navier{Stokes equations which belongs to (L2uloc;xL2t ((0; T ) IR ))d, we nd that ~u0 = limt!0t~u is well de nedd(Theorem 11.2)( and satis es et~u0 = 1) d ~u + B (~u; ~u). Hence, e ~u0 belongs to ET and ~u0 2 (bmo ) . Proof of Theorem 16.1:
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166
Mild solutions Point b) is a direct consequence of Lemma 16.3, due to the Picard contraction principle. Point c) is quite obvious: we have et(f (:))(x) = (e tf )(x); hence, Z tZ jesf (x)j2 ds dx kf (x)k2 = sup sup 1 2
bmo(
1)
0
2
x0 2IRd
td=2 0 jx
p
x0 j t
Thus, the maps f 7!1 f (:) are equicontinuous on bmo( 1) for 2 (0; 1); since we have for f 2 L kdf (:)kbmo C, we see that whenever f belongs to the closure of D(IR ) in bmo( 1), then we have lim!0 kf (:)kbmo = 0. Now, if T 2 (0; 1) and if ~u0 2 (bmo( 1))d is such that r~ :~u0 = 0 and lim!0 k~u0(:)kbmo = 0, we nd that for some small enough > 0, we haved k~u0 (:)kbmo < T ; point ii) gives that we have a solution ~v on (0; T ) IR of the Navier{Stokes problem with initial value ~u0 (:) ; then, ~u = 1 ~v( t ; x ) is a solution on (0; 2T ) IRd of the Navier{Stokes problem with initial value ~u0 . Existence of global solutions for the Navier{Stokes equations may be proved in the same way, just replacing the \local" space bmo( 1) by the \global space" BMO( 1) . Theorem 16.2: (Global mild solutions) There exists a positive constant 1 so that for all ~u0 2 (BMO( 1) )d with k~u0kBMO < 1 and r~ :~u0 = d0 there exists a weak solution ~u of the Navier{ Stokes equations on (0; 1) IR : ~ :~u ~u @t~u = ~u IPr ~r:~u = 0 (
1)
(
(
(
1)
1)
1)
2
(
1)
so p that d d i) t ~u(t; x) 2 (L1 ((0 R R; 1) IR ) 1 ii) sup0
~ :~u ~u @t~u = ~u IPr ~r:~u = 0
so p that d d i) t ~u(t; x) 2 (L1 ((0 R R; 1) IR ) 1 ii) sup0
From Lemma 16.3, we know that B is bounded from E1d E1d to E1d and this will produce the desired result. Proof:
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The Koch and Tataru theorem 167 We conclude by describing some properties of the iterates ~u(n) which appear in the (Picard algorithm when solving the Navier{Stokes equations for ~u0 2 (BMO 1) )d: ~ :~u0 = 0. Let Corollary: Let ~u0 2 (BMO( 1) )d with k~u0 kBMOR < 1 and r t (t s) ~ t IPr:(~u(n) ~u(n)) ds. ~u(n) be de ned by ~u(0) = 0 and ~u(n+1) = e ~u0 0 e Then the sequence w ~ (n) = ~u(n+1) ~u(n) stais es for some positive constants C and with < 1 (which depend on ~u0 but not on n): i) for all n 2 IN, kw ~p(n) (t; x)kE1 Cn ii) for all n 2 IN, k t w ~ (n) (t; x)kL1((0;1)IRd ) Cn iii) for all n 2 IN , supt>0 kw ~ (n) (t; x)kBMO (IRd ) Cn . Proof: i) is of course a direct consequence of the contraction principle. ii) is contained in i). iii) is a consequence of i): indeed, we have that w~ n+1 = IPr~ : R0t e(t s)(w~ n ~un+1 + ~unR w~ n) ds. A proof similar to the proof of Lemma 16.3 gives that (f; g) 7! 0t e(t s)fg ds is bounded from E1 E1 to L1 ((0; 1) IRd). Thus, we nd that sup kw~ k d C kw ~ (n) kE1 (k~u(n) kE1 + k~u(n+1) kE1 ) C 0 n+1 0
(
(
1)
1)
1)
We shall use mainly point ii) in the persistency theorems in the chapters that follow; at times we shall use point iii) as well. We thus introduce the following de nition : De nition 16.2: Let 0 < T 1. If ~u0 2 (BMO( R1) )d , we say that the ~ :(~u(n)
iterated sequence de ned by ~u(0) = 0 and ~u(n+1) = et u~0 0t e(t s) IPr 1 = 2 1 ~u(n) ) ds converges exponentially on (0; T ) (in t L ) if p
lim nsup k t(~u(n+1) !1
~u(n) )k1L=n 1 ((0;T )IRd ) < 1:
We have moreover exponential convergence in B_ 11;1 if
lim nsup k~u !1 (n+1)
~u(n)k1L=n 1 ((0;T );B_ 11;1 (IRd )) < 1:
3. Fourier transform, Navier{Stokes, and BMO( 1)
For the study of space analyticity of solutions to the Navier{Stokes equations, it may be more convenient to study the Fourier transform of solutions. (See Chapter 24.) We thus consider a version of the Koch and Tataru theorem adapted to the Fourier transform.
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168
Mild solutions
Lemma 16.4: i) Let T 2 (0; 1]. Let ET f 2 ET
8 <
,:
L2uloc;x L2t ((0; T ) IRd be de ned by sup0
supx 2IRd sup0
t
d=2
R t R and
0 jx
p jf (s; x)j2 ds dx1=2 < 1
x0 j< t
with norm Z tZ p kf kET = sup tkf k1 + sup d sup t d=2
0
0
x0 2IR
0 jx
x0
jf (s; x)j2 ds dx 1=2 : p j< t
Then the norm k:kET is equivalent to the norm k:kET de ned by
kf kET
= sup
0
p
tkf k1 +
Z t
sup d sup
x0 2IR
0
0
et(jf (s; :)j2 )(x0 ) ds 1=2 :
ii) Let f 2 ET be a real-valued function so that the spatial Fourier transform f^(t; ) is a nonnegative locally integrable function on (0; T ) IRd . Then
p kf kET = (21)d sup0
8 < :
1
NT (f ) = (21)2d sup0
R t R R with tjj2 ^
0
e
f (s; )f^(s; )(x0 ) ds d d 1=2
iii) Let FT be the space of real-valued f 2 ET so that the spatial Fourier transform f^(t; ) is a locally integrable function on (0; T ) IRd and so that F 1 jf^(t; )j 2 ET . Then kf kET kF 1jf^(t; )jkET . Moreover, the map f 7! kF 1 jf^(t; )jkET is a norm on FT , which is then a Banach space.
i) is easy. ii) is a simple consequence of the fact that when f^() is a nonnegative locally integrable function, then f is bounded if and only if f^ is 1 ^ integrable (and, then, kf k1 = (2)d kf k1). iii) is easy as well. The only point to check is the triangle inequality for the norm kf kFT = kF 1jf^(t; )jkET . But we have jf^(t; ) + g(t; )j jf^(t; )j + jg^(t; )j and point ii) then gives that kF 1 (jf^(t; ) + g^(t; )j)kET kF 1 (jf^(t; )j + jg^(t; )j)kET and we conclude by using the triangle inequality on k:kET . Proof:
R p A(f; g) = 0t e(t s)(fg) ds is bounded from ET ET to ET and from FRT FT to FT . ~ :~u ~v ds is bounded from ii) The bilinear operator B (~u; ~v ) = 0t e(t s) IPr d d d FT FT to FT .
Lemma 16.5: i) The bilinear operator
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The Koch and Tataru theorem 169 Proof: The proof of the boundedness of A on ET follows the same line as the proof of Lemma 16.3. (The same estimates are valid on the kernel p e(t s) of A and on the kernel e(t s)IPr~ : of B.) Since A has a nonnegative symbol: Z tZ j j FA(f; g)(t; ) = (2)d e (t s)jj f^(s; )^g(s; ) ds d 0 (where F is the spatial Fourier transform), the boundedness on FT is a direct consequence of the boundedness on ET : jFA(f; g)j FA(F 1 (jF f j); F 1 (jF gj)) and thus kA(f; g)kFT kA(F 1 (jF f j); F 1 (jF gj))kFT = kA(F 1(jF f j); F 1 (jF gj))kET C kF 1 (jF f j)kET kF 1 (jF gj))kET = C kf kFT kgkFT . The Riesz transforms (in space variable x) operate boundedly on FT (since jF (Rj f )(t; )j jf^(t; )j). Since B is composed from the scalar operator A and of Riesz transforms, it is obvious that B is bounded on FTd . 2
Theorem 16.3:
a) [Local existence] For 0 < T < 1 there exists a T > 0 so that for all ~u0 2 (bmo( 1) )d with F 1 (jF ~u0 j) 2 bmo( 1) and with kF 1(jF ~u0 j)kbmo( 1) < ~ :~u0 = 0, there exists a weak solution ~u of the Navier{Stokes equations T and r on (0; T ) IRd : ~ :~u ~u @t~u = ~u IPr r~ :~u = 0
so p that i) t F ~u(t; ) 2 (L1 L1((0; T ) IRd )d R t R R ttjj2 ii) supt
~ :~u ~u @t~u = ~u IPr r~ :~u = 0
so p that d d 1 j) t F ~u(t; ) 2 (L1 t L ((0; 1) IR ) RtR R 2 jj) supt>0 0 e tjj jF ~u(s; )jjF ~u(s; )j ds d d < 1 jjj) limt!0 ~u = ~u0 .
This is direct application of the Picard contraction principle according to Lemma 16.5. Proof:
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Chapter 17 Generalization of the
Lp
theory:
Navier{Stokes and local measures
The results of Chapter 15 may be generalized in a direct way to the setting of spaces of local measures, such as Morrey{Campanato spaces or Lorentz spaces. 1. Shift-invariant spaces of local measures
We consider in this chapter a special class of shift-invariant spaces of distributions: the spaces which are invariant under pointwise multiplication by C0 functions. We have introduced those spaces in Chapter 4 as the shift-invariant spaces of local measures. Let us recall rst the de nition of shift-invariant spaces: De nition 17.1: (shift-invariant Banach spaces of distributions and of local measures) A) A shift-invariant Banach space of test functions is a Banach space E so that we have the continuous embeddings D(IRd ) E D0 (IRd ) and so that: (a) for all x0 2 IRd and for all f 2 E , f (x x0 ) 2 E and kf kE = kf (x x0 )kE . (b) for all > 0 there exists C > 0 so that for all f 2 E f (x) 2 E and kf (x)kEd C kf kE . (c) D(IR ) is dense in E B) A shift-invariant Banach space of distributions is a Banach space E , which is the topological dual of a shift-invariant Banach space of test functions E () . The space E (0) of smooth elements of E is de ned as the closure of D(IRd ) in E . C) A shift-invariant Banach space of local measures is a shift-invariant Banach space of distributions E so that for all f 2 E and all g 2 S (IRd ) we have fg 2 E and kfg kE CE kf kE kg k1 , where CE is a positive constant (which depends neither on f nor on g ).
We recall that we proved the following useful lemma on shift-invariant spaces of local measures:
Lemma 17.1: Then:
Let E be a shift-invariant Banach space of local measures.
171 © 2002 by CRC Press LLC
172
Mild solutions
a) the elements of E are local measures (i.e., are distributions of order 0). 1 of uniformly locally nite More precisely, they belong to the Morrey space Muloc measures. b) E , its predual E () and the space of smooth elements E (0) in E are stable under pointwise multiplication by bounded continuous functions.
We now prove an easy result on real interpolation on local measures:
Proposition 17.1: [Real interpolation on local measures] Let E be a shift-invariant space of local measures. Let us de ne E (1=2) = [E; L1 (IRd )]1=2;1 and E (1=2;) =[E; Cb(IRd)]1=2;1 . Then i) E (1=2) is a shift-invariant space of local measures and is continuously embedded into L2loc (IRd ). Moreover, if D(IRd ) E with continuous embedding and dense range, then E (1=2) = E (1=2;) . ii) If f 2 E (1=2;) and g 2 E (1=2;) , then we have fg 2 E and kfg kE C kf kE(1=2;) kgkE(1=2;) . Moreover, we may replace E (1=2;) by E (1=2) if D(IRd ) E with continuous embedding and dense range or if pointwise multiplication by L1 functions operate boundedly on E . Proof: To prove point i), let E () be the pre-dual of E . We have E (1=2) = ( ([E ) ; L1 ]1=2;c0 )0 (where [P A; B ];c0 is the space of elements of A + B , which may be decomposed into f = k2ZZ fk with supk2ZZ max(2 k kfk kA; 2k(1 ) kfk kB ) < 1 and limjkj!1 supk2ZZ max(2 k kfk kA; 2k(1 ) kfk kB ) = 0). Thus, E (1=2) is a shift-invariant Banach space of local measures. (1=2;) and g 2 E (1=2;) , we may write f = P ii) is very easy. If f 2 E P f with f 2 E \C , k f k P b k kE kfk k1 1 and k2IN jk j C kf kE (1=2;) , k2IN k k and the same for g = k2IN k gk . Then, we write kfk gp kE C kfk kE kgpk1 p and kfk gp kE C kfP k k1 kgp kE , thus P kfk gp kE C kfk kE kgp k1 kfk kE kgp k1 C and kfgkE C ( k2IN jk j)( p2IN jp j). We now list examples of shift-invariant Banach spaces of local measures: i) Lebesgue spaces: E = Lp (IRd ) for 1 < p 1; we then have E () = Lq with 1=p +1=q = 1; moreover, E (0) = Lp when p < 1 and E (0) = C0 (IRd ) for p = 1. ii) bounded Borel measures: E = M (IRd); we have E () = C0 (IRd ) and E (0) = L1 . iii) Morrey spaces of uniformly locally Lp functions: E = Lpuloc (IRd ) is deR ned for 1 < p < 1 by supx0 2IRd jx x0j<1 jf (x)jp dx < 1; the predual of E is then the Wiener space E () = W Lq with 1=p + 1=q = 1, de ned by R P q 1=q (0) p k2ZZd [0;1]d jf (x + k )j dx R < 1; moreover, E = E is the subspace of p p Luloc characterized by limx0 !1 jx x0j<1 jf (x)j dx = 0. iv) Morrey{Campanato spaces M p;q (1 < p q < 1) of locally Lp functions f so that Z sup sup Rd(1=q 1=p) ( jf (x)jp dx)1=p < 1; d 0
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Local measures
173
the predualPof E is then the space of functions f , which may be decomposed as a series n2IN n fn with fn supported by a ball B (xn ; Rn ), Rn 1, fn 2 P Lp=(p 1) , kfnkp=(p 1) Rnd(1=q 1=p) , and n2IN Rjn j < 1; moreover, E (0) is the subspace of M p;q characterized by limx0 !1 jx x0 j<1 jf (x)jp dx = 0 and R limR!0 supx0 2IRd Rd(1=q 1=p) ( jx x0 j
Now, it is quite obvious how Kato's theorem may be generalized to shiftinvariant Banach spaces of local measures. We assign a useful criterion for the boundedness of the bilinear operator B on (L1 ((0; T ); E ))d :
Theorem 17.1: [Local measures and Navier{Stokes] Let E be a shift-invariant Banach space of local measures. Assume that E is continuously embedded into L2loc (IRd ) and that there exists a a shift-invariant Banach space of local measures F so that the pointwise product maps boundedly E E to F : for f 2 E and g 2 E , kfgkF C kf kE kgkE . Then:
A) If, moreover, i) F is boundedly embedded into B1;1 (IRd ) for some p <2 ii) [F; Cb (IRd )]1=2;1 E : for f 2 F \ Cb , kf kE C kf kF kf k1 then, de ning ET = L1 ((0; T ); E d ), the bilinear operator B de ned by B (~u; ~v ) = R t (t s) ~ :(~u ~v) ds is bounded from ET ET to ET : IPr 0e
kB (~u; ~v)kET CT
1 2
4
(1 + T ) 4 k~ukET k~vkET
for a positive constant C that does not depend on T . Moreover, B (~u; ~v ) 2 C ((0; T ]; E d) and converges *-weakly to 0 as t goes to 0. ~ :~u0 = 0), the initial value problem for the Thus, for all ~u0 2 (E d ) (with r Navier{Stokes equations with initial data ~u0 has a solution ~u = et~u0 B (~u; ~u) 4 with ~u 2 ET with T min(1; E k~u0 k 2 ).
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B) Similarly, if i) F is boundedly embedded into B12;1 (IRd ) ii) [F; Cb (IRd )]1=2;1 E then, de ning ET = L1 ((0; T ); E d ), the bilinear operator B de ned by B (~u; ~v ) = R t (t s) ~ :(~u ~v) ds is bounded from ET ET to ET : IPr 0e
kB (~u; ~v)kET C (1 + T )k~ukET k~vkET for a positive constant C that does not depend on T . Moreover, B (~u; ~v ) 2 C ((0; T ]; E d) and converges *-weakly to 0 as t goes to 0. Thus, there exists a positive constant E so that for all ~u0 2 (E d ) (with ~r:~u0 = 0) with k~u0 kE < E , the initial value problem for the Navier{Stokes equations with initial data ~u0 has a solution ~u = et~u0 B (~u; ~u) with ~u 2 ET with T = 1. C) Similarly, if i) F is boundedly imbedded into B_ 12;1 (IRd ) ii) [F; Cb (IRd )]1=2;1 E then, then,R de ning E1 = L1 ((0; 1); E d ), the bilinear operator B de ned by ~ :(~u ~v) ds is bounded from E1 E1 to E1 : B (~u; ~v) = 0t e(t s) IPr
kB (~u; ~v)kE1 C k~ukE1 k~vkE1 for a positive constant C . B (~u; ~v ) 2 Cb ((0; 1]; E d ) and converges *-weakly to 0 as t goes to 0. Thus, there exists a positive constant E so that for all ~u0 2 (E d ) (with r~ :~u0 = 0) with k~u0 kE < E , the initial value problem for the Navier{Stokes equations with initial data ~u0 has a global solution ~u = et~u0 B (~u; ~u) with ~u 2 E1 . (t s) ~ :(~u ~v)kF C p 1 kukE kvkE , which Proof: A) We just write ke 2 IPr t s gives ( ke(t s)IPr~ :(~u ~v)kF C pt1 s kukE kvkE ke(t s)IPr~ :(~u ~v)k1 C pt1 s 1+jtjt sjsj =2 kukE kvkE
hence,
ke(t s)IPr~ :(~u ~v )kE C ke(t s)IPr~ :(~u ~v )k[F;Cb(IRd )] = ; C 0 pt1 s 1+jtjt sjsj =4 k~ukE k~vkE
1 2 1
and thus, for 0 < t < T ,
k
Z t
0
~ :(~u ~v) dskE CT 12 e(t s) IPr
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4
(1 + T ) 4 sup k~ukE sup k~vkE :
0<s
0<s
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This is enough to grant that we have a solution in L1((0; T ); E d) as soon as T 4 is small enough: T min(1; k~u0Rk 2R ) where depends only on E . ~ :(~u ~v) ds into GA + HA with B) We split the integral I = 0t 0t e(t s) IPr
R ~ :(~u ~v) ds GA = 0A e(t s) IPr R t (t s) ~ :(~u ~v) ds IPr HA = A e
We then use the inequalities (
ke(t s)IPr~ :(~u ~v)kF C pt1 s kukE kvkE ke(t s)IPr~ :(~u ~v)k1 C pt1 s 1+jtjt sjsj 1 kukE kvkE
We obtain, for 0 < t < T , kGA kF CA1=2 sup0<s
ke(t s)IPr~ :(~u ~v)kF C pt1 s kukE kvkE ke(t s)IPr~ :(~u ~v )k1 C jt s1j = kukE kvkE 3 2
Examples: (A) may be applied to the Lebesgue spaces Lp, p > d, to the Lorentz spaces Lp;q , d < p < 1, 1 q 1, to the Morrey{Campanato spaces M p;q , 2 p q and d < q and to the multiplier spaces Xr , 0 < r < 1. (B) may be applied to the spaces Ld;1 and MLd;1 and (C) to the space Ld;1.
3. Kato's theorem for local measures: the role of
B11;1
We generalize Kato's theorem to shift-invariant Banach spaces of local measures by using the regularizing e ect of the heat kernel:
Theorem 17.2: [The second theorem on local measures and Navier{Stokes] Let E be a shift-invariant Banach space of local p measures. a) Let EE = L1 ((0; 1); E d ) and B1 = f~uR = t ~u 2 (L1 ((0; T ); Cb (IRd ))d g. ~ :(~u ~v) ds is bounded The bilinear operator B de ned by B (~u; ~v ) = 0t e(t s) IPr from EE B1 to EE and from B1 EE to EE . Moreover, p B (~u; ~v) 2 C ((0; 1]; E d) and converges *-weakly to 0 as t goes to 0. If limt!0 tk~uk1 = 0, if ~u 2 B1 and ~v 2 EE , then the convergence is strong.
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b) If E is continuously embedded in the Besov space B11;1 , then the bilinear operator B is bounded as well from (EE \ B1 ) B1 to B1 and from B1 (EE \ B1 ) to B1 . Hence, there exists a constant E > 0 so that ~ :~u0 = 0) with k~u0 kE E then the initial value for all ~u0 2 (E d ) (with r problem for the Navier{Stokes equations with initial data ~u0 has a solution ~u = et~u0 B (~u; ~u) with ~u 2 EE \ B1 . c) If E is continuously embedded in the Besov space B11;1 , then there exists a positive constant CE sop that, de ning for ~u0 2 E d and T > 0 the T (~u0 ) = sup for the semi-norm N1 0
C k~u k DE N1 (~u0 ) 1 + ln+ ( E 0 E ) < 1 N1 (~u0 )
where DE is a positive constant.
Remark: A suÆcient condition for E to be continuously embedded in the Besov space B1 ;1 (for some 0) is the existence of a constant C 0 so that for all f 2 E we have sup1 kf (x)kE C kf kE . Similarly, a suÆcient condition for E to be continuously embedded in the Besov space B_ 11;1 is the existence of a constant C 0 such that for all f 2 E we have sup>0 kf (x)kE C kf kE . Proof: The boundedness of B from EE B1 to EE and from B1 EE to EE is ~ :kL(Edd;Ed) C p 1 (according to the estimate on obvious, since ke(t s)IPr t s the Oseen kernel (see Chapter 11) and the invariance of E under convolution with L1 kernels). Thus, we get ~ g)kE C kB (f;~
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pt 1 sps ds sup kf~kE sup psk~gk1 0<s
Z t
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which gives the result, since Z t
Z
1
pt 1 sps ds = p1 1 sps ds = : 0 0 ~ g) 2 (C ((0; 1); E ))d and that we have, We now prove more precisely that B (f;~ for 0 < t1 < t2 < 1,
r ~ g)(t1 ) B (f;~ ~ g)(t2 )kE C t2 t1 sup kf~kE sup psk~gk1 : kB (f;~ t1 0<s
We write, for s < t1 , e(t2 s) e(t1 s)
R = t2 t1 e
0
2
d e(t1 s) , which gives
k(e(t s) e(t s) )IPr~ :kL(Edd;Ed) C min( pt 1 s ; jt t2 stj31=2 ) 1 1 2
1
~ g)(t2 ) B (f;~ ~ g)(t1 ) = A1 + A2 with Thus, we write B (f;~ 8 < A1
R = 0t1 (e(t2 s)
~ :(f~ ~g) ds e(t1 s) )IPr and R ~ :(f~ ~g) ds A2 = tt12 e(t2 s) IPr
: R
We have kA1 kE C 0t1 min( pt11 s ; jt1t2 sjt31=2 ) pdss kf~kEE k~gkB1 ; for s < t1 =2, q we write min( pt11 s ; jt1t2 sjt31=2 ) p1s 2(t2t1 t1 ) pt1 1 sps , while for s t1 =2 q q q we write min( pt11 s ; jt1t2 sjt31=2 ) p1s 2(t2t1 t1 ) min( tt21 ts1 ; tt21 ts1 ) t11 s , which q gives kA1 kE C 0 2(t2t1 t1 ) kf~kEE k~gkB1 . On the other hand, we have kA2 kE R C tt12 pt21 s pdss kf~kEE k~gkB1 , which gives (by writing p1s p1t1 ) that kA2 kE q 2C t2t1t1 kf~kEE k~gkB1 . ~ g) goes weakly to 0 when t decreases to We may easily check that B (f;~ ~ 0: since we know that B (f;~g) is bounded in E d for 0 < t < 1, hence is bounded in (S 0 )d , and since, for every ! 2 S , Sj ! goes to 0 strongly in S as j goes to +1 (where Sj is the j th partial sum operator in the Littlewood{Paley ~ g) = 0. decomposition), it is enough to chek that, for all j 2 IN, limt!0 Sj B (f;~ j ~ ~ But pthis is obvious: since jjSj IPr:jj1 C 2 , we nd that kSj B (f;~g)kE C 2j tkf~kEE k~gkB1 . Thus, point a) is proved. ~ :(f~ ~g)k1 Point b) is obvious. We write, for s > t=2, ke(t s)IPr 1 1 ( t s ) ~ ~ :(f~ ~g)k1 p IPr C t s s kf kB1 k~gkB1 and, for s < t=2 and t < 1, ke C t 1 s p1s kf~kB1 k~gkEE . In order to prove point c) with T = 1, we introduce the set K = ff~ 2 EE \ B1 = kf~kEE 2k~u0kE and kf~kB1 2ket~u0 kB1 g. For f~ and ~g 2 K ,
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Mild solutions
~ f~) B (~g;~g)kE C R0t p ds p kf~ ~gkEE (kf~kB1 + k~gkB1 ) we write kB (f; t s s 4C k~u0k1 kf~ ~gkEE . Now, to estimate the L1 norm, we write
1 1 ~ 1 1 ~ p (kf kEE + k~gkEE )) (kf kB1 + k~gkB1 ); !(t; s) = min( p s t s s t s and
ke(t s)IPr~ :(f~ f~ ~g ~g)k1 C!(t; s)kf~ ~gkB1 : We then write = ket~u0 kB1 and = k~u0 kE ; we have 8 =2 d 2 1 p ; min( p ) ds p !(t; s) ds p cos + sin (1 s ) s s 1 s t 0 t 0 0 Since E B11;1 , we have ket~u0 kB1 C (E )k~u0 kE ; hence, we write = = tan 0 with 0 2 [0; arctan C (E ) = E ] and we nd Z
Z t
Z
(1+j ln j) 4 (1+cos )(1+sin ) 0 !(t; s) ds pt ln (1 cos )(1 sin ) CE pt CE0 (1+lnpt ( )) :
Rt
0
0
0
0
0
+
~ f~) is a contraction Thus, if (1 + ln+ ( )) is small enough, f~ 7! et~u0 B (f; on K . This proves point c) for T = 1. T (~u0 ) N 1 (~u0 ) The same estimates are valid for 0 < T 1, since N1 1 k u ~ + 0 kE T T (~u0 )(1 + C (E )k~u0 kE . Thus, if N1 (~u0 )(1 + ln ( N1T (~u0 ) )) is small enough (N1 ln+ ( Nk1u~T 0(k~uE0 ) )) < E , where the positive constant E does not depend on T ), ~ f~) is a contraction on the set ff~ 2 (L1 ((0; T ); E ))d= then f~ 7! et~u0 B (f; p T (~u0 ))g. ~ sup0
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Chapter 18 Further results on local measures
In this chapter, we restate the results of Chapter 17 by describing more precisely the role of the space bmo( 1) in existence theorems for mild solutions in spaces of local measures (Sections 1 and 2). We also give some alternate proofs for the existence of global solutions (Section 3). 1. The role of the Morrey{Campanato space
bmo( 1)
M 1;d
and of
In Chapter 17 (Theorem 17.2), we studied the special case of shift-invariant Banach spaces of local measures that are continuously embedded in B11;1 . As a matter of fact, we may easily characterize the maximal space of this type as being the Morrey{Campanato space M 1;d:
Theorem 18.1: (M 1;d and local measures) Let E be a shift-invariant Banach space of local measures on IRd . Then E is continuously embedded in B11;1 if and only if E is continuously embedded in the Morrey{Campanato space M 1;d . Proof: We easily check that M 1;d is continuously embedded in B11;1 . Indeed, we write for 0 < t < 1 that etf (x) =
Z
y W ( p )f (x y) dy = d= 2 t t 1
Z
p
W (y)f (x
ty) dy
hence,
jet f (x)j
X
sup
k2ZZd y2k+[0;1]
We then write Z
jf (x d
k+[0;1]
p
jW (y)j d
ty)j dy = t d=2
Z
jf (x d
k+[0;1]
jf (z )j dz Ct p p x k t+ t[0;1]d
and we get the inclusion M 1;d B11;1 . 179 © 2002 by CRC Press LLC
Z
p
ty)j dy:
d=2 kf k 1;d (pt)d M
1
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Mild solutions
Conversely, if E B11;1 , we obtain sup!2C0 ;k!k11 k!f kB11;1 C kf kE . Let ' 2 D be equal to 1 on B (0; 1) and let ! 2 D be supported in B (x0 ; 1). We 1 have jhf j!i = jh! f j'(x x0 )i C k!f kB11;1 k'kB11;1 . This gives kf kMuloc C sup!2C0 ;k!k11 k!f kB11;1 . Now, we may de ne the norm of B11;1 with p help of the heat kernel by kf kB11;1 = sup0
Remark: The Morrey{Campanato space M 1;d may be precisely de ned as the spaces of locally nite measures d so that p the positive measure djj is a tempered distribution and so that sup0
Theorem 18.2: Let E be a shift-invariant Banach space of local measures that is continuously embedded in B11;1 . Then E is continuously embedded in bmo 1 . Moreover, there exists a constant CE so that
kf kbmo CE kf kB1 ;1 1
s
1 + ln+ (
1
kf kE kf kB1 ;1 ) 1
Proof: According to Theorem 18.1, we may assume that E = M 1;d. We want to estimate
kf kbmo
Z Z
1
1 t = ( sup sup d=2 jesf (x)j2 ds dx)1=2 : 0
But for f 2 B11;1 and 0 < s < 1, we have that kesf k1 es f is smooth on (0; 1) IRd . Thus, we have
k jes f j2 kM ;d C 1
R
C p1s kf kB1 ;1 ; 1
kf kM ;d kf kB1 ;1 ps ; 1
1
p
hence, k 0t jes f j2 dskM 1;d C kf kM 1;d kf kB11;1 t. This gives kf kbmo 1 q C kf kM 1;d kf kB11;1 C 0 kf kM 1;d . To prove the more precise estimate given in Theorem 18.2, we may assume that f 6= 0 and that we have kf kB11;1 < kf kM 1;d =2 (in the other case, we
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would have kf kM 1;d 2kf kB11;1 1 + ln+ ( kkffkkM 1;d1;1 ) 1=2 ). We de ne = B1 kf k2B 1;1 d 1 kf k2M 1;d 2 (0; 1=4) and we write, for t 2 (0; 1) and x0 2 IR , R tR
s 2 0 jx x jpt je f (x)j ds dx s 2 0 jx x jpt je f (x)j ds dx + I1 (t; x0 ) +
R t R
0
Rt R
p sf (x)j2 ds dx I2 (t; x0 ): kf k 1;d kf k 1;1 We estimate I1 with the inequality k jesf j2 kM 1;d C M ps B1 , and kf k2 1;1 we estimate I2pwithpthe inequality k jes f j2 kM 1;d C Bs1 . Thus, we get I1 (t; x0 ) C t( t)d 1 kf kM 1;d kf kB11;1 = Ctd=2 kf k2B11;1 and I2 (t; x0 ) p C ln 1 ( t)d kf k2B11;1 = 2Ctd=2kf k2B11;1 ln( kkffkkM 1;d1;1 ). = =
0
t jx x0 j t je
B1
2. A persistency theorem
A consequence of Theorem 18.2 is that we must study the size of the norm of ~u0 in (bmo( 1) )d to prove the existence of a mild solution for the Navier{Stokes equations associated with ~u0 . When ~u0 belongs more precisely to E d, where E is a shift-invariant Banach space of local measures, the solution ~u provided by the Koch and Tataru theorem is then expected to remain in E d . This is no longer a theorem on the existence of solutions; this is rather a regularity theorem for the solutions associated with regular data. What is important is that this point may be drastically separated from the problem of existence: if E is any shift-invariant Banach space of local measures and if ~u0 2 (E \ bmo( 1) )d , then if the Picard contraction principle gives a solution in the path space associated with bmo( 1), this solution will remain in E d, however big the norm of ~u0 is in E d . We shall call such a result a persistency theorem (following Furioli, Lemarie-Rieusset, Zahrouni and Zhioua [FURLZZ 00]).
De nition 18.1: (Exponential convergence of the Navier{Stokes Picard sequence) ~ :~u0 = 0. The Navier{Stokes Picard sequence Let ~u0 2 (bmo( 1) )d with r associated with ~u0 is de ned by induction as ~u(0) = et ~u0 and ~u(n+1) = et~u0 R t (t s) ~ :(~u(n) ~u(n) ) ds. IPr 0e We say that the Navier{Stokes Picard sequence associated with ~u0 is exponentially convergent on (0; T ) if we have, for all T0 2 (0; T ), p lim sup k t(~u(n+1) ~u(n) )k1L=n 1 ((0;T0 )IRd ) < 1: n!1 ~ :~u0 = 0 is such that the Navier{ If ~u0 2 (bmo( 1) )d with r Stokes Picard sequence associated with ~u0 is exponentially convergent on (0; T ) Lemma 18.1:
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Mild solutions
IRd , then the sequence converges to a solution ~u of the Navier{Stokes equations.
Proof: Of course, ~u(n) is a Cauchy sequence in (L1 ((t0 ; T0 ) IRd))d for every 0 < t0 < T0 < T ; hence, we may consider the limit ~u of ~u(n) as n goes to 1. We thus obtain, from @t ~u(n+1) = ~u(n+1) IPr~ :~u(n) ~u(n) , that @t~u = ~ :~u ~u. ~u IPr Remark: The solution provided by the Koch and Tataru theorem (for ~u0 2 (bmo( 1) )d ) is given as the limit of an exponentially convergent Navier{Stokes Picard sequence. Theorem 18.3: (Persistency theorem for local measures) Let E be a shift-invariant Banach space of local measures and let ~u0 2 E d ~ :~u0 = 0. If ~u0 2 (bmo( 1) )d is such that the Navier{Stokes Picard with r sequence associated to ~u0 is well de ned and exponentially convergent on (0; T ) IRd , then the sequence converges to a solution ~u of the Navier{Stokes equations, which satis es the regularity property ~u 2 \0
p
Remark: A suÆcient condition to grant that limt!1 tk~u(t; :)k1 = 0 is that ~u0 2 (B1 ;1 )d for some < 1 (see Chapter 20) or that ~u0 2 (cmo( 1) )d , where cmo( 1) is the space of smooth elements in bmo( 1) (see Chapter 16). Proof: The proof is based on the identity relating w~ (n) = ~u(n+1) ~u(n) to ~u(n) : w~ (n) = B (w~ (nR 1) ; ~u(n)) B (~u(n 1) ; w~ (n 1) ), where B is the bilinear operator ~ g)(t; x) = 0t e(t s)IPr ~ :f~ ~g ds. B (f;~ We de ne Mn = max0jn sup0
kw~ (n) (t)kE CE
pt1 s p1s pskw~ (n 1) (s)k1 (k~u(n) kE + k~u(n 1)kE ) ds: 0
Z t
Due to the exponential convergence of the Navier{Stokes Picard sequence, we know there exists a positive constant C0 and a positive 2 (0; 1) so that, for
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p
every n 2 IN, sup0
Mn M1 = M1
1 Y j =0
(1 + C C0 j ) < 1
P and thus, 1 n=0 n < 1. Now, we check the continuity of ~u. The continuity at t = 0 of ~u et~u0 is obvious, since we have for a positive constant CE
k~u et~u0kE = kB (~u; ~u)kE CE
sup k~ukE sup
0<s
0<s
psk~u(s; :)k : 1
The continuity for t > 0 of ~u is easy as well: we already know that t 7! et~u0 is continuous from (0; +1) to E (Proposition 4.4); for w~ = B (~u; ~u), we write, for 0 < t < T0 and t=2 < < min(3 t=2; T ), de ning 1 = tR 2jt j, w~ ( ) w~ (t) = R ( s0) ( ) ( t ) 1 1 ~ :~u ~u ds + t e(t s) IPr ~ :~u ~u ds, (e e )w~ ( 1 ) 1 e IPr 1 which gives
kw~ ( ) w~ (t)kE C j ln t j
sup kw~ kE + C
0<s
0
r
j1 t j
p
sup k~ukE sup sk~uk1 :
0<s
0
0<s
0
Finally, we note that the *-weak continuity at t = 00 is provided by the weak continuity in S 0 and the boundedness in E = (E () ) . 3.
Some alternate proofs for the existence of global
solutions
We rst check that the proof based on maximal functions given for Ld by C. Calderon [CAL 93] and by Cannone [CAN 95] may easily be adapted to the case of Lorentz spaces Ld;q (1 q 1). Moreover, the persistency theorem states that the Koch and Tataru solutions satis es similar maximal estimates when ~u0 belongs to a homogeneous Morrey{Campanato spaces M_ p;q (1 < p q).
Theorem 18.4: (Maximal functions and Navier{Stokes equations) Let E be a Lorentz space Lp;q (IRd ) (1 < p < 1, 1 q 1) or a homogeneous Morrey{Campanato space M_ p;q (1 < p q ). Let EE be the space of measurable functions F (t; x) de ned on (0; 1) IRd so that there exists a function F 2 E with jF (t; x)j F (x) almost everywhere on (0; 1) IRd , normed with kF kEE = minjF jF kF kE . A) f 2 E , et f 2 EE . B) Let E be a Lorentz space Ld;q (IRd ). Then EE is an admissible path space ~ g)(t; x) forRthe Navier{Stokes equations: the bilinear operator B de ned by B (f;~ t (t s) ~ ~ d d d = 0e IPr:f ~g ds is continuous from E E to E . In particular, there
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Mild solutions
~ :~u0 = 0 and k~u0kE < exists a positive constant d;q so that, if ~u0 2 E d with r
d;q , then the Navier{Stokes problem with initial value ~u0 has a solution ~u 2 (EE )d . C) Let E be a Lorentz space Lp;q (IRd ) (1 < p < 1, 1 q 1) or a homogeneous Morrey{Campanato space M_ p;q (1 < p q ). Let ~u0 2 E d with r~ :~u0 = 0. If ~u0 2 (BMO11;1 )d is small enough to satisfy the conclusion of the Koch and Tataru theorem for global solutions, then the Koch and Tataru solution ~u of the Navier{Stokes equations satis es the regularity property ~u 2 (EE )d . Proof: The proof of point A) relies on the observation that there exists a constant C0 so that for any f in the Morrey space L1uloc of uniformly locally integrable functions and for any positive t, we have jetf j C0 Mf (x), where Mf is the Hardy{Littlewood maximal function of f . Thus, it is enough to show that Mf 2 E whenever f 2 E . We know (Theorem 6.2) that f 2 Lp and p > 1 imply Mf 2 Lp ; hence, by interpolation of sublinear operators we nd that f 2 Lp;q and 1 < p < 1 imply Mf 2 Lp;q . Thus, we must only study the case of homogeneous Morrey-Campanato spaces E = M_ p;q , R 1 1 < p q. We simply write Mr f (x) = sup0<
j~h(t; x)j C
Z tZ
0
IRd
1
1
jt sj(d+1)=2 1 + jpxt zsj 1+d F (z )G(z ) dz ds
Since F and G belong to Ld;q , the product F G belongs to Ld=2;q . If we look at the integral with respect to s, we have t Z jx zj2 d 1 1 ds = 1+d p p d 1 j x z j 0 t s + jx z j 0 1 + 1+d
Z t
jx Czdjd 1
R with Cd = 0+1 pd1+d < 1. Thus, j~h(t; x)j Cd jxj1d 1 (F G). Since 1+ 1d 1 2 L d d 1 ;1 and since L d d 1 ;1 Ld=2;q Ld;q , we get the required estimate. jxj We now proveppoint C). The Picard sequence ~u(n) satis es, for some 2 (0; 1), supn2IN n tk~u(n+1) ~u(n) k1 < 1. By induction on n, we are able to
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prove that ~u(n) 2 E d : let f~ 2 EEd , supt>0 jf~(t; x)j F (x) and let ~g be such that p ~ g); we have t ~g 2 (L1 ((0; 1) IRd ))d and let ~h = B (f;~
j~h(t; x)j C
Z tZ
1 1 F (z ) dz d= 2 j x 0 IRd jt sj 1 + pt zsj 1+d
p pt dssps sup tkg(t; :)k1 t>0
thus, writing M p maximal function R F for the Hardy{Littlewood R of F , we nd that j~h(t; x)j C 0 0t MF (x) pt dssps supt>0 tkg(t; :)k1 with 0t pt dssps = . But we already know that F 2 E and p > 1 imply MF 2 E (see the proof of point A). Then, the proof follows the same line as for Theorem 18.3: we may use the identity on w~ (n) = ~un+1 ~un to get for some positive constant C (depending only on E and ~u0) k~un+1 ~un kE Cn (k~u(n) kE + k~u(n 1)kE ). We now describe a curious result of Gallagher [GAL 99], which deals with ~u0 2 R(RL2loc(IRd))d , depending only on the two rst variables x1 , x2 and such that j~u0 (x1 ; x2 ; x0 )j2 dx1 dx2 < 1 (where x0 = (x3 ; : : : ; xd )).
Theorem 18.5: Let E be the space E = ff 2 L2loc (IRd )= for j = 3; : : : ; d; @x@ j f = 0 RR and jfR(xR1 ; x2 ; x0 )j2 dx1 dx2 < 1g (where x0 = (x3 ; : : : ; xd )) normed with 1=2 0 2 kf kE = jf (x1 ; x2 ; x )j dx1 dx2 . Let Cx1 ;x2 be the space Cx1 ;x2 = ff 2 d Cb (IR ) = for j = 3; : : : ; d; @x@ j f = 0g, normed with the norm k:k1 of uniform
convergence. Then: i) E M 1;d (IRd ); ii) pointwise multiplication is a bounded operator from E Cx1 ;x2 to E ; iii) the convolution is a bounded operator from E L1 (IRd ) to E and from Cx1 ;x2 L1 to Cx1 ;x2 ; iv) e maps boundedly E to Cx1 ;x2 ; v) for all f 2 E , lim!0+ kf (x)kbmo( 1) = 0. ~ :~u0 = 0, the solution to the Navier{Stokes equations Given ~u0 2 E d with r with initial value ~u0 may be solved through the following methods: j) the Picard contraction principle converges on some interval [0; T ] to a solution ~u 2 C ([0; T ]; E d); this solution may be uniquely continued to a solution ~u 2 C ([0; +1); E d); jj) the solution ~u of the molli ed equations is de ned on (0; +1) IRd and converges (strongly in C ([0; T ]; E d) for every T > 0) to the unique solution ~u of the Navier{Stokes equations in Cb ([0; 1); E d ).
Proof: Points i) to iv) are obvious. In order to prove point v), we use Theorem 18.2: it is then enough (since kf (x)kE =pkf kE ) to check that lim!0 kf (x)kB11;1 = 0. This is equivalent to limt!0 tketf k1 = 0: since f may be considered as a function de ned on IR2 : f (x1 ; x2 ; x0 ) = g(x1 ; x2 ) with 2 2 g 2 L2(IR2 ) and since etf (x1 ; x2 ; x0 ) = et(@1 +@2 ) g(x1 ; x2 ) and the test functions are dense in L2 (IR2 ), this limit is easily computed.
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We now turn to point j). Since lim!0+ k~u0(x)kbmo( 1) = 0, the Koch and Tataru theorem grants that the Picard iteration sequence will converge on (0; T ) IRd for some positive T . Due to points i) to v), we nd that the iterates ~u(n) belong (for all t 2 (0; T )) to the space Cx1;x2 and that we may apply the persistency theorem (Theorem 18.3) to get that ~u(n) converge to a solution in (C ([0; T ]; E ))d. In order to get a global solution, we could conclude directly with the above analysis in the case that k~u0 kBMO( 1) is small enough to apply the Koch and Tataru theorem. But we want to prove global existence for all ~u0 , not only for the small ones. Global existence is then proved through energy estimates. For ~ 00 = (@1 ; @2 ); ~u 2 (C ([0; T ]; E ))d, ~u = ((u1 ; u2; ~u0 )), we write ~u00 = (u1 ; u2 ) and r then the Navier{Stokes equations are split in two systems: ~ 00 )~u00 r ~ 00 p and r ~ 00 :~u00 = 0 @t~u00 = ~u00 (~u00 :r and
~ 00 )~u0 : @t~u0 = ~u0 (~u00 :r The system on ~u00 are the two-dimensional Navier{Stokes equations and we know that we may nd a solution ~u00 in (Cb ([0; 1); L2 ))2 \ (L2t H_ 1 )2 and we easily check that the linear equations on ~u0 have a solution ~u0 in (Cb ([0; 1); L2 ))d 2 \ (L2t H_ 1 )d 2 . Point jj) is quite obvious. We have uniform energy estimates on ~u: ZZ Z 1Z Z 2 sup sup j~uj dx1 dx2 < 1 and sup (j@1 ~uj2 + j@2~uj2 ) dx1 dx2 < 1 >0 t>0 >0 0 The limiting process then gives the existence of a sequence that converges d weakly to a solution; this solution belongs to (L1 t E ) and its partial derivatives 2 d belong to (Lt E ) : this limit then belongs to Cb ([0; 1); E d ) and thus is equal to ~u. We now estimate the size of ~u ~u. We use the energy equality: R k~u(tR) RR~u(t)k2E + 2 0t kr~ (~u(s) ~u (s))k2E ds = t ~ ](~u ~u) dx1 dx2 ds ~u:[(~u ! ~u):r R 0tRR ~ (~u ~u ) dx1 dx2 ds + 0 ~u: [(~u ~u) ! ]:r On [0; T ], we split, for a given > 0, ~u in ~u = ~ + ~ where k ~ kL1E < and where 2 (L1 ((0; T ) IRd))d . This gives R k~u(t) ~u(t)k2E + 2 0t kr~ (~u(s) ~u (s))k2E ds ~ (~u ! ~u)kL2 E ) R0t kr ~ (~u ~u )k2E ds 21 C k~u0 kE (k~u ! ~ukL1E + kr R ~ (~u ~u )k2E ds +C k~u0k4E + C 0t kr R R t ~ (~u ~u )k2E ds +C k ~k21 0 k~u ~uk2E ds + 0t kr Now, if is small enough and if < 0 ( ; T; ~u) this gives
k~u(t) ~u (t)k2E C0 k~u0k4E + C1 k ~ k21
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Z t
0
k~u ~uk2E ds
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for two constants C0 and C1 which do not depend on . Thus, we nd that for every > 0, there exists 0 = 0( C0 k ~u0 k4 ; T; ~u) > 0 so that for < 0 , we have E k~u(t) ~u(t)k2E eC1tk ~k21 . The space E considered by Gallagher is not a space of local measures, since it is not invariant through pointwise multiplication by bounded continuous functions, but the scheme of the proof is the same as for local measures. We may consider spaces of functions that are invariant through pointwise multiplication by functions whose Fourier transforms are integrable (a proper subclass of bounded continuous functions):
Theorem 18.6: (Wiener algebra and Navier{Stokes equations) Let E be a shift-invariant Banach space of distributions. We assume that there exists a positive constant CE so that, for all T 2 E and all ' 2 S (IRd ), k'T kE CE k'^k1kT kE (so that pointwise multiplication of distributions in E by functions in the Wiener algebra F L1 is well de ned). Let ~u0 2 E d with r~ :~u0 = 0. If ~u0 2 (bmo( 1) )d is such that F ~u0 is locally integrable and F ( 1) (jF ~u0 j) 2 (bmo( 1) )d and if the Navier{Stokes Picard sequence associated with ~u0 is exponentially convergent on (0; T ) IRd in the Wiener norm, i.e., there exists two positive constants C and with < 1 so that, for all n 2 IN, kF (~u(n+1) ~u(n) )k1 Cn , then the sequence converges to a solution ~u of the Navier{Stokes equations which satis es the regularity property ~u 2 \0
Proof: The same proof as for the persistency theorem (Theorem 18.3) may be used.
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Chapter 19 Regular initial values 1. Cannone's adapted spaces
In his book [CAN 95], Cannone studied Banach spaces X so that the bilinear operator B de ned by
~ g)(t) = B (f;~
Z t
0
~ :f~ ~g ds e(t s) IPr
is bounded from L1 ((0; T ); X d) L1 ((0; T ); X d) to L1 ((0; T ); X d). (In our presentation, X is an adapted value space with L1 ((0; T ); X d) as admissible path space.)
De nition 19.1: (Cannone's adapted spaces) According to Cannone, a Banach space X is adapted to the Navier{Stokes
equations if the following assertions are satis ed: a) X is a shift-invariant Banach space of distributions. b) the pointwise product between two elements of X is still well de ned as a tempered distribution c) there is a sequence of real numbers j > 0; j 2 ZZ, so that X
j 2ZZ
2 jjj j < 1
and so that
8j 2 ZZ; 8f 2 X; 8g 2 X j j (fg)j X j j f j X j gj X Theorem 19.1: (Cannone's theorem for adapted spaces) Let X be a Banach space adapted (according to Cannone) to the Navier{ ~ :u~0 = 0, there is a T = Stokes equations. Then, for all u~0 2 X d so that r T (j u~0 j X ) and a unique solution ~u of the Navier{Stokes equations on (0; T ) 189 © 2002 by CRC Press LLC
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Mild solutions
with initial value u~0 so that ~u et u~0 2 C ([0; T ); X d ). More precisely, we have ~u etu~0 2 C ([0; T ); (B_ X0;1 )d ) and we have the inequality:
8t 2 (0; T ) j ~u et u~0j B_ X; C (t 0 1
X
4j t1
2j j +
X
4j t>1
2 j j ) j u~0 j 2X
Proof: The proof is based on the Picard iteration scheme. The main point is to prove that the bilinear transform B is continuous on L1 (X d). It is enough to notice the following properties of this operator: ~ : ) is a matrix of convolution operators; ) e(t s) IP(r ) since X is a shift-invariant Banach space of distributions, we have for f 2 X and g 2 L1 kf gkX kf kXPkgk1;
) for all j we have j = ( kk==jj+22 k ) j ; Æ) we may conclude that
j e(t s) IP(r~ :j (~u ~v ))j X C min(2j ; 4 j (t s) 3=2 ) j j (~u ~v)j X The continuity of the bilinear transform is then easily established, writing
j B (~u; ~v)j X j B (~u; ~v)j B_ X; 0 1
R and estimating the integral 0t
P
Z t X
0 j2ZZ
j 2ZZ
j e(t s) IP(r~ :j (~u ~v))j X ds
min(2j ; 4 j (t s) 3=2 ) j ds.
If we look accurately at the de nition of an adapted space, we may see that the theorem is not really a result of existence of solutions but a result of regularity at t = 0 of mild solutions. Indeed, the Navier{Stokes equations may be solved in a greater space: all Cannone's adapted spaces are included in the Morrey{Campanato space M 2;d, and we know the existence of a local solution ~ :~u0 = 0 for the Navier{Stokespequations with initial value ~u0 2 (M 2;d )d with r provided that limt!0 tket~u0 k1 = 0 (see Chapter 17). We easily check the following inclusion result:
Proposition 19.1: Let X be an adapted space (according to Cannone) for the Navier{Stokes
equations and assume that i) S is *-weakly dense in X ; ii) there is a Banach space F continuously embedded into S 0 (IRd ) so that (f; g) 7! fg is bounded from X X to F . p Then X is continuously embedded in M 2;d and, for all f 2 X , limt!0 tketf k1 = 0.
Proof: The rst point is to check that X is embedded into L2loc. Let K be a compact subset of IRd and !K a compactly supported nonnegative C 1 function
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on IRd so that !K is identically equal to 1 in a neighborhood of K . There R exists a constant DK so that for all f 2 S , we have IRn jf (x)j2 !K (x) dx DK kf 2kF CDK kf k2X (we consider real-valued functions). We may conclude (according to Cauchy{Schwarz) that for all f 2 S and all ' 2 DK we have
j
Z
IRd
p
f (x) '(x) dxj CDK kf kX k'k2
Now, if T 2 X , we approximate T by a sequence of functions in the Schwartz class and get that for all ' 2 DK we have p
j hT j'i j CDK kT kX k'k2 Thus, X L2loc (IRd ; dx). Since the norm of X is shift invariant, the embedding X L2loc is equivalent to X L2uloc. We may de ne the following numbers:
(r) =
Z
sup jf (y)j2 dy d j f j X 1; x2IR jx yjr
We estimate (r) for r 1. We select a smooth nonnegative function , which is equal jxj 1 and to 0 for jxj 2, and we write f 2 = P to 1 on 2 2 S0 (f ) + ( j0 j (f )) = g + h. It is easy to see thatR g is bounded (since f 2 2 L1uloc and that S0 mapsPL1uloc into L1) so that j g(y) ( y r x ) dy j Crd j f j 2X . We may write h as 1jd @j hj , where hj R2 X and j hj j X C j f j 2X ; R y x ) dy j P 1=r j hj (y) @j ( y r x ) dy j we then have j h ( y ) ( r p p 1j d R C rd=2 1 (2r)j f j 2X C 0 rd=2 1 (r)j f j 2X ; since jx yjr jf (y)j2 dy p R (g(y)+h(y)) ( y r x ) dy , we obtain that for r 1, (r) C (rd + (r)rd=2 1 ), which gives (r) Crd 2 ; thus, X is embedded in the Morrey{Campanato space M 2;d. p This implies that X B11;1 (and also that sup0p t1 t j etf j 1 C j f j X ). We are now going to show that we have limt!0 tj etf j 1 = 0. Let us de ne Cj = sup j j f j 1 : j f j X 1 Since X is invariant through dyadic dilations, we see that Cj is slowly varying: Cj 1 9 > 1 8j 2 ZZ P
Cj+1 . If fj satis es j fj j X = 1 and jj fj (0)j 1=2 Cj , then fj fj+3 = j+1kj+5 k (fj fj+3 ); hence, X
1=4 Cj Cj+3 (
j +1kj +5
Ck ) (
X
j lj +6
l )
Thus, we get limj!+1 2 j Cj = 0. Then, write for f 2 Xp f = SN f + P t j N j f = fN + gN ; for t 2 (0; 1] and N 1 we have: t j e f j 1
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Mild solutions
p p p t j et fN j 1 + t j et gN j 1 t j fN j 1 + C j gN j B1 ;1 C (2N t + supjN 2 j Cj )j f j X . p
1
We have shown that the spaces studied by Cannone are subspaces of one space, the space M 2;d. Moreover, for all the examples quoted in [CAN 95] [Sobolev spaces Hps (p < d; s > dp 1), Morrey{Campanato spaces M 2;p (p > d), Lebesgue spaces Lp; (p > d)], we have the property that (19:1)
8f; g 2 X \ L1 kfgkX C (kf kX kgk1 + kgkX kf k1);
we know that the bilinear transform B is bounded p on 1ETd , where ETd is the 1 2 ;d admissible path space L ((0; T ); M ) \ ff = tf 2 L ((0; T ) IR )g, and we obviously nd (using inequality (19.1)) that it is bounded on FTd , where FT is the admissible path space ET \ L1 ((0; T ); X ). This gives another proof of Theorem 19.1 (under the assumption that (19.1) is valid). 2. Sobolev spaces and Besov spaces of positive order
While Cannone developped an abstract theory of adapted spaces, all the examples which have been discussed [CAN 95] were either shift-invariant Banach spaces of local measures (as, for example, Morrey{Campanato spaces Kato [KAT 92], Taylor [TAY 92]) or Besov spaces or potential spaces with positive regularity over local measures, such as H_ ps (Kato and Ponce [KATP 88]). See also for Besov spaces Planchon [PLA 96] and Terraneo [TER 99]. All the latter examples may be treated in a much easier way using the regularity property given by inequality (19.1):
De nition 19.2: (Pointwise product regularity) Let E be a shift-invariant Banach space of distributions. Then E has the pointwise product regularity property if
(19:2)
8f; g 2 E \ L1 kfgkE C (kf kE kgk1 + kgkE kf k1);
Lemma 19.1:
Let F be a shift-invariant Banach space of local measures. Then, for any positive s, the Besov spaces BFs;1 and B_ Fs;1 have the pointwise product regularity property.
Proof: Just apply Lemma 4.2 (i.e., use paraproduct operators). Theorem 19.2: [Besov regularity and Navier{Stokes] (A) Let E be a shift-invariant Banach space of distributions with the pointwise product regularity property. Then:
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p
193
a) Let EE = L1 ((0; 1); E d ), B1 = f~u = t ~u 2 (L1 ((0; 1); Cb (IRd ))d g and let FE = EE \ B1 . The bilinear operator B de ned by
B (~u; ~v) =
Z t
0
~ :(~u ~v) ds e(t s)IPr
is bounded from FE FE to EE . B p(~u; ~v) 2 C ((0; 1]; pE d) and converges *weakly to 0 as t goes to 0. If limt!0 tk~uk1 = limt!0 tk~v k1 = 0, then the convergence is strong. b) If E is continuously embedded in the Besov space B11;1 , then the bilinear operator B is bounded as well from FE FE to B1 . Hence, there exists a ~ :~u0 = 0) with k~u0kE E , the constant E > 0 so that, for all ~u0 2 E d (with r initial value problem for the Navier{Stokes equations with initial data ~u0 has a solution ~u = et ~u0 B (~u; ~u) with ~u 2 EE \ B1 . c) If moreover E is continuously imbedded in the Besov space B_ 11;1 , then ~ :~u0 = 0) with there exists a constant ÆE > 0 such that, for all ~u0 2 E d (with r k~u0kE E , the initial value problem for the Navier{Stokes equations with initial data p ~u0 has a global solution ~u = et~u0 B (~u; ~u) in L1 ((0; 1); E d) with sup0
Proof: The proof is similar to the proof in the case of local measures (Theorem 17.2). The boundedness of B from FE FE to EE is obvious, since ke(t s)IPr~ :kL(Edd;Ed) C pt1 s :
pt 1 sps ds sup (kf~kE psk~gk1 + k~gkE pskf~k1 ): 0<s
Z t
2
1
1
2
Thus, point a) is proved. ~ :(f~ ~g)k1 Point b) is obvious. Indeed we write, for s > t=2, ke(t s)IPr 1 1 ( t s ) ~ ~ C pt s s kf kB1 k~gkB1 and, for s < t=2 and t < 1, ke IPr:(f~ ~g)k1 C t 1 s p1s (kf~kB1 k~gkEE + k~gkB1 kf~kEE ). Point c) is treated in the same way. Now, we prove point B). We just notice that, when B is any shift-invariant Banach space we have, for 2 ( 1; 1) and b 2 B dd, the p of distributions, ~ :bkB C ;B kbkB (t s) (1+ )=2 . We interpoestimate k( ) e(t s)IPr _ _ ~ :bkB late, since [HB ; HB ]1=2;1 = B_ B0;1 (see Chapter 4), and get ke(t s)IPr
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Mild solutions
C ;B kbkB (t s) 1=2 . We conclude the proof as follows: for E = HFs or E = BFs;q , we have E BFs;1 ; BFs;1 has the pointwise product regularity property, so that we may apply the results obtained in point A); moreover, we have a more precise result, and nd that the bilinear operator B maps F F to E with F = L1 ((0; 1); (BFs;1 )d ) \B1 and E = L1((0; 1); (B_ B0;s;1 1 )d ); we then F easily conclude, since B_ B0;s;1 1 BB0;s;1 1 = BFs;1 E . The case of E = H_ Fs or F F E = B_ Fs;q is similar, using the embeddings B_ B0_;s;1 1 = B_ Fs;1 E B_ Fs;1 . F
We now present an example of Iftimie [IFT 99], which may be seen as a generalization of Gallagher's construction.
Proposition 19.2: We de ne the partial Littlewood-Paley decomposition operators Sj[d 2] and [jd 2] as the tensor products Sj[d 2] = IdS 0 (IR2 ) Sj(d 2) and [jd 2] = IdS 0 (IR2 ) (jd 2) , where Sj(d 2) and (jd 2) are the LittlewoodPaley operators on S 0 (IRd 2 ). Let B be the Banach space of distributions de ned P d 2 by f 2 B , j 2ZZ 2j 2 k[jd 2] f k2 < 1. Then B is contained in B_ 11;1 and has the pointwise product regularity property.
Proof: The pointwise regularity property is easily proved by using a d dimensional paraproduct decomposition of the pointwise product.
2-
3. Persistency results
When we look for global solutions in Besov spaces BEs;q , we apply the same method as for global solutions in shift-invariant Banach spaces of local measures: we have a persistency theorem that states that the Koch and Tataru solutions retain their initial regularity (see Furioli [FUR 99] and Furioli, Lemarie{Rieusset, Zahrouni and Zhioua [FURLZZ 00]).
Theorem 19.3: (Persistency theorem for Besov spaces or potential spaces over local measures) Let F be a shift-invariant Banach space of local measures and let E be a space of regular distributions over F : for some positive s and for some q 2 [1; +1], E = HFs = (Id ) s=2 F , E = BFs;q or E = B_ Fs;q . Let ~u0 2 E d ~ :~u0 = 0. If ~u0 2 (BMO 1 )d is small enough to satisfy the conclusion with r
of the Koch and Tataru theorem for global solutions, then the Koch and Tataru solution ~u of the Navier{Stokes equations satis es the regularity property ~u 2 (L1 ((0; 1); E ))d . More precisely, in that case, p ~u 2 C ((0; T ); E d) and is *-weaklytconvergent to ~u0 in E as t ! 0. If limt!0 tk~u(t; :)k1 = 0, then limt!0 k~u e ~u0 kE = 0. In particular, a suÆcient condition to have a global solution in E is that ~u0 2 (BMO 1 )d and that k~u0kBMO 1 is small enough to grant that the Picard
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iteration scheme described by the Koch and Tataru theorem works on (0; 1) IRd . This smallness condition is independent of E and of the size of k~u0kE .
The proof relies on the following lemma:
Lemma 19.2: Let F be a shift-invariant Banach space of local measures, let s > 0, and let > 0 satisfy s=(s + 1) < < 1. Let f~ andp ~g belong to the space L1p((0; 1); (B_ Fs;1 \ B_ 11;1 )d ) and be such that supt>0 tkf~k1 < ~ g) 2 L1 ((B_ Fs;1 )d ). More precisely, 1 and supt>0 tk~gk1 < 1. Then B (f;~ p s;1 , N1 (f ) = supt>0 tkf k1 , and if we de ne the norms Ns (f ) = kf kL1 t B_ F ~ ~g), s = Ns (f~ ~g), s = 1;1 , and if we write 1 = N1 (f N~1 = kf kL1 t B_ 1 Ns (f~) + Ns (~g) and A~1 = N~1 (f~) + N~1 (~g), we have:
kB (f;~ f~) B (~g;~g)kL1t B_ Fs; C (; s; E )( 1 1
1 ( s A~1 ) )
s + ( 1 s )
Proof: We begin by proving that if f 2 B_ Fs;1 \ B_ 11;1 and g 2 B_ Fs;1 \ L1 , then we may write fg = u + v where kukB_ Fs;1 C (; s; F ) kf kB_ Fs;1 kgk1 and kvkB_ Fs ;1 C;s;F (kf kB_ Fs;1 kgk1)1 (kf kB_ 11;1 kgkB_ Fs;1 ) . In order to prove it, we use the paraproduct operators of Bony. We use the Littlewood{ Paley decomposition of f and of g and write fg =
X
j 2ZZ
Sj 2 f j g +
X
j 2ZZ
Sj+3 gj f = $_ (f; g) + _(f; g):
We easily obtain the classical estimate k_(f; g)kB_ Fs;1 C (s; F ) kf kB_ Fs;1 kgk1 (see Lemma 4.2). To prove the estimate on $_ (f; gP ), we must estimate, for all j 2 ZZ, 2j(s ) kSj 2 f j gkF . Writing Sj 2 f = kj 3 k f , we use the estimates:
kk f j gkF CF kk f kF kj gk1 C 2 ks kf kB_ Fs;1 kgk1 and
kk f j gkF CF kk f k1kj gkF C 2k kf kB_ 1 ;1 2 js kgkB_ Fs;1 1
which give (writing kk f j gkF = (kk f j gkF )1 (kk f j gkF ) )
kk f j gkF C 2k((s+1) s) 2 and summing over k j
js (kf k s;1 B_ F
kgk1)1
(kf kB_ 11;1 kgkB_ Fs;1 )
3 we get (since (s + 1) s > 0)
kSj 2 f j gkF C 2j( s) (kf kB_ Fs;1 kgk1)1
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(kf kB_ 11;1 kgkB_ Fs;1 ) :
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Mild solutions
~ f~) B (~g;~g) = B (f; ~ f~ ~g)+ Now, the proof is easy. We write B (f; ( t ) ~ ~ B (f ~g;~g). Then we notice that the operator e IPr: maps the Besov space (B_ F;1 )dd to the Besov space (B_ F;1 )d for 1 < 1 with an operator norm O((t ) 1=2 ( )=2 ); hence, (by interpolation) maps (B_ Fs;1 )dd to (B_ Fs;1 )d with an operator norm O((t ) 1=2 ) and (B_ F ;1 )dd to (B_ Fs;1 )d with an ~ f~ ~g)(t)kB_ s;1 operator norm O((t ) 1=2 =2 ). Thus, we nd kB (f; F Rt 1 p 1 C (s; F )A + C (s; F; )B with A = 0 pt p kf~( ) ~g( )k1 kf~( )kB_ Fs;1 d R and B = 0t (t ) 1=2 =2 1=2+=2 !( ) d with p !( ) = kf~k1B_ s;1 kf~kB_ 11;1 kf~ ~gkB_ s;1 ( kf~( ) ~g( )k1 )1 : F F The lemma is proved. Proof of Theorem 19.3: We may assume that E is a homogeneous space H_ Fs or B_ Fs;q , since we know that we control the norm of ~u in F d (persistency _ s;1 E B_ Fs;1 . theorem for local measures). Thus, we may assume that B RF t ~ :(~u(n)
Let ~u(n) be de ned by ~u(0) = 0 and ~u(n+1) = e ~u0 0t e(t s)IPr ~u(n) ) ds, and let (w~ n ) be the sequence de ned by w~ (n) = ~u(n+1) ~u(n) The proofs relies on the identity relating w~ (n) to ~u(n): w~ (n) = B (w~ (n 1) ; ~u(n) ) B (~u(n 1) ; w~ (n 1) ): We de ne Mn = max0jn supt>0 k~u(j) kE and n = supt>0 kw~ (n) kE . We apply 0 B with Lemma 19.2 and get the estimate supt>0 kw~ (n) kE CE0 A + CE; p A = sup tkw~ (n 1) k1 (sup k~u(n)kE + sup k~u(n 1)kE ) 0
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Chapter 20 Besov spaces of negative order 1.
Lp (Lq )
solutions
In Chapter 14, we saw the uniqueness criterion of Serrin expressed in terms of Lp Lq -norms with 1p = 12 2dq . Existence of solutions in Lp Lq has been discussed by Fabes, Jones, and Riviere [FABJR 72] and more extensively by Giga [GIG 86].
Theorem 20.1: (A) Let q 2 (d; 1) and pq be de ned by p1q = 12 2dq Let p pq . Then, for ~ :~u0 = 0, there exist all ~u0 2 (S 0 (IRd ))d with et ~u0 2 (Lpt Lqx ((0; 1) IRd ))d and r p a positive T and a (unique) weak solution ~u 2 \T
Z t
1 ~ gkq ds: d kf kq k~ 1 0 (t s) 2 + 2q
q+d 2q ~ g) belongs to Since t 7! 1t>0 t 2q belongs to L q+d ;1 (IR), we nd that B (f;~ Lr Lq with r1 = p2 + ( 12 + 2dq ) 1 1p since p1 p1q = 12 2dq . Finally,
~ g)kLpLq ((0;T )IRd ) C T 1=p 1=r kf~kLpLq ((0;T )IRd ) k~gkLpLq ((0;T )IRd ) kB (f;~ and this gives the existence of a solution for the Navier{Stokes equations on the interval (0; T ) for 4CT 1=p 1=r ket~u0 kLpLq ((0;T )IRd ) < 1. This condition is full lled for T = T (~u0) small enough: this is obvious for p > pd since limT !0 T 1=p 1=r = 0; for p = pd , the condition is full lled since we have limT !0 ket~u0kLpLq ((0;T )IRd ) = 0. 197 © 2002 by CRC Press LLC
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Mild solutions
We now consider the regularity of ~u. We know, according to the regularity criterion proved in Chapter 15 (Proposition 15.1), that it is enough to prove that ~u is bounded on every strip (t1 ; t2 ) IRd with 0 < t1 < t2 < T . We rst prove that this is true in the strips close to 0 (i.e., with t2 < (~u0 )). The condition et~u0 2 (Lpt Lqx((0; 1) IRd ))d is equivalent to ~u0 2 (Bq 2=p;p (IRd))d . For p pq , 2=pq ;pq B 1;1 ; we have, according to the Bernstein inequalities, Bq 2=p;p 1 p Bq we may thus assume that p = pq and we have sup0
ET = ff 2 Lpq Lq ((0; T ) IRd ) =
p
p
sup tkf k1 < 1 and tlim !0 tkf k1g t
and we easily check that the bilinear operator B is bounded from ETd ETd to ~ g)k1 by splitting the integral on s 2 (0; t) in ETd . Indeed, we Restimate kB (f;~ R t=2 (t s) ~ ~ ~ :(f~ ~g) ds. two parts: I1 = 0 e IPr:(f ~g) ds and I2 = t=t 2 e(t s) IPr We then Rget: C 0 t1 2=pq kf~k pq q k~gk pq q i) jI1 j 0t=2 (t s)1C=2+d=q kf~kq k~gkq ds t1=2+ L L L L d=q R p p C 0 sup pskf~k sup psk~gk ii) jI2 j t=t 2 (t Cs)1=2 skf~k1 sk~gk1 dss p 1 s
p t ketw~ 0 kLpq Lq ((0;T )IRd ) + sup tke w~ 0 k1 0
=p;p
2
We propagate this estimate along the interval (0; T ), following May [MAY 02] who uses a compactness argument of Brezis [BRE 94]. We rst check that ~u belongs to (Bq 2=p;p )d ; if < T < T , we write, for 0 < t < T ,
~u(t + ) = et~u()
Z t
0
~ :(~u(s + ) ~u(s + )) ds; e(t s) IPr
since ~u 2 (LpRLq ((0; T ) IRd))d , we have ~u(t + ) 2 (Lp Lq ((0; T ) IRd ))d , ~ :(~u(s + ) ~u(s + )) ds (by the boundedness of the and so does 0t e(t s)IPr bilinear operator B on (Lp Lq )d ); hence, et~u() 2 (Lp Lq ((0; T ) IRd ))d , which shows that ~u() 2 (Bq 2=p;p )d . In particular, we get uniqueness on (0; T ) of the solution which belongs to \0
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which in turn is controlled (according to the Navier{Stokes equations and the boundedness of B ) by
k~u(t + t1 ) ~u(t + t2 )kLpLq (Q ) (1 + k~u(t + t1 )kLpLq (Q ) + k~u(t + t2 )kLpLq (Q ) ) Since the test functions are dense in Lp Lq , we see that 7! ~u(t + ) is uniformly
continuous from [0; T 2] to (Lp Lq (Q ))d and this gives the continuity of the map t 7! ~u(t) from [0; T ) to (Bq 2=p;p )d . We now apply May's proof: if T < T , the range of t 7! ~u(t), 0 t T , is a compact subset of (Bq 2=p;p )d ; thus, there are two xed p positive numbers Æ and D so that for each 2 [0; T ], we have for t 2 (0; 2Æ] tk~u(t + )k1 D, nding that ~u is controlled by Dt 1=2 on (0; Æ] and by DÆ 1=2 on [Æ; T ]. Thus, ~u is smooth. (B) is a direct consequence of the proof given for (A). The norm of et~u0 in (Lpd Lq ((0; 1) IRd))d is equivalent to the norm of ~u0 in the homogeneous Besov space (B_ q 2=pq ;pq )d . We note that, for d = 3 or 4, this yields a new proof of the Kato theorem:
Proposition 20.1: (Kato's theorem) ~ :~u0 = 0, there exist Let d = 3 or d = 4. For all ~u0 2 (Ld (IRd ))d such that r d d a positive T and a weak solution ~u 2 C ([0; T ); (L ) ) for the Navier{Stokes equations on (0; T ) IRd so that ~u(0; :) = ~u0 . This solution may be chosen so that for all T 2 (0; T ), we have ~u 2 (L4t L2xd ((0; T ) IRd ))d . With this extra condition on the L4 L2d norm, such a solution is unique and, moreover, it is smooth on (0; T ) IRd . This solution is equal to the Kato solution provided by Theorem 15.3.
Proof: We borrow the proof from Planchon's thesis [PLA 96]. If we look for a solution in (L4t L2xd ((0; T ) IRd))d , we see that ~u0 must belong to (B2d1=2;4 )d . We know that Ld(IRd ) Bd0;d B2d1=2;d (for d 2), and then Ld(IRd ) Bd 1=2;4 when d = 3 or d = 4. Thus, when d = 3 or d = 4, when ~u0 2 (Ld)d , we may construct a solution so that for some positive T and for all T 2 (0; T ) we have ~u 2 (L4t L2xd ((0; T ) d IRd ))d . We prove now that ~u belongs to (Ld )d : if '~ (x) 2 (L d 1 (IRd ))d , we have Z
Z
Z
t t j ~u(t; x):~'(x) dxjk~u0 kdk'~ k d d +C ( ke(t s)r~ '~ k2d d ds)1=2 ( k~uk42dds)1=2 0 0 1
1
R1
Rt
~ '~ k2 d ds)1=2 by ( 0 ketr ~ '~ k2 d ds)1=2 , and we may control ( 0 ke(t s)r d 1 d 1 ~ '~ in (B_ d1;2 )dd; we use the embedding or equivalently by the norm of r d 1 d 0 ; 2 _ d 1 L B d to get the inequality d 1 Z
Z t
j ~u(t; x):~'(x) dxj k'~ k d d k~u0kd + C ( k~uk42d ds)1=2 : 1
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0
200
Mild solutions
d Thus, we have ~u 2 ((L d 1 )0 )d = (Ld )d . In particular, we have proved that B maps (L4t L2xd)d (L4t L2xd)d to (L1 Ld)d ; since the test functions are dense in L4t L2xd, we may improve this result and conclude that B maps (L4t L2xd )d (L4t L2xd)d to (Cb ([0; T ]; Ld)d . This proves the proposition. Since we know that the solutions in (L4t L2xd)d are bounded by Ct 1=2 , we nd that they are the same as the solutions provided by Kato's theorem. 2. Potential spaces and Besov spaces
The results on Lp Lq solutions may be easily generalized to the setting of initial data in Besov spaces with negative exponent over shift-invariant spaces of local measures. The case of Besov spaces on Lp has been investigated by Cannone [CAN 95] and the case of Banach spaces on Morrey{Campanato spaces has been investigated by Kozono and Yamazaki [KOZY 97]. A useful tool will be again the regularizing properties of the heat kernel:
Theorem 20.2: Let F be a shift-invariant Banach space of local measures and let 2 ( 1; 0) and q 2 [1; 1]. (A) Let E =pBF;q , E = ff~ 2 L1 ((0; 1); F d ) = t =2 kf~kF 2 (Lq \L1 )((0; 1); dtt )g, B1 = f~u = t ~u 2 (L1 ((0; 1); Cb (IRd ))d g and F = E \ B1 . Then: a) The bilinear operator B de ned by B (~u; ~v ) =
Z t
0
~ :(~u ~v) ds e(t s)IPr
is bounded from F F to E . b) If F is continuously embedded in the Besov space B11 ;1 (or, equivalently, if E is continuously embedded in the Besov space B11;1 ), then the bilinear operator B is bounded as well from F F to B1 . Hence, there exists a ~ :~u0 = 0) with k~u0 kE E , the constant E > 0 so that, for all ~u0 2 E d (with r initial value problem for the Navier{Stokes equations with initial data ~u0 has a solution ~u = et ~u0 B (~u; ~u) with ~u 2 E \ B1 . (B) Similarly, if F is continuously embedded in the homogeneous Besov space B_ 11 ;1 and if E = B_ F;q (so that E is continuously embedded in the Besov space B_ 11;1 ), then there exists a constant ÆE > 0 so that, for all ~u0 2 E d ~ :~u0 = 0) with k~u0 kE ÆE , the initial value problem for the Navier{ (with r Stokes equations with initial data ~u0 has a global solution p ~u = et~u0 B (~u; ~u) dt = 2 q 1 with t k~ukF 2 (L \ L )((0; 1); t ) and sup0
We begin the proof with an easy lemma:
LemmaR 20.1: For 2]0; 1=2[ and q 2 [1; 1], the operator f F (t) = 0t pt 1sps st f (s) ds is bounded on Lq ((0; 1); dtt ).
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7! F
where
Singular Besov spaces
201
R R Proof: For q = 1, this is obvious since 0t pt 1sps st ds = 01 p1 1sps sds < 1; similarly, for q = 1, we just write Z +1 Z +1 pt 1 sps st dtt = ptt 1 dtt < 1: s 1 s For 1 < q < 1, the result follows by interpolation.
Proof of Theorem 20.2: We prove only point (A), since point (B) is proved in exactly the same way. To prove (a), we note that the boundedness of B from F F to E is a direct consequence of Lemma 20.1; more precisely, B is bounded from E B1 to E and from B1 E to E . ~ g)k1 by using the inequality, for 0 < s < To prove (b), we estimate kB (f;~ t s (1+ ) =2 kf k 1 ;1 ): t < 1, ke 2 f k1 C (t s) B1 Rt C ~ gk1 ds ~ g)(t)j 0 jB (f;~ 1+ kf kF k~ 2 R t (tC s) 0 (t s)1+ 2 s 1ds2 kf~kE k~gkB1 = C t 1=2 kf~kE k~gkB1 with C < 1 since 2 ( 1; 0). Finally, we discuss the approach of global solutions by the use of maximal functions (as developped for Ld by Calderon [CAL 93] and Cannone [CAN 95]):
Proposition 20.2: For d < p < 1 and = 1 d=p, there exists 0 (d; p) > 0 (which depends only on p and on the dimension d) so that if ~u0 2 (H_ p (IRd ))d ~ :~u0 = 0 and k~u0kH_ < 0 (d; p), there exists a weak solution ~u for the with r p Navier{Stokes equations on (0; +1) IRd with initial value ~u(0; :) = ~u0 so that sup0
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Mild solutions
and the proposition is proved. 3. Persistency results
One more time, Theorem 20.2 is not a new existence result, since we could apply the Koch and Tataru theorem:
Proposition 20.3: Let 2 ( 1; 0) and let F be a shift-invariant Banach space of local measures. Then: (A) If F is continuously embedded in the Besov space B11 ;1 , then F is more d precisely continuously embedded in Rthe Morrey{Campanato space M = M 1; 1+ (de ned by supx0 2IRd ;0
Proof: We prove only (A), (B) can be proved in the same way. We mimic the proof of Theorem 18.1, starting from the inequality
kf kMuloc C 1
sup
!2C0 ;k!k1 1
k!f kB1
1
;1 ;
and we write
1 = sup0<<1 k1+ f (x)kMuloc C sup0<<1 sup!2C0 ;k!k11 k1+ !(x)f (x)kB11 ;1 = C sup0<<1 sup!2C0 ;k!k11 k1+ !(x)f (x)kB11 ;1 C 0 sup!2C0 ;k!k1 1 k!f kB11 ;1 C 00 sup!2C0 ;k!k11 k!f kF C 000 kf kF : ;1 B 1;1 , we write for x 2 IRd This gives F M . Now, for f 2 E = BM 0 1 and 0 < t < 1
kf kM
R
R
t s 2 jx x0 j
p R C ( t)d 1 0t kesf kM kes f k1 ds R p C ( t)d 1 kf k2E 0t s=2 s 1=2 ds C 0 td=2kf k2E ;
;1 bmo 1 . this proves that BM
Theorem 20.2 is primarily a regularity result for the Koch and Tataru solutions when the initial value belongs more precisely to a Besov space BF;q . Again,
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203
this regularity can be handled with a persistency theorem (Furioli, Lemarie{ Rieusset, Zahrouni and Zhioua [FURLZZ 00]):
Theorem 20.3: (Persistency theorem for Besov spaces with negative exponents) Let F be a shift-invariant Banach space of local measures and let E be a homogeneous Besov space of singular distributions over F : for some negative ~ :~u0 = 0. If ~u0 2 s and for some q 2 [1; +1], E = B_ Fs;q . Let ~u0 2 E d with r (BMO 1 )d is small enough to satisfy the conclusion of the Koch and Tataru theorem for global solutions, then the Koch and Tataru solution ~u of the Navier{ Stokes equations satis es the regularity property t s=2 k~ukF 2 Lqt ((0; 1); dtt ). Proof: Let the sequence ~u(n) be de ned by ~u(0) = 0 and ~u(n+1) = et~u0 R t (t s) ~ :(~u(n) ~u(n) ) ds, and let (w~ n ) be the sequence de ned by w~ (n) = IPr 0e ~u(n+1) ~u(n) The proof relies, as usual, on the identity relating w~ (n) to ~u(n): w~ (n) = B (w~ (n 1) ; ~u(n) ) B (~u(n 1) ; w~ (n 1) ): We then de ne the norm kf (t; x)kF;s = kt s=2 kf kF kLq ( dtt ) and the quantities Mn = max0jn k~u(j) kF;s and n = kw~ (n) kF;s . Starting from the estimate
kw~ (n) kF C
Z t
pt1 s kw~ (n 1) (s)k1 (k~u(n)(s)kF + k~u(n 1)(s)kF ) ds; 0
we apply Lemma 20.1 to get the estimate
p kw~ (n) kF;s CE sup tkw~ (n 1) k1 (k~u(n) kF;s + k~u(n 1)kF;s ) 0
But we know that for a global Koch and Tataru solution given by a Picard iterative scheme we have lim sup sup kw~ (n) k1 1=n < 1: n!1 t>0
We nd that for some constants C0 and , which depend on E and ~u0 , with 0 C0 and 0 < 1, we have
n C0 n 1 Mn : Q P1 j We get that Mn M1 = M1 1 j =0 (1 + C0 ) < 1 and thus n=0 n < 1.
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Chapter 21 Pointwise multipliers of negative order
In the classical Lp LR q Runiqueness criterion of Serrin for Leray solutions, one estimates integrals 0t IRd u(s; x)v(s; x)@j w(s; x) dx where w 2 L2t H_ x1 (so 2 that @j w 2 L2t L2x) and v 2 L2t H_ x1 \ L1 t Lx ; one then uses the embeddings 2 r r d t L2t H_ x1 \ L1 t Lx Lt H_ x with 1= = r=2 and H_ p(IR ) L with 1=t = 1=2 r=d; q thus, the integral is well de ned when u 2 Lt Lx with 1 = 1=2 + 1=t + 1=q = 1 r=d + 1=q (hence q = d=r) and 1 = 1=2 + 1= + 1=p = r + 1=p (hence 2=p = 1 d=q = 1 r). One direct way to generalize this result is to study functions u(s; x) 2 Lp X_ r with 2=p = 1 d=q = 1 r and with the space X_ r de ned by the condition that the pointwise product between a function in X_ r and a function in H_ r belongs to L2 . An equivalent requirement is that the pointwise product between a function in X_ r and a function in L2 belongs to H_ r . Since we have a loss of regularity, we shall say that elements in X_ r are pointwise multipliers of negative order. 1. Multipliers and Morrey{Campanato spaces
We rst recall the de nition of Morrey{Campanato spaces:
De nition 21.1: (Morrey{Campanato spaces) a) For 1 < p q 1 the Morrey{Campanato space M p;q is de ned by: f 2 M p;q if and only if f is locally Lp on IRd and kf kM p;q < 1 where
kf kM p;q =
sup sup Rd=q d=p kf (y)1B(x;R)(y)kLp(dy)
x2IRd 0
For p = 1, we require that f is a locally nite measure so that
sup sup
x2IRd 0
Rd(1=q
1)
Z
B (x;R)
djf j(y) < 1
b) The closure of S in M p;p is called Ep .
It is easy to check that (for p < 1) f belongs to EpRif and only if f is locally Lp and f vanishes at in nity in the sense that limx0 !1 jx x0j<1 jf (x)jp dx = 0. 205 © 2002 by CRC Press LLC
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Mild solutions
The space Xr (IRd ) of pointwise multipliers which map L2 into H r is de ned in the following way:
De nition 21.2: (Pointwise multipliers of negative order.) For 0 r < d=2, we de ne the space Xr (IRd ) as the space of functions, which are locally square-integrable on IRd and such that pointwise multiplication with these functions maps boundedly H r (IRd ) to L2 (IRd ). The norm of Xr is given by the operator norm of pointwise multiplication:
kf kXr = supf kfgk2 = kgkH r 1g We have the following comparison between Xr and Morrey{Campanato spaces:
Theorem 21.1: (Comparison theorem) For 2 < p d=r and 0 < r we have M p;d=r Xr M 2;d=r . Proof: The injection Xr M 2;d=r is easy: indeed, since H r :Xr L2 , we have that Xr L2loc. Choosing ! equal to 1 on the unit ball of IRd, we have for R < 1: kf (y)1B(x;R)(y)kL2 (dy) kf kX k!((x y)=R)kH r (dy) C kf kX Rd=2 r r
r
For the injection M p;d=r Xr , we consider a dual formulation of the problem, introducing the predual of M p;d=r . 0 0
De nition 21.3: (The predual N0 p 0;q ) 0 For 1 q0 p0 , we de ne N p ;q as the subspacePof Lq (IRd) of functions f , which can be decomposed into an atomic series f = k2IN gk where the func0 tions gk are in Lp with compact support and satisfy the following inequalities for 0 d=p0 P the diameter dk of the support of gk : dk 1 and k2IN dd=q kgk kp0 < 1. k0 0 P 0 ;q0 d=q d=p p P N is normed by kf kN p0;q0 = inf f = kgk kp0 . k2IN dk k2IN gk 0 0 00 p00 ;q0 Remark: 0 ;1 0 It is obvious that dfor1=pq0 1<=p00p < p , we have N q q L L since kgk kp0 (C dk ) kgk kp00 . 0 0
Lemma 21.1: The dual space of N p ;q for 1 1=p + 1=p0 = 1 and 1=q + 1=q0 = 1. 0
0 0
q0 p0
N p0 ;q0
is M p;q where
0 0
Proof: It is obvious since Lpcomp N p ;q and thus (N p ;q ) Lploc . According to Lemma 21.1, we may reformulate Theorem 21.1 in the following proposition:
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207
Proposition 21.1: Pointwise multiplication is a bounded bilinear operator 0 0 from L2 (IRd ) H r to N p ;q for 1 q0 p0 < 2 and 1=q0 = 1 r=d. 0
Pointwise multiplication by L2 maps obviously H r to Lq because of the Sobolev embedding theorem (H r Ls with 1=s = 1=2 r=d0 ); it is well known that it maps more precisely H r into the Lorentz space Lq ;1 since the sharp Sobolev embedding theorem states that H r Ls;2 . Now, we must prove a still sharper Sobolev embedding estimate. We are going to introduce an atomic decomposition for functions in the Sobolev space H r , which will be transformed, by multiplication with a function in L2 , into an 0 ;q0 p atomic decomposition in N . The keyp idea for decomposing v 2 H r is to start from an atomic decompostion of ( )r v 2 L2 , where L2 is to be considered as thepreal Hardy space H2 (Stein [STE 93]), then to apply integration ( ) r to each p fractional r atom of the decomposition of ( ) v to arrive at an atom of the decomposition of v. However, in order to preserve the compactness of the supports of the atoms, it will be more convenient to work with the decomposition of v on compactly supported orthonormal wavelets (Kahane and Lemarie-Rieusset [KAHL 95], Meyer [MEY 92]), and to de ne p the atomic decomposition through the wavelet coeÆcients instead of using ( )r v. Let us recall that, if 2 L2 (IR) is a compactly supported orthonormal scaling function so that ; 0 ; : : : ; (m) are bounded functions for some m > r, if ' = : : : is the d-dimensional scaling function associated with and if ; = 1; : : : ; 2d 1 are the associated separable wavelets, then we have, decomposing f into
f=
X
k2ZZd
ck '(x k) +
X X
X
j 0 k2ZZd 12d
1
cj;k; 2dj=2 (2j x k)
and letting Qj;k be the dyadic cube Qj;k = fx= 2j x k 2 [0; 1]dg, the following equivalence of norms:
kf kH r k(ck )kl
2
k(ck )kl
2
+
+(
X X
X
j 0 k2ZZd 12d
X
12d 1
k
X X
1
j 0 k2ZZd
4jr jcj;k; j2 )1=2
4jr jcj;k; j2 2dj 1Qj;k (x) 1=2 k2
and
8p 2 (1; 1) kf kp k(ck )klp +
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X
12d 1
k
X X
j 0 k2ZZd
jcj;k; j2 2dj 1Qj;k (x) 1=2 kp
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Mild solutions In order to prove Proposition 21.1, we take u 2 L2 and v 2 H r and write X
X
X
XX
v = w0 + w = ck '(x k)+ cj;k; 2dj=2 (2j x k) 12d 1 12d 1 j0k2ZZd k2ZZd
0 0
Lemma 21.2: u w0 2 N p ;q . Proof: This is obvious. We have ' 2 Ls where 1=s + 1=2 = 1=p0, and thus, writing K = Supp ': X
jck j ku '(x k)kp0 k'ksk(ck )kl
k2ZZd
XZ 2
k2ZZd
juj2 1K (x k) dx 1=2 C kuk2kvkH r
0 0
Lemma 21.3: For = 1; : : : ; 2d 1, u w 2 N p ;q . Proof: To avoid notational burden, we drop the index . P P 1=2 . For N 2 ZZ, jr 2 dj We de ne W (x) = j 0 k2ZZd 4 jcj;k j 2 1Qj;k (x) we consider EN = fx= W (x) > 2N g. Then, 2j(r+d=2) jcj;k j > 2N implies Qj;k EN . We write FN = f(j; k) 2 IN ZZd = Qj;k EN g and we get cj;k 6= 0 ) 9N Qj;k 2 FN and Qj;k 2= FN +1 .Thus, we may write: w=
X
X
N 2ZZ Qj;k 2FN FN +1
cj;k 2dj=2 (2j x k) =
X
N 2ZZ
wN
Last, we consider for a given N the collection HN of maximal dyadic cubes in FN FN +1 , and we decompose wN into
wN =
X
X
Q2HN
Qj;k 2FN FN +1 ;Qj;k Q
cj;k 2dj=2 (2j x k) =
X
2HN
wN;Q
This is the atomic decomposition we are looking for. We rst notice that: X
Qj;k 2FN FN +1 ;Qj;k Q
4jr jcj;k j2 2dj 1Qj;k (x) 1=2 C 2N 1Q~ (x)
where Q~ j;k = fx= 2j x k 2 [ M; M ]dg with a constant M , depending only on the support of . In particular, we get that Supp wN;Q Q~ and that for 1 < s < 1 we have kwN;Qks Cs 2N drQ+d=s where dQ is the diameter of Q. We then x s; t < 1 so that 1=2 + 1=s + 1=t = 1=p0 and we write (taking as 1= = 1=2 + 1=s) 0 d=p0 0 d=p0 r+d=t dd=q ku wN;Q kp0 C ku 1Q~ k 2N dd=q dQ = C ku 1Q~ k 2N ddQ(1 1=) Q Q
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Multipliers of negative order
209
and thus XX
d
dQq0
N Q2HN
d p0
ku wN;Q kp0 C
XX
ku 1Q~ k 2N (2 ) 1
XX
N Q2HN
ddQ 22N 1
ddQ 22N 1
1
N Q2HN
This latter inequality may be rewritten as XX
d d dQq0 p0 ku wN;Q kp0 C ku
XX
1 2N (2 ) 1Q~ k
XX
N Q2HN
N Q2HN
N Q2HN
1
and thus we get
ku wkN p0 ;q0 C kuk2 k = C kuk2 k
XX
N Q2HN
X X
4N 2 (s+2) 1Q~ 1=2+1=s ks
=s 24N=(s+2) 1Q~ k11+=2+1 s=2
N Q2HN
XX
N Q2HN
X X
N Q2HN
ddQ 22N 1=2 1=s
ddQ 22N 1=2 1=s
We then use the classical result that for any sequence of positive real P numbers j and any sequence of cubes Qj we have k j j 1Q~ j k1+s=2 P C k j j 1Qj k1+s=2 : indeed, let LR be the dual space of L1+s=2 , then we have for g 2 L and x 2 Qj : jQ1j j Q~ j jg(y)j dy Cg (x) (where g is the Hardy{Littlewood maximal function of g), hence: Z X
j
j 1Q~ j (y) jg(y)j dy C
Z X
j
X j 1Qj g(y) dy C 0 k j 1Qj k1+s=2 kgk
j
Now, we use the inequalitiesR N Q2HN 24N=(s+2) 1Q C W (x)2=(1+s=2) and P P d 2N 2 N Q2HN dQ 2 C W (x) dx to conclude: P
P
ku wkN p0 ;q0 C kuk2 kW k2 C 0 kuk2 kwkH r Thus, Theorem 21.1 is proved. As a direct corollary, we deal with homogeneous spaces:
Proposition 21.2: Let us de ne the homogeneous spaces M_ p;q for p q 1 and X_ r for 0 r < d=2) by R i) kf kM_ p;q = supx2IRd sup0
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Mild solutions
Proof: We have the homogeneity properties kd=2 f (x)k2 = kf (x)k2 and kd=2 r f (x)kHr _ = kf (x)kH_ r . This gives a homogeneity property for the norm r _ of Xr : k f (x)kX_ r = kf (x)kX_ r . We will achieve a more precise result. Since H r = L2 \ X_ r and kf kH_ r kf kH r , we see that X_ r Xr and kf kXr kf kX_ r ; by homogeneity, we have sup>0 kr f (x)kXr kf kX_ r . Conversely, we have, for every g 2 H r , the inequality kfgk2 kr f (x)kXr kd=2 r g(x)kH r ; since we have lim!+1 kd=2 r g(x)kH r = kgkH_ r and since H r is dense in H_ r , we get the equality sup>0 kr f (x)kXr = kf kX_ r Proposition 21.2 is clearly a direct consequence of Theorem 21.1 and of the property sup>0 kd=q f (x)kM p;q = kf kM_ p;q . The embedding X_ r M_ 2;d=r is not an isomorphism, at least when 2=r 2 IN (May [MAY 02], following an idea of Meyer [MEY 99]):
Proposition 21.3: Let r 2 (0; d=2) and assume that 2=r = k 2 IN. Let ' 2 L2 (IRk ) and ' 6= 0. Then the function de ned by (x1 ; : : : ; xd ) = '(x1 ; : : : ; xk ) (i.e. = ' 1) belongs to M_ 2;d=r (IRd ) and not to X_ r . Proof: Obviously, writing x = (x1 ; : : : ; xd ), x0 = (x1 ; : : : ; xk ) and x00 = (xk+1 ; : : : ; xd ), we have Z
B (x;R)
j (y)j2 dy
Z
B (x0 ;R)
j'(y0 )j2 dy0
Z
B (x00 ;R)
dy00 CRd k k'k22
2d so that 2 M_ 2; k (IRd ). We now check that 2= X_ r with r = k=2 by looking the operation of pointwise multiplication with on functions f (x) = '(x0 )g(x00 ) = ' g, where g 2 L2 (IRd k ). If 2 X_ r , we should have the existence of a constant Cr so that for all g 2 L2 (IRd k) the inequality k (' g)kH_ r = kj'j2 gkH_ r C0 kgk2 holds. Using Parseval equalities, we would have ZZ R j '^(0 )'^(0 0 ) d0 j2 jg^( 00 )j2 d 0 d 00 C 2 (2)k Z jg^( 00 )j2 d 00 0 (j 0 j2 + j 00 j2 )r
which is equivalent to Z
R
j '^(0 )'^(0 0 ) d0 j2 d 0 C 2 (2)k 0 j 0 j2r
In the neighborhood of 0 = 0, we have the equivalence R j '^(0 )'^(0 0 ) d0 j2 k'k22 = k'k22 j 0 j2r j 0 j2r j 0 jk which gives
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R
R
j '^(0 )'^(0 0 ) d0 j2 0 d = 1. Thus, 2= X_ r . j0 j2r
Multipliers of negative order
2. Solutions in
211
Xr
If we want to study the existence of solutions for the Navier{Stokes equations in (C ([0; T ); Xr ))d or (Lpt Xr )d , we may directly apply the results of the preceding chapters. Indeed, Xr is a Banach space of shift{invariant local measures:
Proposition 21.4: Let r 2 (0; d=2). (A) i) Let us de ne the space Yr in the following way: f 2PS 0 (IRd ) belongs to Yr if and only if f can be decomposed as a series f = n2IN fn gn with P fn 2 L2, gn 2 HPr and n2IN kfnk2 kgnkH r P < 1. We de ne the norm of Yr as kf kYr = inf f n2IN kfnk2 kgnkH r = f = n2IN fn gn g. Then Yr is a shiftinvariant Banach space of test functions. ii) Xr is a shift-invariant Banach space of local measures. More precisely, Xr is the dual of the space Yr . (B) Similar resultsPhold for Y_r and X_ r , where the Banach space Y_ r is P de ned by kf kY_ r = inf f n2IN kfnk2 kgnkH_ r = f = n2IN fn gng. Proof: We rst prove that k:kYr and k:kY_ r are norms. With the inequalities kgkq Cr kgkH_ r for 1=q = 1=2 d=r and kgkH_ r kgkH r , we get that, for 1=p = 1 d=r, kf kp Cr kf kY_ r Cr kf kYr . Therefore, if the distribution f belongs to Yr and has a norm equal to 0, it must be the null distribution f = 0 (and the same for Y_r ). Since the test functions are dense in L2, H r and H_ r , we nd that they are dense in Yr and Y_r . So our spaces are shift-invariant Banach spaces of test functions. P Xr operates on Yr : if f 2 Yr is decomposed as f = n2IN fn gn , we saw that f 2 Lp (1=p = 1 d=r), hence is a measurable function; if h 2 Xr , h 2;d=r ; moreover, we have jhf j is P a measurable function too, since Xr M n2IN jh(x)fn (x)gn (x)j; since khfn gn k1 kfn k2 khgn k2 kfn k2 kgn kHRr khkXr , we nd that hf is P integrable over IRd, and we P may de ne hhjf i = hf dx. We have jhhjf ij n2IN khfngn k1 khkXr n2IN kfnk2 kgnkH r . This gives jhhjf ij khkXr kf kYr . Thus, Xr (Yr )0 and kf kYr0 kf kXr . Conversely, let T 2 (Yr0 ). Let K be a compact subset of IRd and !K be a compactly supported smooth function equal to 1 in a neighborhood of K ; then, if ' 2 DK , we have
jhT j'ij = jhT j'!K ij kT kYr0 k'k2 k!K kH r R
Thus, T is de ned by a locally square-integrable dx. R function h: hT j'i = h' If ' and are two test functions, we get j h' dxj kT kYr0 k'k2k kH r . This gives that the function h belongs to L2 with a norm less or equal to kT kYr0 k kH r . Since D is dense in H r , this inequality extends to the whole of
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212
Mild solutions
H r , and this proves that h equality of norms.
2 Xr
and khkXr
kT kYr0 .
Thus, Xr = Yr0 with
Thus, we know how to solve the Navier{Stokes equations for initial values in Xr with r 1 (Chapter 17) or in BXs;qr with r s +1 and with s > 0 (Chapter 19) or 1 < s < 0 (Chapter 20). Similarly, we may complete the uniqueness criterion of Serrin (Theorem 14.2):
De nition 21.3: A vector eld ~u on (0; T ) IRd will be labeled r-regular for some r 2 [0; 1] if ~u belongs to L ((0; T ); (Xr )d ) with 2= = 1 r (0 < r < 1) or to C ([0; T ]; (X1(0))d ) (r = 1), where X1(0) is the space of smooth elements in X1 (i.e. the closure in X1 of the space of the test functions). A smooth r-regular vector eld is an r-regular vector eld so that, for almost every t, ~u belongs to (Xr(0))d ) where Xr(0) is the space of smooth elements in Xr . r-regular solutions of the Navier{Stokes equations are provided by initial values in (X1(0) )d or in (BXr2=; )d (0 < r < 1). Theorem 21.2: (Serrin's uniqueness theorem) ~ :~u0 = 0. Assume that there exists a solution Let ~u0 2 (L2 (IRd ))d with r ~u of the Navier{Stokes equations on (0; T ) IRd (for some T 2 (0; +1]) with initial value ~u0 so that: i) ~u 2 L1 ((0; T ); (L2(IRd )d ); ii) ~u 2 L2((0; T ); (H 1 (IRd )d ); iii) For some r 2 (0; 1], ~u is r-regular. Then, ~u is the unique Leray solution associated with ~u0 on (0; T ). Proof: Let ~v be another solution associated with ~u0 on (0; T ) (with associated pressure q) so that ~v 2 L1 ((0; T ); (L2(IRd )d ) \ L2 ((0; T ); (H 1(IRd )d ). As in Chapter 14, we use a smoothingR Rfunction (t; x) = (t) (x) 2 D(IRd+1 ) where is supported in [ 1; 1], with dx dt = 1, and de ne for > 0, (t; x) = d 1 t x d+1 ( ; ). Then, ~u and ~v are smooth functions on (; T ) IR and we may write @t ( ~u):( ~v) = ( @t~u):( ~v) + ( ~u):( @t~v). We then obtain:
@t ( ~u):( ~v) = ~r: ( [r ~ ~u]):( ~v ) ( [r ~ ~u]):( [r ~ ~v]) r~ : ( [~u ~u]):( ~v) + ( [~u ~u]):( [r~ ~v ]) r~ : ( p)( ~v) ~ : ( ~u):( [r ~ ~v]) ( [r ~ ~u]):( [r ~ ~v]) +r ~r: ( ~u):( [~v ~v]) + ( [r ~ ~u]):( [~v ~v]) r~ : ( q)( ~u)
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Multipliers of negative order
213
We integrate this equality against a test function (t)'(x=R) with ' equal to 1 in a neighborhood of 0, and we let R go to 1. The terms, which are written r~ :F~ with F~ 2 (L1 ((0; T ) IRd))d , will have a null contribution. Since ~u and ~v belong to (L1 L2 )d \ (L2 H 1 )d (L4 H 1=2 )d (L4 Lq )d with 1=q = 1=2 1=(2d), we nd that the pressures p and q belong to L2 Ld=(d 1), and so do p and d d 2d 1 q; since ~u and ~v belong to (L1 t Lx ) \ L ((0; T ) IR )) , we nd that they belong to (L2 Ld)d , so that ( p)( ~v) and ( q)( ~u) belong to (L1 ((0; T ) IRd ))d . Hence, we get the equality in D0 (; T ): R
@t ( ~u):( ~v) dx =
R ~ ~u]):( [r ~ ~v]) dx 2 R ( [r ~ ~v]) dx + R ( [~u ~u]):( [r ~ + ( [~v ~v]):( [r ~u]) dx
We may rewrite the last summand as Z
~ ~u]) dx = ( [~v ~v ]):( [r
Z
~ :(~v ~v )]):( ~u) dx ( [r
The next step is to check that ~u ~u 2 (L2 ((0; T ) IRd ))dd . We write 2=r r 2 2 Lt H 1 \ L1 t Lx Lt Hx for 0 r 1; we then write that the pointwise 2 =r product maps L H r L2=(1 r)Xr to L2 L2 . We now let go to 0. When ! 0 and f 2 L2 ((0; T ) IRd ), f is strongly convergent toR f in L2 ((0; T ) IRd). Thus, we have the following convergences in R R 0 DR (0; T ): @t ( ~u):(R ~v) dx ! @t ~u:~v dx, ( [r~ ~u])R:( [r~ ~v]) dx ! r~ ~u:r~ ~v dx and ( [~u ~u]):( [r~ ~v ]) dx ! ~u ~u:r~ ~v dx = R ~ )~v dx. Thus, we must study the convergence of the last summand ~u:(~u:r R ~ :(~v ~v)]):( ~u) dx. ( [r ~ :(~v ~v )] as [(~v:r ~ )~v]. We know (from We begin by rewriting [r Proposition 21.4) that pointwise product maps H r L2 in the predual Yr of ~ )~v ] converges Xr and that smooth functions are dense in Yr ; thus, [(~v :r d ~ )~v in (L r Yr ) while ~u converges weakly to ~u in (L r Xr )d . strongly to (~v:r 0 ((0; T ))) R ( [r ~ :(~v ~v)]):( ~u) dx ! This gives the convergence (in D R ~ ~u:(~v:r)~v dx. We have thus obtained the following equality in D0 (0; T ): 2 1+
@t
Z
~u:~v dx = 2
Z
1
r~ ~u:r~ ~v dx +
Z
~ )~v dx ~u:(~u:r
Z
2
~ )~v dx ~u:(~v:r
R ~ :(~u ~u) ds, Since ~u ~u 2 (L2 ((0; T ) IRd ))dd and ~u = et~u(0) IPR 0t e(t s)r 2 d t 7! ~u is continuous from [0; T ] to (L (dx)) and t 7! ~u:~v dx is continuous. Thus, we may integrate our equality and obtain the equality Z
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~u(t; x):~v (t; x) dx + 2
Z tZ
0 IRd
r~ ~u:r~ ~v dx ds =
214
Mild solutions Z tZ
Z tZ
~ )~v dx ds ~ )~v dx ds ~u:(~v :r k~u0 k22 + ~u:(~u:r 0 IRd 0 IRd Of course, this equality holds as well for ~v = ~u. Now, if we assume that ~v satis es the Leray inequality
k~v(t)k22 + 2
Z t
0
kr~ ~vk22 ds k~u0k22 ;
we get the following inequality for ~u ~v:
k~u(t; :) ~v(t; :)k22
2
Z tZ
jr~ (~u ~v)j2 dx ds d
2
0 IR
Z tZ
~ ~v dx ds ~u: (~u ~v):r
0 IRd
R R ~ ~u dx ds = 0. Moreover, we have the antisymmetry property 0t IRd~u: (~u ~v):r For r < 1, we quickly conclude as in Serrin's theorem (Theorem 14.3) since we have R R
j t IRd~u: (~u ~v):r~ (~v ~u) dx dsj R R R Cr ( t k~ukXrr ds)(1 r)=2 ( 0t k~v ~uk2H ds)1=2 ( t k~v ~ukHr r ds) r 1
2
2
2
1
If ~u = ~v on [0; ], we nd that Z t
sup k~u ~vk22 Cr00 ( 0<st
k~ukXrr ds)1=2 1
2
sup k~u ~vk22
0<st
and uniqueness is valid on a bigger interval. By weak continuity of t 7! ~v, we nd ~u = ~v. For r = 1, we easily conclude as in Von Wahl's theorem (Theorem 14.4). If T0 < T , then for each > 0 we may split ~u on [0; T0 ] in ~u = ~ + ~ with ~ 2 (L1 ((0; T0 ) IRd )d and k ~ kL1X1 < . Then we write
R R
j 0t IRd~u: (~u ~v)R:r~ (~v ~u) dx dsj C k ~ kL1X 0t k~v ~uRk2HR ds R tR ~ (~v ~u)j2 dx ds)1=2 ( 0t IRdj~v ~uj2 dx ds)1=2 +k ~k1 ( 0 IRdjr R R R tR 2C 0 IRdjr~ (~v ~u)j2 dx ds + ( C4 k ~ k21 + C) 0t IRdj~v ~uj2 dx ds: 1
1
Choosing so that 2C < 1, we get Z t
4 ~ 2 k~v(t; :) ~u(t; :)k22 ( C k k1 + C) k~v(s; :) ~u(s; :)k22 ds: 0 The Gronwall lemma gives then that ~u = ~v.
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Multipliers of negative order
215
3. Perturbated Navier{Stokes equations
We now consider perturbated Stokes equations or Navier{Stokes equations, where the perturbations are de ned with the help of r-regular vector elds:
Theorem 21.3: Let T < 1, r1 ; r2 ; r3 2 [0; 1], let V~ be a smooth r1 -regular vector eld on ~ be a smooth r2 -regular vector eld on (0; T ) IRd. Let P (0; T ) IRd and let W be the perturbated Stokes equations ( = 0) or Navier{Stokes equations ( = 1): 8 u < @t ~
(P )
~ :(~u ~u) = ~u r
r~ :(V~ ~u) ~r:~u = 0 ~u(0; :) = ~u0
:
with
r~ :(~u W~ ) r~ p
8 <
2 f0; 1g ~r:V~ = r ~ :W ~ =0 : ~r:~u0 = 0 ~u0 2 (L2 (IRd ))d Then the problem (P ) has a solution ~u 2 L1 ((0; T ); (L2)d ) \ L2((0; T ); (H 1 )d ) and limt!0 k~u ~u0 k2 = 0. If = 0 (linear case), then the solution ~u is unique and we have the energy equality:
k~u(t)k22 + 2
Z t
kr~ ~uk22 ds = k~u0k22 + 2
Z Z t
~ ) ~u):W ~ dx ds ((~u:r 0 If = 1 (non-linear case), then one may choose ~u such that we have the
(21:1)
0
energy inequality:
k~u(t)k22 + 2
(21:2)
Z t
0
kr~ ~uk22 ds k~u0k22 + 2
Z Z t
0
~ ) ~u):W ~ dx ds ((~u:r
Moreover, if ~u and ~v are two solutions of (P ) with the same initial value, if ~u and ~v belong to L1 ((0; T ); (L2)d ) \ L2 ((0; T ); (H 1)d ), if ~u satis es inequality (21.2) and ~v is r3 -regular, then ~u = ~v (Serrin's uniqueness criterion).
Proof: Step 1: Elimination of the pressure If we take the divergence of equation (P ), we obtain ~ r ~ ):(~u ~u) (r ~ r ~ ):(V~ p = (r
~u)
~ r ~ ):(~u W ~) (r
For general classes of solutions, this allows elimination of the pressure by the ~p=r ~ 1 p. We just adapt the proofs of Theorems 11.1 and 11.2 to formula r
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Mild solutions
~ , we will not consider the class deal with equation (P ). In order to de ne ~u W d 2 2 d (Luloc;xLt ((0; T ) IR )) , but the smaller class (L2t L2uloc;x((0; T ) IRd ))d (as in Furioli, Lemarie-Rieusset and Terraneo [FURLT 00]). Indeed, the product of a function in L2uloc by a function in Xr is uniformly locally in H r and since a function f in the SchwartzP class will satisfy, for any 2 D(IRd) with P k2ZZd (x k ) = 1, the inequality k2ZZd k (x k )f kH r < 1, we get for j < 0 ~ :(~u W ~ )k1 C 2j kS0(~u W ~ )k1 C 0 2j k~ukL2 kW ~ kXr the inequality kIPj r uloc ~ :(~u W ~ ) is well de ned. Thus, we get from (P ) the equations: so that IPr ~ :(~u ~u) IPr ~ :(V~ lim1(Id Sj )(@t ~u ~u) = IPr
j!
~u)
~ :(~u W ~) IPr
To get rid of Sj , we must assume that ~u vanishes at in nity, hence we replace L2uloc by the subspace E2 . We then obtain the following result:
Lemma 21.4: (Equivalent formulations for the Navier{Stokes equations) Let ~u 2 \T
~ :(~u ~u) = ~u r
r~ :(V~ ~u) ~r:~u = 0 ~u(0; :) = ~u0
r~ :(~u W~ ) r~ p
~ satisfy the hypotheses of Theorem 21.2. where ; V~ and W ~ :~u = 0; @t~u = ~u IPr ~ :( ~u ~u + V~ ~u + ~u W ~) (B) ~u is a solution of r 3 0 3 t ~ (C) there exists ~u0 2 (S (IR )) , r:~u0 = 0, so that ~u = e ~u0 R t (t s) ~ :( ~u ~u + V~ ~u + ~u W ~ ) ds. e IPr 0 Step 2: The molli ed equations We follow the lines of Chapter 13 and introduce a modi ed equation, choosd ), with R ! dx = 1 and a function 2 D(IRd+1 ) with ing a function ! 2 D (IR d IR R IRd+1 dt dx = 1 and solving for > 0 the equations (P; ) given by 8 ~ :((~u ! ) ~u) r ~ :((V~ ) ~u) r ~ :(~u (W ~ )) r ~p u =~u r < @t ~ ~r:~u = 0 : ~u(0; :) = ~u0 where ! = 1d !( x ) and = d1+1 ( t; ; x ) (we extend V~ to the whole IR IRd by taking V~ = 0 for t < 0 or t > T (0 r < 1) or, for r = 1 and for a
2 D(IR) so that (0) = 1, V~ = (t)V~ (0) for t < 0 and V~ = (t T )V~ (T ) for t > T ). Following Lemma 21.4, we are led to solve the xed-point problem ~u = et~u0 B (~u) A (~u), where the bilinear operator B is given by
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Multipliers of negative order
217
Rt
~ :((~u ! ) ~v) ds and the linear operator A is given B (~u; ~v) = 0Re(t s) IPr t (t s) ~ ~ ~ )) ds. Now, we notice by A (~u) = 0 e IPr:((V ) ~u + ~u (W 2 d t 2 that for ~u0 2 (L ) , we have p e ~u0 2 C ([0; T ]; (L )d ) \ L2([0; T ]; (H 1)d ). Moreover p kB (~u; ~v )(t)k2 C t sup0<s
k~u(t)k22 + 2
Z Z t
0
jr~ ~uj2 dx ds = k~u0k22 + 2
Z Z t
0
~ ) ~u):W ~ dx ds ((~u:r
~ is bounded, we nd that Since W 2
Z Z t
0
~ ) ~u):W ~ dx ds ((~u:r
Z Z t
0
jr~ ~uj2 dx ds + C
Z t
0
k~uk22 ds
~ k1 and not on t nor on T. The Gronwall lemma where C depends only on kW then gives the existence of ~u on the whole [0; T ] IRd . Step 3: The weak limit We call ~u the solution of (P; ). We have seen that
k~u(t)k22 + 2
Z Z t
0
jr~ ~uj2 dx ds = k~u0 k22 + 2
Z Z t
0
~ ) ~u):W ~ dx ds: ((~u :r
Since the norm of Xr is invariant under translation, we have that kW~ kL ((0;T );(Xr )d) kW~ kL ((0;T );(Xr )d ) and thus 2
Z Z t
0
~ ) ~u ):W ~ dx ds C k~ukL2 (I;(H_ 1 )d ) k~ukL(I;(H r )d ) kW ~ kL (I;(Xr )d) ((~u :r
where I = (0; T ), 2= = 1 positive :
k~ukL ((H )d ) k~ukL((H r )d ) 2
1
r and 1= = r=2. If r < 1, we write for every r 1 r k~uk1+ L ((H )d ) k~u kL1(;(L )d ) (1+2r) k~uk2L ((H )d ) + 2 (1r =r) r k~uk2L1((L )d ) 2
2
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1
1
2
(1+ ) (1
)
2
218
Mild solutions
For small enough, this gives (by the Gronwall lemma) that we have RT sup>0 sup0
Z T
0
kV~ ~u + ~u W~ k22 ds < 1 and RT
sup
Z T
0
k~u ~uk2d d ds < 1; 1
hence, we get sup 0 k@t~uk2H ds < 1 with = 1 if = 0 and = d=2 1 if = 1. Now, we use the following lemma (see Theorem 13.3).
Lemma 21.5: a) Let > 0. If (f)0<<1 is a family of functions on (0; T ) IRd so that for all 2 D((0; T ) IRd ) we have: ( f )0<<1 is bounded in L2 ((0; T ); H 1) and ( @t f)0<<1 is bounded in L2((0; T ); H ), then there exists f1 2 L2loc ((0; T ) IRd ) and a sequence n converging to 0 so that for all 2 D((0; T ) IRd ) the sequence fn converges to f1 strongly in L2((0; T ) IRd ) and weakly in L2 ((0; T ); H 1). b) If, moreover, for all 2 D(]0; T [IR3 ) we have that ( f )0<<1 is bounded in L1 ((0; T ); L2), then the sequence fn converges to f1 strongly 2d in Lp ((0; T ); L2) for all p 2 [1; 1) and strongly in L3((0; T ); L d 1 ). We apply the lemma to ~u. We nd a sequence n and a vector eld ~u so that for all 2 D((0; T ) IR3 ) we have convergence of ~un to ~u, strongly in L2 ((0; T ); (L2)d ) and weakly in L2 ((0; T ); (H 1)d ). Moreover, p may be written ~ r ~ :(V~ ~u + ~u W ~ ) and r = 1 r ~
as p = q + r with q = 1 r ~r:~u ~u: we have sup>0 kq kL2((0;T )IRd ) < 1 and sup>0 kr k d < L2 ((0;T );L d 1 ) 1; hence, we may impose in the same way weak convergence of pn to p. Then, we have that each term of equations (P; ) applied to ~u converges in D0 ((0; T ) IRd ) (we write instead of n ): ~u converges to ~u; hence, @t ~u ~ p converges to @t ~u and ~u converges to ~u; p converges to p; hence, r ~ p; moreover V~ converges strongly to V~ in L ((0; T ); (Xr )d ) (with converges to r 2= = 1 r) while ~u converges strongly in (L2loc ((0; T ) IRd)d ), and this gives local strong convergence of V~ ~u in L2=(2 r) ((0; T ); (H r )dd ); thus, ~ :(V~ ~u + ~u W ~ ); nally, ~u ~u converges r~ :(V~ ~u + ~u W~ ) converges to r d 3 =2 d d ~ :((~u !) ~u) converges d 1 strongly in (Lloc ((0; T ); L (IR ))) to ~u ~u; hence, r ~ to r:(~u ~u).
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Moreover, for all ' 2 D(IRd ),p ' @t~u is bounded in L2 ((0; T ); (H )d ) and thus k (~u (t) ~u0 )kH C (') t where C (') does not depend on t nor on ; this proves that ~u(t) converges to ~u0 in H . Thus, ~u is a solution of (P ).
Step 4: Energy estimates If = 0, we nd that ~u 2 L2 ((0; T ); (H 1)d ) and @t~u 2 L2((0; T ); (H 1 )d ), so that @t j~uj2 = 2h@t~uj~ui and equality (21.1) is obvious. If = 1, we start from the equality obtained in Step 2: Z Z t Z Z t 2 2 2 ~ ~ ) ~u ):W ~ dx ds k~u(t)k2 + 2 jr ~uj dx ds = k~u0k2 + 2 ((~u :r 0 0 The strong convergence of ~un in (L2loc ((0; T ) IRd ))d shows (through a diagonal extraction process) that we may nd a subsequence (still written (n )) so that we have pointwise convergence almost everywhere on (0; T ) IRd; hence, we have, for almost every t, pointwise convergence almost everywhere on IRd . Since ~ ~un in (L2 ((0; T ) IRd ))dd , we get that almost we have weak convergence of r everywhere on (0; T ) we have
k~u(t)k22 +2
ZZ t
Z Z t
~ ) ~un ):W ~ dx ds jr~ ~u j2 dx ds k~u0 k22 +2 lim sup ((~un :r n !0 0 0 ~ ) ~un ):W ~ n to ((~u:r ~ ) ~u):W ~ . We We now study the convergence of ((~un :r 2 2 d ~ ~un to r ~ ~u in L ((0; T ); (L ) d) and should use the convergence of r 2 =r r d of ~un to ~u in L 2 ((0; T ); (H 2 ) ), but both are weak convergences, and, thus,we cannot deal directly with the bilinear term. However, the solution ~ in X~ + Y~ , is easy. Let us assume r2 < 1. For any positive , we may split W where kX~ kL2 ((0;T );(Xr2 )d < and Y~ 2 (D((0; T ) IRd ))d (since we work in RRt ~ ) ~un ):Y~n dx ds converges to the subspace Xr(0) ((~un :r 2 ); we have that RRt RT 0 ~ ~ ~ 0 ((~u:r) ~u):Y dx ds while sup 0 j((~u :r) ~u):X~ j dx ds < C , and letting ~ in go to 0, we arrive at the desired convergence result. If r2 = 1, we split W X~ + Y~ where kX~ kC([0;T ];(X1)d < and Y~ 2 (D([0; T ) IRd))d ; then, we split Y~ ~ 0kL4 ((0;T );(X1=2 )d < and Y~ 0 2 (D((0; T ) IRd ))d and the in X~ 0 + Y~ 0 where kX end of the proof is similar to the case r2 < 1. We attained inequality (21.2) for almost every t; we then conclude that it is valid for every t since the continuity of ~u in (Hloc (IRd ))d and the boundedness of ~u in (L2 )d imply the weak convergence of ~u(t0 ) to ~u(t) in (L2 )d when t0 goes to t. Step 5: Uniqueness The proof is exactly the same as the proof of the classical theorem of Serrin (see Theorem 21.3). Since ~v is r3 -regular, we obtain the energy inequality R k~u(t) ~v(t)k22 + 2 0t kr~ R R (~u ~v )k22 ds R tR ~ (~u ~v) dx ds + 2 0t ((~v ~u):r ~ ) (~v ~u):W ~ dx ds 2 0 IRd~v: (~v ~u):r and we conclude as for Theorem 21.3.
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Chapter 22 Further adapted spaces for the Navier{Stokes equations 1. The analysis of Meyer and Muschietti
In his book, Y. Meyer presents a generalization of Cannone's adapted spaces to deal with the case of some Besov spaces with null or negative regularity exponent [MEY 99]. In order to nd global solutions, they restrict their analysis to the case of homogeneous spaces (kf (x)kE = kf kE ).
De nition 22.1: (Meyer and Muschietti's adapted spaces) According to Meyer and Muschietti, a Banach space X is adapted to the Navier{Stokes equations if the following assertions are satis ed: a) X is a shift-invariant Banach space of distributions. b) the norm of X is homogeneous of exponent 1: kf (x)kX = kf kX ~ f 2 X d, then f is a pointwise multiplier of X : c) if f 2 L1 and r (22:1)
kfgkX C (kf k1 +
d X i=1
k@i f kX )kgkX :
The theorem of Meyer and Muschietti then states the following result:
Theorem 22.1: (Meyer and Muschietti's theorem for adapted spaces) Let X be a Banach space adapted (according to Meyer and Muschietti) to the Navier{Stokes equations. Then, there exists a positive X so that for all u~0 2 ~ :u~0 = 0 and k~u0kX < X , there is a solution ~u of the Navier{ X d such that r Stokes equations on (0; 1) with initial value u~0 so that ~u 2 Cb ((0; 1); X d). Proof: Let M (X ) be the Banach space of pointwise multipliers of X , normed by kf kM (X ) = supg2X;kf kX 1 kfgkX . Of course, M (X ) is a Banach algebra: kfgkM (X ) kf kM (X ) kgkM (X ). The proof then follows the line of Theorem 17.2. We introduce as admissible path space X associated to X the space EX \ BX p where EX = L1 ((0; 1); X d) and BX = f~u = t ~u 2 (L1 ((0; 1); (M (X ))d g. 221 © 2002 by CRC Press LLC
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Clearly, ~u0 7! (et~u0 )t>0 maps boundedly X d to X (according to inequality (22.1)). The boundedness of B from EX BX to EX is obvious, since we ~ :kL(X dd;X d) C p 1 . We get a more precise estimate, for have ke(t s)IPr t s 0 < t1 < t2 , r ~ g)(t1 ) B (f;~ ~ g)(t2 )kX C t2 t1 sup kf~kX sup psk~gkM (X ) kB (f;~ t1 0<s
2
~ g) from (0; 1) to X d . which shows the continuity of t 7! B (f;~ Now, we prove that B is bounded from BX (BX \ EX ) to BX : we write, ~ :(f~ ~g)kM (X ) C p 1 1s kf~kBX k~gkBX and, for s < t=2 for s > t=2, ke(t s)IPr t s ~ :(f~ ~g)kM (X ) C t 1 s p1s kf~kBX k~gkEX . and t < 1, ke(t s)IPr Thus, if k~u0 kX is small enough, we get a global solution ~u 2 BX \ EX . Meyer and Muschietti have given examples of spaces adapted to the Navier{ Stokes equations. A trivial example is given by the shift-invariant Banach spaces of local measures with homogeneous norms (with homogeneity exponent 1): Ld , Lorentz spaces Ld;q , homogeneous Morrey{Campanato spaces M_ p;d. Another example is given by the Besov spaces B_ q 1+d=q;1 for 1 q < 2d. We give a little more precise statement:
Proposition 22.1: (A) Let X be a Banach space adapted (according to Meyer and Muschietti) to the Navier{Stokes equations. Then X is continuously embedded into B_ M_ 12=;22d;1 . (B) Let E be a shift-invariant Banach space of local measures. Assume that E is continuously embedded into L2loc(IRd) and that there exists a shift-invariant Banach space of local measures F so that the pointwise product maps boundedly E E to F : for f 2 E and g 2 E , kfgkF C kf kE kgkE . Assume, moreover, that: i) the norm of E is homogeneous with homogeneity exponent for some 2 (1=2; 1] (i. e. k f (x)kE = kf kE ) p ii) [F; Cb (IRd )]1=2;1 E : for f 2 F \ Cb , kf kE C kf kF kf k1 then, for 1 q 1, the space X = B_ E 1+ ;q is a Banach space adapted (according to Meyer and Muschietti) to the Navier{Stokes equations. Proof: (A) is clear. We start from the embedding X B_ X0;1 = B_ B_ 11==22;;11 . We X want to prove B_ X1=2;1 M_ 2;2d. We have kf kM_ 2;2d = supk2ZZ 2k=2 kf (2k x)kL2uloc and kf kB_ 1=2;1 = 2k=2 kf (2k x)kB_ 1=2;1 ; thus it is enough to prove the continX X uous embedding B_ X1=2;1 L2uloc, and even B_ X1=2;1 L2loc since the norm of B_ X1=2;1 is shift-invariant. We have seen that it is suÆcient to check that the pointwise product of two elements in B_ X1=2;1 is de ned as a distribution (see
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the proof of Proposition 19.1). Since B_ X0;1 X B_ X0;1 , we have B_ X1=2;1 = [X; B_ X1;1 ]1=2;1 , thus we may write u P 2 B_ X1=2;1 as a sum u = Pn2IN n un with supn2IN kunkX kun kB_ X1;1 1 and n2IN jn j < 1. If u and v satisfy kukX kukB_ X1;1 1 and kvkX kvkB_ X1;1 1, we use inequality (22.1), which gives kfgkX C kf kB_ X1;1 kgkX to get p
q
p
q
kuvkX = kuvkX kuvkX C kukB_ X; kvkX kvkB_ X; kukX C: 1 1
1 1
Thus, pointwise product maps boundedly B_ X1=2;1 B_ X1=2;1 to X . To prove (B), we use the paradi erential calculus of Bony and write
fg =
P
j 2ZZ Sj
P
P
P
2 f j g + j2ZZ Sj 2 gj f + j2ZZ jj kj2 j f k g = _ (f; g) + _ (g; f ) + _(f; g):
We control the paraproduct _ (f; g) by
k_ (f; g)kX C k2j( 1) kSj C 00 kf k1k2j(
2 f j gkE kq C 0 k2j( 1) kSj 2 f k1kj gkE kq 1) kj gkE kq = C 00 kf k1kgkX ;
and we control the paraproduct _ (g; f ) by
C k2j( 1) kSj 2 gj f kE kq C 0P k2j( 1) kSj 2 gk1 kj f kE kq 00 C P k( kj 3 2k kk gkE )kj f kX kq ~ j f kX kq C 000 k( kj P 3 2k kk gkE )2 j kr 0000 k j k ( 1) C kr~ f kX k kj 3 2 2 kk gkE kq 00000 ~ C krf kX kgkX : Then, by writing the Bernstein inequality kj f kE p we control the remainder p C kj f kF kj f k1 C 0 kj f kF 22j kj f kF = C 0 2j kj f kF : k_(f; g)kX C k2j( 1) j k_(f; g)kE kq 0 k2j(2 1) j k_(f; g)kF kq CP 00 j (2 1) C k2 P kj 4 kk gkE k Pjl P kj2 l f kE kq 000 j (2 1) k ( 1) C k2 k jl kj2 l f kX kq kj 4 kk g kE 2 C 0000 kr~ f kXPk2j(2 1) Pkj 4 kk gkE 2 k kq C 00000 kr~ f kX k kj 4 2(k j)(1 2 ) 2k( 1) kk gkE kq C 000000 kr~ f kX kgkX k_ (g; f )kX
since 2 1 > 0. Whenever X is adapted in the sense of Meyer and Muschietti, we have the inequality kfgkX C kf kB_ X1;1 kgkX . This provides an easy way to generalize the theorem of Meyer and Muschietti:
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De nition 22.2: A Banach space X is adapted in the generalized sense of Meyer and Muschietti to the Navier{Stokes equations if the following assertions are satis ed: a) X is a shift-invariant Banach space of distributions. b) the norm of X is homogeneous with exponent 1: kf (x)kX = kf kX c) for a 2 (0; 1) X satis es the regularity property (P ): if f and g belong to B_ X1;1 \ B_ X;1 , then
kfgkB_ X;1 C (kf kB_ X; kgkB_ X;
(P )
1 1
1
+ kgkB_ X1;1 kf kB_ X;1 )
Theorem 22.2: (A) If X is adapted in the sense of Meyer and Muschietti, then X satis es (P ) for all 2 (0; 1). (B) Let E be a shift-invariant Banach space of local measures. Assume that E is continuously embedded into L2loc(IRd) and that there exists a shift-invariant Banach space of local measures F so that the pointwise product maps boundedly E E to F : for f 2 E and g 2 E , kfgkF C kf kE kgkE . Assume that: i) the norm of E is homogeneous with homogeneity exponent for some 2 (0; 1] (i. e. k f (x)kE = kf kE ) p ii) [F; Cb (IRd )]1=2;1 E : for f 2 F \ Cb , kf kE C kf kF kf k1 then, for 1 q 1, the space X = B_ E 1+ ;q is a Banach space adapted (in the generalized sense of Meyer and Muschietti) to the Navier{Stokes equations. More precisely, X satis es (P ) for all 2 (0; 1) so that + 2 1. (C) Let X be a Banach space adapted (in the generalized sense of Meyer and Muschietti) to the Navier{Stokes equations. Then, there exists a positive ~ :u~0 = 0 and k~u0kX < X , there is a X so that for all u~0 2 X d so that r solution ~u of the Navier{Stokes equations on (0; 1) with initial value u~0 so that ~u 2 Cb ((0; 1); X d). Proof: We rst prove (A). Proving inequality (P ) is equivalent to proving the inequality on dyadic blocks. On dyadic blocks, (P ) reads as 2l kl (j f k g)kX
C (2j kj f kX 2k kk gkX + 2k kk gkX 2j kj f kX ) But we know that l (j f k g) = 0 if l max(j; k) + 3. On the other hand, if l k + 2, we have 2l kl (j f k g)kX C 2k k(j f k g)kX C 0 2k 2j kj f kX kk gkX : To prove (B), we want to estimate
2l kl (j f k g)kX 2l(+ 1) kl (j f k g)kE C 2l(+ 1) min 2l kl (j f k g)kF ; kj f k gkE 0 l ( + 1) min 2l kj f kE kk gkE ; kj f k1kk gkE ; kj f kE kk gk1 C 2 C 00 2l(+ 1) min 2l 2(j+k)(1 ) ; 2j 2k(1 ) ; 2j(1 ) 2k kj f kX kk gkX
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Recall that l (j f k g) = 0 unless jj kj 2 and l max(j; k) + 3 or min(j; k) max(j; k) 3 and jl max(j; k)j 3. In the case jj kj 2, we write 2l(+2 1) 2(j+k)(1 ) kj f kX kk gkX C 2j kj kX 2k kk gkX 2(l j)(+2 1) while we write in the case k j
3
2l(+ 1) 2j 2k(1 ) kj f kX kk gkX C 2j kj f kX 2k kk gkX and a similar estimate when j k 3. This gives the inequality (P ) provided that + 2 1 0. The proof for (C) follows the line of Theorem 22.1. We introduce as admissible path space X associated p to X the space EX \ BX , where EX = L1 ((0; 1); X d) and BX = f~u = t ~u 2 (L1 ((0; 1); (B_ X1;1 )d g. Clearly, we have that ~u0 7! (et~u0)t>0 maps boundedly X d to X . The boundedness of B from X X to X is obvious: if ~u 2 X , we have k~ukB_ X;1 C k~ukB_ 1;1 k~uk1B_ 0;1 Ct =2 k~ukX . Thus, if ~u and ~v belong X X to X , we nd that ~u(t) ~v(t) belongs to (B_ X;1 )dd with a norm bounded by Ct (1+)=2 k~ukX k~vkX . We have, for 2 1; 1,
ke(t s) IPr~ :kL((B_ X;1)dd;(B_ X ;1 )d ) C (t s)1(1+)=2 +
This gives by interpolation:
kB (~u; ~v)kB_ X; C 0 1
and
Z t
1 1 dsk~ukX k~vkX (1 ) = 2 (1+ s )=2 0 (t s)
Z t
1 1 dsk~ukX k~vkX (2 ) = 2 (1+ s )=2 0 (t s) and nally kB (~u; ~v)kX C k~ukX k~vkX . We now prove more precisely that B (~u; ~v) R2 (C ((0; T ); X ))d. For 0 < t1 < t2 , we write, for s < t1 , e(t2 s) e(t1 s) = 0t2 t1 e d e(t1 s) , which gives
kB (~u; ~v)kB_ X; C 1 1
k(e(t s) e(t s) )IPr~ :kL((B_ X;1)dd;(B_ X; )d ) C min( 1 (t1 s) 2
1
0 1
1
2
;
t2 t1 ) jt1 sj 3 2
which then gives kB (~u; ~v)(t1 ) B (~u; ~v)(t2 )kX C t2 t t1 1
Thus, Theorem 22.2 is proved.
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1+ 2
k~ukX k~vkX :
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Mild solutions
2. The case of Besov spaces of null regularity
We previously discussed a persistency theorem in the case of initial data given in a Besov space over local measures when the Besov regularity exponent was positive (Chapter 19) or negative (Chapter 20). We have as well a persistency result for the Koch and Tataru solutions in the case of a null Besov exponent (Furioli, Lemarie-Rieusset, Zahrouni and Zhioua [FURLZZ 00]):
Theorem 22.3: (Persistency theorem for Besov spaces with null regularity exponents) Let E be a shift-invariant Banach space of local measures and assume that D is dense in E . Let ~u0 2 (B_ E0;1 \ BMO 1 )d with r~ :~u0 = 0. If ~u0 2 (BMO 1 )d is small enough to satisfy the conclusion of the Koch and Tataru theorem for global solutions, then the Koch and Tataru solution ~u of the Navier{ Stokes equations satis es the regularity property ~u et~u0 2 (L1 ((0; 1); B_ E0;1 ))d . Proof: In Chapter 17, we introduced the space E (1=2) = [E; Cb (IRd)]1=2;1 . We obviously have B_ E0;1 \ B_ 11;1 B_ E(11==22);1 . We know that E (1=2) is a shiftinvariant Banach space of local measures. We may apply the persistency theorem Theorem 20.3 and conclude that ~u satis es the estimate supt>0 t1=4 k~ukE(1=2) < 1. Moreover, we know that the pointwise product maps E (1=2) E (1=2) to (t s) C ~ :~u ~ukE E . We thus get that ke 2 ( )1 1=4 IPr (t s)1=4 ps . This gives k( ) 1=4 B (~u; ~u)kpE C t1=4 and k( )1=4 B (~u; ~u)kE C + t p1=4 , and then kB (~u; ~u)kB_ E0;1 C k( ) 1=4 B (~u; ~u)kE k( )1=4 B (~u; ~u)kE C C + . 3. The analysis of Auscher and Tchamitchian
In their paper, Auscher and Tchamitchian gave another generalization of Cannone's adapted spaces [AUST 99]. We present in detail two results from this paper: i) a remark on the range of convergence for the Picard method in the case of bilinear equations; (see Theorem 22.4 bis below) ii) an existence result for the Navier{Stokes equations with a criterion based on some estimates for the pointwise product of dyadic blocks in the LittlewodPaley decomposition (see Theorem 22.5 below). We rst recall the Picard contraction principle (Theorems 13.2 and 15.1):
Theorem 22.4: (The Picard contraction principle) Let E be a Banach space and let B be a bounded bilinear transform from E E to E : kB (e; f )kE CB kekE kf kE :
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Then, if 0 < Æ < 4C1B and if e0 2 E is such that ke0kE Æ, the equation e = e0 B (e; e) has a solution so that kekE 2Æ. This solution is unique in the ball B (0; 2Æ) and depends continuously on e0 : if kf0 kE Æ, f = f0 B (f; f ) and kf kE 2Æ, then ke f kE 1 41CB Æ ke0 f0 kE .
The proof is based on the fact that whenever ke0 kE Æ < 4C1B the map e 7! e0 B (e; e) is a contraction on the ball B (0; 2Æ), so that the solution may be constructed recursively by choosing f0 2 B (0; 2Æ) and de ning fn+1 = e0 B (fn ; fn ): the unique solution e = e0 B (e; e) is the limit of the sequence fn . Auscher and Tchamitchian extended this result to the ball B (0; 2C1B ) and ke0k 4C1B , though the map is no longer a contraction:
Theorem 22.4 bis: (The full Picard contraction principle) Let E be a Banach space and let B be a bounded bilinear transform from E E to E : kB (e; f )kE CB kekE kf kE : Then, if e0 2 E is so that ke0kE 4C1B , the equation e = e0 B (e; e) has a solution such that kekE 2C1B . This solution is unique in the ball B (0; 2C1B ). Moreover, the solution may be constructed recursively by choosing f0 2 B (0; 2C1B ) and de ning fn+1 = e0 B (fn ; fn ): the unique solution e = e0 B (e; e) is the limit of the sequence fn . Proof: Let F be the map F (e) = e0 B (e; e). If ke0 kE 4C1B , then F maps the ball B (0; 2C1B ) to itself: kF (e)kE ke0 kE + CB kek2E 4C1B + CB ( 2C1B )2 = 1 2CB . We want to show that F has a xed point e and that the sequence fn is convergent to this xed point e, whatever the choice of f0 . In particular, this xed point is unique (if e0 is another xed point, look at the sequence fn associated to f0 = e0 ). We rst look at the case E = CI and B (z; w) = zw=4. We want to solve z = z0 + z 2=4 when jz0 j 1.p The discriminant of the equation is = 1 z0 ; we de ne the square Z for RZ 0 by writing Z = ei with p root p 2 p i= [ =2; =2] andp Z = e 2 . The solutions are then z + = 2(1p+ 1 z0 ) and z = 2(1 1 z0 ). We have z +z = 4z0 and jz +j 2(1+ R 1 z0 ) 2 (with equality if and only if z0 = 1 and z + = zp = 2), and thus jz j 2 Thus, the unique solution in D (0; 2) is z = 2(1 1 z0 ). For jz0 j < 1, we can develop z in a Taylor series 1 1 X (k 1=2) k X z =z+ z0 = ck z0k : k=2 (1=2) (k + 1) k=1 P1 P k Since thepTaylor coeÆcients are positive, we have k=1 ck = limt!1 1 k=1 ck t = 2(1 1 1) = 2. Thus, the Taylor expansion is still valid for jz0 j = 1. This expansion will be generalized to the Banach space setting. We are going to expand in a Taylor series the solution of e = ze0 B (e; e) for jz j < 1
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Mild solutions
and then show that the expansion is valid for z = 1. We rst look at the analytic functions fn (z ) de ned by f0 = 0 and fn+1(z ) = ze0 B (fn(z ); fn (z )). The Picard contraction principle (Theorem 22.4) shows that the sequence fn (z ) is uniformly convergent on any closedPdisk D (0; ) with < 1. Thus, the limit k f1 (z ) is analytic. We write f1 (z ) = 1 k=1 z k and, substituting the series in the identity f ( z ) = ze B ( f ( z ) ; f ( z )), we get 1 = e0 and, for 2 n, 0 1 1 Pn 1 1 n = k=1 B (k ; n k ). Moreover, we may easily compare kk kE to the k th coeÆcient ck in the Taylor expansion of z . We can prove by induction that we write kn+1 kE kPk kE 4cCkB : we have k1kE =Pke0kE 4C1B = 4Cc1B ; now, n kB ( ; n ck cn+1 k = 1 Pn ck cn+1 k = cn+1 . ) k C k n +1 k E B k=1 P k=1 4CB 4CB 4C1B k=1 4P1 4CB P Since 1 k=1 ck < 1, we nd that limt!1 f1 (z ) = k=1 k . If e = k=1 k , we have e = limt!1 te0 B (f1 (t); f1 (t)) = e0 B (e; e). Hence, the map F has at least one xed point in B (0; 2C1B ). de ne fn+1 = e0 B (fn ; fn ): we We now select f0 2 B (0; 2C1B ) and P1 must prove the convergence of f to attempt n =1 k . By induction,Pwe Pn Pk1 n . We 1 to prove that kfn k c . Let g = n k=1 k E k=n+1 k k=1 k 4CP B 1 c
have kf0 g0 kE = kf0kE 2C1B = 4kC=1B k ; thus, the inequality is true for n = 0. We split fn+1 gn+1 into fn+1 gn+1 = hn B (fn gn ; gn) B (gn ; fn gn ) B (fn gn ; fn Pgn) with hn = e0 B (gn ; gn ) gn+1 . We havePe0 B (gn ; gn ) gn+1 = kn;qn;k+qn+2 T (k ; q ); hence, khn kE CB kn;qn;k+qn+2 4cCkB 4CcqB . On the other hand, the induction hypothesis gives Pn 8 ck 1 P1 < kB (fn gn ; gn )kE CB 4CB Pk=n+1 ck Pk=1 4CB 1 n 1 kB (gn; fn gn )kE CB 4CB k=n+1 ck k=1 4cCkB : 2 kB (fn gn ; fn gn )kE CB 4C1B P1 k=n+1 ck and this gives 1 1 X X X ck cq 1 X ck ]2 = c: kfn+1 gn+1 kE CB [ 4C1 4CB k=n+2 k B k=1 k+qn+1 4CB 4CB P This yields the convergence of fn to the (unique) xed point 1 k=1 gk . We now consider the generalization of Theorem 22.2 to spaces closer to B_ 11;1 . We replace (P ) (controlling the pointwise product in B_ X1;1 \ B_ X;1 by a condition on the pointwise product in the whole B_ X1;1 . We rst consider the inequality (P1 ):
Lemma 22.1: Let X be a shift-invariant Banach space of distributions. Then X satis es the regularity property (P1 ): (P1 ) kfgkB_ X1;1 C kf kB_ X1;1 kgkB_ X1;1 if and only if we have for all j; k; l 2 ZZ and all f; g 2 X :
kj (k f l g)kX C 2k+l j kk f kX kl gkX
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Further adapted spaces
229
Proof: This is obvious. This property may be precised in the case of Besov spaces of local measures:
Proposition 22.2: Let E be a shift-invariant Banach space of local measures. Assume that E is continuously embedded into L2loc(IRd) and that there exists a shift-invariant Banach space of local measures F so that the pointwise product maps boundedly E E to F : for f 2 E and g 2 E , kfgkF C kf kE kgkE . Assume that: i) the norm of E is homogeneous with homogeneity exponent for some 2 (0; 1] (i. e. k f (x)kE = kf kE ) p ii) [F; Cb (IRd)]1=2;1 E : for f 2 F \ Cb , kf kE C kf kF kf k1 then, for 1 q 1, the space X = B_ E 1+ ;q satis es (P1 ) and satis es more precisely the inequalities:
kj (k f l g)kX C 2 2 jl max(k;j)j 2k+l j kk f kX kl gkX : Proof: We have kj f kX 2j( 1) kf kE . If k j 3, we just write that j (k f l g) = 0 if jj lj 4 while kk f l gkE C k k f k1kl gkE C 0 2k kk f kX 2l(1 ) kl gkX . If jk lj 2, we write that j (k f l g) = 0 if j l + 6 while kj (k f l gkE C 2j kk f l gkF C 0 2j kk f kE kl gkE C 00 2(j k l) 2k kk f kX 2l kl gkX . The theorem of Auscher and Tchamitchian is discussed below:
Theorem 22.5: (Auscher and Tchamitchian's adapted spaces) Let X be a shift-invariant Banach space of distributions.such that: i) the norm of X is homogeneous of exponent 1: kf (x)kX = kf kX ii) there exists a sequence of positive numbers (n )n2IN so that we have the inequalities for all j; k; l 2 ZZ and all f; g 2 X :
kj (k f l g)kX jl max(k;j)j 2k+l j kk f kX kl gkX : Then: P (A) If n2IN n < 1, the bilinear operator B is bounded on EX \BX where EX = L1 ((0; 1); (B_ X0;1 )d ) and BX = f~u = t3=4 ~u 2 (L1 ((0; 1); (B_ X3=2;1 )d g. ~ :u~0 = 0 and Thus, there exists a positive X so that for all u~0 2 X d so that r k~u0kX < X , there is a solution ~u of the Navier{Stokes equations on (0; 1) with initial value u~0 so that ~u 2 EX \ BX .
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Mild solutions P
(B) If n2IN nn < 1, the bilinear operator B is bounded on FX \ BX where FX = L1 ((0; 1); X d ) and BX = f~u = t3=4 ~u 2 (L1 ((0; 1); (B_ X3=2;1 )d g. ~ :u~0 = 0 and Thus, there exists a positive X so that for all u~0 2 X d so that r k~u0kX < X , there is a solution ~u of the Navier{Stokes equations on (0; 1) with initial P value u~0 so that ~u 2 L1 ((0; 1); X d). (C) If n2IN nn < 1 and if 0 maps X to C0 , X is continuously embedded into BMO( 1) .
Proof: The proof of (A) is quite easy. If f~ 2 EX \ BX , we have kf~(t)kB_ X0;1 kf~kEX and kf~(t)kB_ 3=2;1 kf~kBX t 3=4 ; hence,
X
kj f~(t)kX min(kf~kEX ; kf~kBX (2j p1t)3=2 ): ~ g) is still in EX \ BX , we In order to prove that for f~ and ~g in EX \ BX , B (f;~ ~ must estimate the size of kj (B (f;~g)(t))kX . We de ne the norm of X = EX \ BX as
kf~kX =
sup max(1; 23j=2 t3=4 )kj f~(t)kX :
t>0;j 2ZZ
~ g) = B1 (f;~ ~ g) + B2 (f;~ ~ g) + B3 (f;~ ~ g) with We then write B (f;~ 8 > <
~ g) = R0t e(t s)IPr ~ : Pj2ZZ Sj 2 f~ j ~g ds B1 (f;~ ~ g) = R0t e(t s)IPr ~ : Pj2ZZ j f~ Sj 2~g ds B2 (f;~ > R : ~ g) = 0t e(t s)IPr ~ : Pj2ZZ Pjj kj2 j f~ k~g ds: B3 (f;~ ~ g)(t)) = R0t e(t s)IPr ~ :j (Pjl jj3 Sl 2 f~ l~g) ds so that We have j (B1 (f;~ ~ j = r ~ 1 j ) (writing r R t C kf~kX k~g kX p
0
t s
~ g)(t))kX kj (B1 (f;~
min(1; 4j (t1 s) ) kj 2k min(1; 32k1 34 ) min(1; 3j1 3 ) ds 2 s 2 2 s4 P
P Writing kj 2k min(1; 23k=12 s3=4 ) min(1; 23j=21s3=4 ) kj 2k 2j1ps = p2s and ~ g)(t))kX C 0 kf~kX k~gkX . On the min(1; 4j (t1 s) ) 1 gives that kj (B1 (f;~ P other hand, we have that k2ZZ 2k min(1; 23k=12 s3=4 ) C p1s and we write P
Rt
1 1 1 0pt sps min(1; 4j (t s) ) min(1R; 2 j= s = ) ds 0 41j 2t 3=4 p1s min(1; 2 j=1s = ) ds + t=t 2 pt 1sps 42j t 3=4 ds C 2 j=1t = : R1
3
3
2 3 4 3
2 3 4
~ g)(t))kX Thus, we get that 23j=2 t3=4 kj (B1 (f;~
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2 3 4
C kf~kX k~gkX .
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231
~ g)(t)). To estimate B3 , we write We have a similar estimate for j (B2 (f;~ R t (t s) P P ~ ~ ~ j j (B3 (f;~g)(t))= 0 e IPr:j ( lj 3 jl kj2 k f~ l~g) ds and IPr ~ j ), so that ~ ( )1 1=4 j = Id Æ (IPr = ( )1=4 IPr Rt
~ g)(t))kX kj (B3 (f;~ P
1 1 k j 2j= (t s) = ) kj 5 4 2 jk jj min(1; 2 k s = ) ds R Since min(2j ; 2j= (t1 s) = ) 2j , jk jj k(n )k1 and 01 4k min(1; 2 k1s = ) ds R ~ g)(t))kX C kf~kX k~gkX . On = 01 min(1; s 1= ) ds < 1, we get that kj (B3 (f;~ 1 j the other hand, we write min(2 ; 2j= (t s) = ) 2j= (t1 s) = and then Rt 1 k 1 0 ( t s) = 4 min(1; 2 k s = ) ds R R 01 2t 3=4 4k min(1; 2 k1s = ) ds + t=t 2 (t s1) = 2t ds Ct 3=4 : P P ~ g)(t))kX Since kj 5 jk jj = n 5 jnj < 1, we get that 23j=2 t3=4 kj (B3 (f;~ C kf~kX k~gkX and (A) is proved.
0 C kf~kX k~gkX
min(2j ;
2
2
3 4
3
3 4
3 2
3
3 2
3 2
2
3 4
3 4
2
3
3
3 4
3 2
3 2
3 4
We now prove (B). Since FX is continuously embedded into EX , it is suf cient (according to (A)) to prove that B maps (EX \ BX )P (EX \ BX ) to FX . In the proof of (A), the estimates for B1 showed that k l2ZZ Sl 2 f~(s)
~ l g(s)kB_ X0;1 C kf kXpks~gkX ; we have seen in several instances that such an ~ g) 2 L1 ((0; 1); (B_ X0;1 )d ). We use the same estiestimate gives that B1 (f;~ mates for B2 . The diÆculty lies P in the third term B3 . We want to show ~ g)kX C kf~kX k~gkX . Since that we have the inequality supt>0 j2ZZ kj B3 (f;~ p 3=2 j ~ ~ kj B3 (pf;~g)kX C kf kX k~gkX (2 t) , we may limit the sum to the j 's such that 2j t 1. We want to estimate R P P I (t) = 2j pt1 0t min(2j ; 2j=2 (t1 s)3=4 ) kj 5 4k 2 j jk jj min(1; 23k1s3=2 ) ds R P P = 2j pt1 t kj 5 4k jk jj min(1; 3k13=2 ) ds
0 2 s 1 = ) ds C min(1; 4k t) and get, writing k = We write that 2 kP s j +n t; 1). The sum on the j 's such n + j , that I (t) C n5 jnj 2j pt1 min(4P n + j that 4 t 1 may be estimated through 4j n t1 4n+j t 4=3, while the p j j +n t > 1 may be estimated through sum P on the j 's such that 2 t 1 and 4 P p p 1=(2n t)<2j 1= t 1 1+n ln t=ln2 3n+1. Thus, I (t) C n2IN (1+n)n . We now prove (C). Due to the characterizations of BMO (Proposition 10.1 and Theorem 10.1) and of BMO( 1) (Proposition 16.1), it is suÆcient to prove that for f 2 X the sequence (2 j j f )j2ZZ is a Carleson measure on ZZ IRd , Rt k min(1; 04P
3
3 2
+
i.e., we must show that sup
x0 2IRd ;R>0
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R d
Z
X
B (x0 ;R) 2j R1
4 j jj f j2 dx < 1:
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Mild solutions R
We write I (x0 ; R) = B(x0 ;R) 2j R1 4 j jj f j2 dx. We choose ' 2 D(IRd ) so that ' P0 and ' =R 1 on B (0; 1) and we de ne 'x0 ;R = '( x Rx0 ). Thus j 'x ;R jj f j2 dx. Since limx!1 j f (x) = 0, we have I (x0 ; R) 2jP 0 R1 4 R that jR j f j2 = l2ZZ l jj f j2 in S 0 . Thus, we may write 'x0 ;R jj f j2 dx = P P 2 ~ ~ l2ZZ l 'x0 ;R l jj f j dx with l = jk lj2 k . We now write P
kl jj f j2kX C 4j 2 ljj lj kf k2X for l j + 3 l jj f j2 = 0 for l j + 4: This gives
I (x0 ; R) C kf k2X
X
X
2j R1 lj+3
~ l 'x0 ;R kX () 2 l jj lj k
where X () is the predual of X . The norm of X () (which is isometrically embedded into the dual X of X ) is shift-invariant and has homogeneity exponent 1 d: kg(x)kX ( () = 1 d kgkX () . The norm of B_ X1(;1 has then ) l d ~ homogeneity exponent d. We then write 2 kl 'x0 ;R kX () CR 1 k'kX () ~ l 'x0 ;R kX () CRdk'kB_ 1;1 (we have ' 2 B_ 10;1 B_ 1(;1 and 2 l k X ) , since X () X B_ 11;1 ). We get
I (x0 ; R) CRdkf k2X
X
X
1 jj lj min(1; l ): 2R
2j R1 lj+3 P P It is then enough to write fj2ZZ= 2j n R1g 2j 1n R 2 and fj2ZZ= 12j R<2n g 1 P (1 + 3n) to get that I (x0 ; R) CRd kf k2X n2IN (1 + n)n .
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Chapter 23 Cannone's approach of self-similarity
In his book [CAN 95], Cannone gave a very clear strategy for exhibiting selfsimilar solutions to the Navier{Stokes equations. We take a shift-invariant Banach space of distributions X whose norm is homogeneous with the good scaling for the Navier{Stokes equations (kf (x)kX = 1 kf kX ). This that p implies 1 ; 1 t _ X B1 , or equivalently that for all f 2 X we have sup0
In his book, Cannone [CAN 95] describes, how Besov spaces are used to provide self-similar solutions for the Navier{Stokes equations. The general strategy may be described by the following theorem: 233 © 2002 by CRC Press LLC
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Mild solutions
Theorem 23.1: Let E be a shift-invariant Banach space of local measures so that the norm of E is homogeneous: for some 2 (0; 1], we have, for all f 2 E and all > 0, kf (x)kE = kf kE . Then: (A) the bilinear operator B is bounded on E d where E is de ned by f
8 < t(1 )=2 f
2 E , : p
2 L1 ((0; +1); E ) and
2 L1 ((0; 1) IRd) p with norm kf kE = supt>0 t(1 )=2 kf kE + supt>0 tkf k1. 1+ ;1 tf
(B) Let E be de ned as E = E if = 1 and E = B_ E Then, for f0 2 S 0 (IRd ), f0 2 E , et f0 2 E . (C) Let C0 be the norm of the bilinear operator B on E d :
C0 =
sup
kf~kE 1;k~gkE 1
if 0 < < 1.
~ g)kE kB (f;~
and let C1 be the norm of the linear operator f~0 7! etf~0 from E d to E d . ~ :~u0 = 0, there exists a solution ~u 2 E d for the Then if k~u0 kE 4C10 C1 and r Navier{Stokes equations on (0; 1) IRd with initial value ~u0 , with k~ukE 2C1 0 . This solution is unique in the closed ball ff~ = kf~kE 2C1 0 g and is smooth on (0; 1) IRd . ~ :~u0 = 0, if > 0 and if ~v0 (x) = In particular, if k~u0kE 4C10 C1 and r ~ :~v0 = 0. Moreover, the solutions ~u and ~u0 (x), then k~v0 kE = k~v0 kE and r ~v of the Navier{Stokes equations on (0; 1) IRd with initial value ~u0 and ~v0 described in point (C) satisfy ~v(t; x) = ~u(2 t; x). ~ :~u0 = 0, if > 0 and if ~u0 is homogeneous: for (D) If k~u0 kE 4C10 C1 and r all > 0, ~u0 (x) = ~u0 (x), then the solution ~u of the Navier{Stokes equations on (0; 1) IRd with initial value ~u0 described in point (C) satis es ~u(t; x) = p1t ~u(1; pxt ).
Proof: Point (A) was proved in Chapters 17 and 20. Point (B) was proved in Chapter 5. Point (C) is based on the Picard contraction principle when k~u0kE < 4C10 C1 ; the case k~u0kE = 4C10 C1 was proved in Chapter 28. The identity ~v(t; x) = ~u(2 t; x) is a consequence of the homogeneity of the Navier{ 2 Stokes equations, of the identity (et~v0 )(x) = (e t~u0 )(x), of the homogeneity of the norms in E and E and of the uniqueness of solutions in the ball B (0; 2C1 0 ). Point (D) is a direct consequence of Point (C). Cannone studied the case of the Besov spaces B_ pd=p 1;1 based on Lebesgue spaces. Such spaces contain nontrivial homogeneous distributions. For instance, d=p 1;1 d 1 (IR ), 1 p 1. (Notice that those jxj belongs to all the spaces B_ p
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Self-similarity
235
spaces form a monotonous scale: B_ pd=p 1;1 B_ qd=q 1;1 for 1 p q 1). Cannone [CAN 95] gave a precise description of homogeneous distributions in B_ pd=p 1;1 . First, we need to de ne Besov spaces on the sphere S d 1. We begin with an easy lemma:
Lemma 23.1: a) Let ' 2 D(IRd ). Then the pointwise multiplication operator M' de ned by M'f = 'f is bounded on every Besov space Bqs;p , s 2 IR, p 2 [1; 1] and q 2 [1; 1]. b) Let 1 and 2 be two di eomorphic open subsets of IRd and h be a smooth di eomorphism from 1 onto 2 . For every ' 2 D( 1 ), the map f 7! ' f Æ h de ned from D0 ( 2 ) to D0 ( 1 ) is continuous in the Bqs;p norm for every s 2 IR, p 2 [1; 1] and q 2 [1; 1]: if K is a compact subset of 2 so that h( Supp ') is contained in the interior set of K and if f 2 Bqs;p (IRd ) has its support contained in K , then k' f Æ hkBqs;p(IRd ) C (K; s; p; q)kf kBqs;p(IRd ) . Proof: a) is very easy. Since pointwise multiplication by a bounded function is a bounded operator on Lp for every p 2 [1; 1], the Leibnitz rule for computing the derivatives of 'f shows that M' is a bounded operator on the Sobolev spaces W k;p for every k 2 IN and every p 2 [1; 1]. By interpolation, we get that M' is a bounded operator on the Besov spaces Bqs;p for every s > 0, every p 2 [1; 1] and every q 2 [1; 1]. When s 0, we may write s = 2N with > 0 and N 2 IN. Then, f belongs to Bqs;p if and only if (Id ) N f belongs tp Bq;p . Thus, we write M'f = M'(Id )N (Id ) N f , and we again use the Leibnitz rule to nd smooth, P compactly supported functions ' ;N so that, for g 2 S 0 , M'(Id )N g = j j2N @ (M' ;N g). We now prove b). For f 2 D0 ( 2 ), the distribution f Æ h 2 D0 ( 1 ) is de ned by hf Æ hj!iD0 ( 1 );D( 1 ) = hf j j det (dh 1 )j ! Æ h 1 iD0 ( 2 );D( 2 ) where dh 1 is the Jacobian matrix of h 1 . In particular, we easily check that @x@ k (f Æ h) = Pd @ @ j =1 @xk hj ( @xj f ) Æ h. Thus, if ' 2 D( 1 ) and if 2 D( 2 ) is identically equal to 1 in a neighborhood of h( Supp '), the mapping f 7! ' ( f ) Æ h is a bounded operator on every Sobolev space W k;p with k 2 IN and p 2 [1; 1]; hence, by interpolation, on every Besov space Bqs;p with s > 0, p 2 [1; 1] and q 2 [1; 1]. Similarly, we nd that the mapping ! 7! ( !) Æ h 1 is bounded on the closure of the test functions in every Besov space Bqs;p with s > 0, p 2 [1; 1] and q 2 [1; 1], where (x) = j det (dh 1 )j ' Æ h 1 and (x) = Æ h; thus, by duality, we get that f 7! ' ( f ) Æ h is bounded on every Besov space Bqs;p with s < 0, p 2 [1; 1] and q 2 [1; 1]. The case s = 0 then follows by interpolation between the case s > 0 and the case s < 0. We then may de ne Besov spaces on a compact smooth manifold:
De nition 23.1: (Besov spaces on a compact manifold) Let M be a compact smooth manifold of dimension Æ. The Besov space Bqs;p (M ) is then de ned by: T 2 D0 (M ) belongs to Bqs;p (M ) if and only if for
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Mild solutions
every ( ; h; ') such that is an open subset of IRÆ , h : 7! M is a smooth di eomorphism from onto h( ) and ' 2 D( ) the distribution T ;h;' de ned by T ;h;'(f ) = T (('f ) Æ h 1 ) belongs to Bqs;p (IRÆ ).
Let ( ; h ) 2A be a nite atlas of M : is an open subset of IRÆ , h is a di eomorphism from to h( ), M = [ 2A h ( ) and A is nite. Let (' ) 2A be a partition of unity subordinated P to the h ( ). We may then de ne the norm of Bqs;p (M ) as kT kBqs;p = 2A kT ;h ;' Æh kBqs;p(IRÆ ) . According to Lemma 23.1, we easily see that the Banach space Bqs;p (M ) does not depend on the choice of the atlas, since all those norms are equivalent.
De nition 23.2: Let F 2 D0 (S d 1 ) and let f
2 \T >0 L1((0; T ); rd 1 dr).
hF ()f (r)j'(x)iD0 (IRd);D(IRd) = hF ()j
1
Z
0
Then the formula
'(r)f(r)rd 1 driD0 (Sd 1 );D(Sd 1)
is well de ned and de nes a distribution F ()f (r) 2 D0 (IRd).
This de nition can be useful when characterizing the homogeneous distributions:
Lemma 23.2: Let T be a distribution on IRd and > d. Then the following assertions are equivalent: (A) T is homogeneous with degree :
8' 2 D(IRd ) 8 > 0 hT (x)jd '(x)i = hT (x)j'(x)i (B) There exists ! 2 D0 (S d 1) such that T (x) = !()r . The distribution ! is then unique. We shall write ! = T jSd . 1
Proof: (A) is obviously a consequence of (B). Conversely, let us de ne a dis0 (S d 1 ) by choosing a real-valued function 2 D((0; 1)) so tribution R 1 ! 2 D + that 0 (r)r d 1 dr = 1: we then de ne ! by h!j'()iD0 (Sd 1 );D(Sd 1) = hT j'( jxxj ) (jxj)iD0 (IRd );D(IRd ) . We rst prove that we have T (x) = !()r in D0 (IRd f0g), i.e., that for all ' 2 D(IRd f0g), de ning 2 D(IRd f0g) by Z 1 x (x) = '(x) (jxj) '(r )r +d 1 dr; j x j 0 we have T ( ) = 0. We write x = r in polar R coordinates; hence, (x) = F (r; ) with F 2 D((0; 1) S d 1). We have 01 F (r jxxj )r +d 1 dr = 0, so that @ G(r; ) with G 2 D((0; 1) S d 1 ). Let r d G(r; ) = F (r; ) = r1 d @r @ + (d + ) = Pd x @ + (d + ). Since T (x); we get (x) = r @r i=1 i @xi
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Self-similarity
237 P
is homogeneous, it satis es the Euler equation di=1 xi @x@ i T = T , and thus hT j i =< Pdi=1 xi @x@ i T + T ji = 0. We nd that the distribution T !()r is supported in f0g, thus its Fourier transform is a polynomial and must be a homogeneous distribution with homogeneity exponent d ; since d < 0, we nd that this must be the null polynomial. A nal lemma will be useful:
Lemma 23.3: a) Let 2 D(IR) with (0) 6= 0. Then for a distribution T 2 D0 (IRd 1 ) and for s 2 IR, p 2 [1; 1] and q 2 [1; 1], the following assertions are equivalent: i) T 2 Bqs;p (IRd 1 ) ii) T 2 Bqs;p (IRd ). b) Let 2 D((0; 1)) with (1) 6= 0. Then for a distribution T 2 D0 (S d 1 ) and for s 2 IR, p 2 [1; 1] and q 2 [1; 1], the following assetions are equivalent: i) T 2 Bqs;p (S d 1 ) ii) T () (r) 2 Bqs;p (IRd). R
Proof: We choose 2 D(IR) so that dx = 1 and we de ne the operators A and B by: T and
2 D0 (IRd 1 ) ! 7 A (T ) = T 2 D0 (IRd )
T 2 D0 (IRd) 7! B (T ) =
Z
T (xd ) dxd 2 D0 (IRd 1 )
(i.e., hB (T )j!i = hT j! i). We have B Æ A = Id; thus, we are able to prove the lemma by showing that A maps boundedly Bqs;p (IRd 1 ) to Bqs;p (IRd ) and that B maps boundedly Bqs;p (IRd ) to Bqs;p (IRd 1 ). Obviously, A maps boundedly the Sobolev space W k;q (IRd 1) to W k;q (IRd) and B maps boundedly the Sobolev space W k;q (IRd ) to W k;q (IRd 1) for every k 2 IN and every q 2 [1; 1]; by interpolation, this gives the required estimates for Besov spaces Bqs;p with positive exponent s. We deal easily with the remaining exponents by noting that, writing d for the d-dimensional Laplacian and d 1 for the (d 1)-dimensional Laplacian, we have, for T 2 D0 (IRd 1), A (d 1 T ) = d 1 A (T ) in D0 (IRd ) and, for T 2 D0 (IRd ), B (d T ) = d 1 B (T )+B d2 (T ) dx2 in D0 (IRd 1). Point b) is a direct consequence of Point a) and Lemma 23.1, by using an atlas on S d 1 . Cannone's result follows:
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Mild solutions
Proposition 23.1: Let p 2 [1; +1] and T be a distribution on IRd which is homogeneous with degree 1. Then the following assertions are equivalent: (A) T 2 B_ pd=p 1;1 (IRd ) (B) T jSd 1 2 Bpd=p 1;p (S d 1). Proof: The proof relies on the characterization of Besov spaces by means of wavelets (see Chapter 9). According to Lemma 23.3, we may replace assertion (B) by (r)T (x)(= r (r)T jSd 1 ()) 2 Bpd=p 1;p (IRd ), where 2 D((0; 1)) with (1) 6= 0. We use d-variate Daubechies compactly supported wavelets. Since T is homogeneous, it is a distribution with nite order K and thus both T and (r)T may be decomposed on a wavelet basis ('(x k))k2ZZd [ ( ;j;k )12d 1;j2IN;k2ZZd , where ( ;j;k )12d 1;j2ZZ;k2ZZd is an orthonormal basis of L2 (IRd ) formed by d-variate Daubechies compactly supported wavelets and where the associated compactly supported scaling function ' is of class C N with N > K . Moreover, we assume N > d 1. P P Let Qj be the projection operator Qj f =P 12d 1 k2ZZd hf j ;j;k i ;j;k and let Pj be the projection operator Pj f = k2ZZd hf j'j;k i'j;k . Then we have the following equivalences for a distribution T of order K < N : T 2 B_ pd=p 1;1 , sup 2j(d=p 1) kQj T kp < 1 and j!lim1 kPj T k1 = 0 j 2ZZ and
(r)T
2 Bpd=p 1;p , (2j(d=p 1) kQj ( T )kp )j2IN 2 lp (IN) and kP0 ( T )kp < 1:
In the case of a distribution T , which is homogeneous with homogeneity exponent 1, we have 2j(d=p 1) kQj T kp = kQ0T kp and 2 j kPj T k1 = kP0 T k1, so that we have: T 2 B_ d=p 1;1 , Q0T 2 Lp and P0 T 2 L1 : p
If T jsd 1 belongs to Bps;p and if we choose 2 D((0; 1)) with = 1 on [1=2; 4], then if Æ is the maximum of the diameters of the supports of the wavelets (1 < 2d) and of the scaling function ', we have Supp (x k) fx 2 IRd = jkj Æ jxj jkj + Æg (and the same for the support of '(x k)), so that if jkj 2Æ and if we write jkj 2 [2j0 ; 22j0 ), we nd that the support of (x k) and the support of '(x k) are contained in the annnulus fx 2 IRd = 12 j 2xj0 j 4g, and we nd
kQ0 T kp C with
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X
1<2d
k hT j (x k)i klp(ZZ) C
X
1<2d
(A + B )
A = k hT j (x k)i 1[0;4Æ](jkj)klp (ZZ) B = k1[2Æ;1)(2j )k hT j (x k)i 1[2j ;22j ) (jkj)klp (ZZ) klp(ZZ) :
Self-similarity
239
There is only a nite number of indices k so that jkj 4Æ; moreover, T = 1 T jS(d 1) (); hence, (since R 1 1 rd 1 dr < 1), the distribution T is conr 0r trolled by the distribution T jS(d 1) and we may write j hT j (x k)i j C (k)kT jS(d 1) kBpd=p 1;p . Thus, A is well controlled. In order to control B , we write for 2j 2Æ and jkj 2 [2j ; 22j ) that hT j (x k)i = h (2 j r)T j (x k)i = 2j(d 1) h (r)T j (2j x k)i; this gives for 2j 2Æ,
k hT j (x k)i 1[2j ;22j ) (jkj)klp (ZZ) 2j(d 1) k h (r)T j (2j x k)iklp (ZZ) C 2j(d=p 1) kQj ( (r)T )kp : We nd that B C kT jS d kBpd=p ;p . The same proof (replacing by ' and p by +1) gives that kP0 T k1 C kT jS d kB1 ;1 C 0 kT jS d kBpd=p ;p . Conversely, let us assume that T 2 B_ pd=p 1;1 . Let 2 D((0; 1)). We want to prove that (r)T 2 Bpd=p 1;p . We de ne for j 0 Kj = fk 2 ZZd = 9 2 f1; : : : ; 2d 1g Supp ;j;k \ Supp (r) = 6 ;g and we similarly de ne K 1 = d fk 2 ZZ = Supp '(x k) \ Supp (r) = 6 ;g. Then, we de ne the distribution (
(
L=
1
1)
X
k2K
1)
(
1
1)
1
hT j'(x k)i '(x k) + 1
X X
X
j 0 k2Kj 12d
hT j
1
;j;k i ;j;k :
We have (r)T = (r)L; since the space Bpd=p 1;p is preserved by pointwise multiplication with smooth functions, hence we shall prove T jSd 1 2 Bpd=p 1;p (S d 1 ) by proving that L 2 Bpd=p 1;p (IRd). This is obvious, since we have by homogeneity of T that X
X X X 2j(d p) kQj Lkpp C j hT j (x k)i jp C 0 kQ0T kpp d j 0 j 0 12 1 k2Kj
since the family Kj satis es supk2ZZd 2. The Lorentz space
P
j 2IN 1Kj (k ) < 1.
Ld;1
Barraza [BAR 96] studied the existence of self-similar solutions for an initial value in the Lorentz space (Ld;1)d . Theorem 23.1 may be applied to E = Ld;1 (with = 0). Moreover, Barraza was able to easily characterize the homogeneous distributions in Ld;1:
Proposition 23.2: Let T be a distribution on IRd which is homogeneous with degree 1. Then the following assertions are equivalent: (A) T 2 Ld;1(IRd )
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Mild solutions
(B) T jSd
1
2 Ld(S d 1 ).
Proof: Let T be a distribution on IRd which is homogeneous of degree 1 and locally integrable for the Lebesgue measure. Then f = T jSd 1 belongs to L1 (S d 1 ). Moreover, T belongs to Ld;1 if and only if supk2ZZ 2kd m(fx 2 IRdP= jT (x)j 2 [2k ; 2k+1 ]g) < 1, while f belongs to Ld(S d 1 ) if and only if k2ZZ 2kd (fx 2 S d 1 = jT (x)j 2 [2k ; 2k+1 ]g) < 1. It is then enough to write that, for r 2 [2j ; 2j+1 ], we have jT (r)j 2 [2k ; 2k+1 ] ) jf ()j 2 [2k+j ; 2k+j+2 ]. Thus, if k = m(fx 2 IRd = jT (x)j 2 [2k ; 2k+1 ]g) and k = (fx 2 S d 1 = jT (x)j 2 [2k ; 2k+1 ]g), then 2kd k C 2kd
X
j 2ZZ
2jd k+j
C 0 kf kdd:
On the other hand, we have X
k2ZZ
2kd k C
X
k2ZZ
m(fx 2 IRd =jxj 2 [2k ; 2k+1 ] and jT (x)j 2 [1=2; 2]g)
so that kf kdd Cm(fx 2 IRd =jT (x)j 2 [1=2; 2]g). As was noted by Cannone [CAN 95], following Taylor [TAY 92] and Federbush [FED 93], homogeneous Morrey{Campanato spaces may provide a good frame for the existence of global solutions and hence for self-similar solutions. Indeed, Theorem 23.1 may be applied to E = M_ p;d (with = 0) for 1 p < d. We may easily extend the results of Proposition 23.2 to the setting of the Morrey{Campanato spaces.
Proposition 23.3: Let T be a distribution on IRd, which is homogeneous with degree 1. Let p 2 [1; d). Then the following assertions are equivalent: (A) T 2 M_ p;d(IRd ) (B) T jSd 1 2 Lp (S d 1 ) (if d 1 p < d) or T jSd 1 2 M p;d 1(S d 1 ) (if 1 p < d 1).
() belongs to M_ p;d with p < d. We see easProof: Let us assume that T (x) = fRR r R ily that f 2 Lp : we have kf kpp d 1 p 01 jf ()jp rd p drr d Cp jxj1jT (x)jp dx. If we assume that p < Rd 1, we must prove that we have moreover sup0 2Sd 1 ;2(0;1=2) p d+1 j 0 j jf ()jp d < 1: R R R p 2(1R 1)d+1 p j 0 j 11+ jf ()jp rd p drr j 0 j jf ()j d C 1 jx j0 2 jT (x)jp dx C kT kdM_ p;d d p 1 : Conversely, let us assume that f belongs to Lp (S d 1) and, if p < d 1, more precisely to M p;d 1(S d 1 ). We want to estimate, for r0 > 0 and x0 2 IRd ,
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Self-similarity p dR
241
p dRR
p d p dr d . I (x0 ; r0 ) = r0 jx x0jr0 jT (x)jp dx = r0 r jr x0jr0 jf ()jr r We write x0 = 0 0 and r = 0 . We have I (x0 ; r0 ) = I (0 ; 00 ). We may assume that x0 = 0 and try to prove that sup0 2Sd 1 ;r0 >0 I (0 ; r0 ) < 1. R We have I (0 ; r0 ) kf kpp r0d p 01+r0 rd p drr ; we get the required inequality for r0 > 1=2. If r0 < 1=2, we write for jr x0 j < r0 that we have r 2 [1 r0; 1+ r0] and j 0 j < 2r0 ; this gives R R I (0 ; r0 ) r0p d j 0 j<2r0 jf ()jp d 11+rr00 rd p drr r0p d (2r0 )max(d p 1;0)kf kp p r0 max((1 r0 )d p 1 ; (1 + r0 )d p 1 ) X
with X p = Lp if p 2 [d 1; d) and X p = M p;d 1 if p 2 [1; d 1].
3. Asymptotic self{similarity
Planchon has shown that the asymptotic behavior of solutions for the Navier{Stokes equations was controlled by the behaviour of the initial value at low frequencies [PLA 96]. We consider initial values with small norms in (B_ pd=p 1;1 (IRd))d . Recall that B is bounded on the space Epd (d < p < 1) 1 d p where Ep = ff 2 L1 loc;t Lx = supt>0 t 2 2p kf kp < 1g, since
ke(t s)IPr~ :~u ~vkp C
1 1 ukEp k~vkEp : d 1 d k~ 1 + (t s) 2 2p s p
In order to describe the asymptotic behavior of solutions as t goes to +1, we introduce the following de nition:
De nition 23.3: d 1 p Let Ep = ff 2 L1 loc;t Lx = supt>0 t 2 2p kf kp < 1g. Two functions f; g 2 Ep 1 d are asymptotic in Ep if limt!+1 t 2 2p kf (t; :) g(t; :)kp = 0. We then have the following easy lemma:
Lemma 23.4: ~ f~) and B (~g;~g) If f~ and ~g belong to Epd and are asymptotic in Ep , then B (f; are asymptotic in Ep . 1 d ~ f~) B (~g;~g) = Proof: Let kf kp;t = t 2 2p kf (t; :)kp . We have the identity B (f; B (f~ ~g; f~) + B (~g; f~ ~g); hence,
kB (f;~ f~) B (~g;~g)kp;t C
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Z
1
1 1 ~ gkp;t )kf~ ~gkp;t d 1 d 1 d (kf kp;t + k~ + 0 (1 ) 2 2p p
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Mild solutions
and we conclude by dominated convergence. We have a \converse" result:
Lemma 23.5: Let Cp be the norm of the bilinear operator B on (Ep )d . Let f~ and ~g belong ~ f~) and ~g + B (~g;~g) are to Epd with kf~kEp < 2C1 p and k~gkEp < 2C1 p . If f~ + B (f; asymptotic in Ep , then f~ and ~g are asymptotic in Ep . 1 d Proof: Let kf kp;t = t 2 2p kf (t; :)kp . De ne (t) = kf~ ~gkp;t and (t) = kf~ + B (f;~ f~) ~g B (~g;~g)kp;t . Writing again B (f;~ f~) B (~g;~g) = B (f~ ~g; f~) + B (~g; f~ ~g), we get for R > 1 (t) (t) + I (R) + J (R) with
I (R)
R ~ :[f~ (f~ ~g) + (f~ ~g) ~g] dskp;t = k 0t=R e(t s)Æ IPr R 1=R 11 d 1 d [kf kp;t + kgkp;t ] (t ) d (R); C 0 1 + 2 2p p
(1 )
C R 1=R 4Cp2 0
1 d , and 1 (1 ) dp dp Rt ~ :[f~ (f~ ~g) + (f~ ~g) ~g] dskp;t J (R) = k t=R e(t s)IPr Æ supt=R<s
where (R) =
1+ 2 2
1
with Æ = Cp (kf~kEp + k~gkEp ) < 1. The main point is that Æ does not depend on R and is less than 1 while limR!1 (R) = 0. Finally, we de ne (t) = sups>t (s) and (t) = sups>t (s). We have for N 2 IN
(RN +1 ) (RN +1 ) + (R) + Æ (RN )
which gives
(RN )
N X
1 ÆN (Rk )ÆN k +
(R) + ÆN (1); 1 Æ k=1
we then let N go to +1 and get lim supt!+1 (t) (R) and then we let R go to +1 to get limt!+1 (t) = 0.
Ep :
Thus, we may describe the asymptotics of the Navier{Stokes solutions in
Theorem 23.2: (Equivalence in Ep ) 1 d p Let p 2 (d; 1). Let Ep = ff 2 L1 loc;t Lx = supt>0 t 2 2p kf kp < 1g and let Cp be the norm of the bilinear operator B on (Ep )d . Let ~u0 and ~v0 in ~ :~u0 = r ~ :~v0 = 0 and such that ket~u0kEp < 4C1 (B_ pd=p 1;1 (IRd ))d be such that r p
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Self-similarity
243
and ket~v0 kEp < 4C1 p ). Let ~u and ~v be the mild solutions of the Navier{Stokes equations with initial value ~u0 and ~v0 (with k~ukEp < 2C1 p and k~vkEp < 2C1 p ). Then the following assertions are equivalent: (A) ~u and ~v are asymptotic in Ep (B) et~u0 and et~v0 are asymptotic in Ep (C) limj! 1 kSj (~u0 ~v0 )kB_ pd=p 1;1 = 0 (D) limj! 1 2j(d=p 1) kj (~u0 ~v0 )kp = 0.
Proof: (A) implies (B) by Lemma 23.4, since we have et~u0 et~v0 = ~u ~v + B (~u; ~u) B (~v ; ~v). (B) implies (A) by Lemma 23.5. j (B) ) (D): We notice that e 4 j is continuous on Lp with an operator norm which does not depend on j ; thus, for t = 4 j , 2j(d=p 1) kj (~u0 ~v0 )kp d 1 Ct 2 2p ket(~u0 ~v0 )kp . (C) , (D) is obvious. (C) ) (B): For f 2 B_ pd=p 1;1 and j 2 ZZ, we have (Id Sj )f 2 B_ pd=p 2;1 ; 1 d hence, t 2 2p ket(Id Sj )f kp is O(1=t) and goes to 0 as t goes to +1. Thus, 1 d we get that lim supt!+1 t 2 2p ketf kp kSj f kB_ pd=p 1;1 . From Theorem 23.3, Planchon's result is more precise:
Theorem 23.3: 1 d p Let p 2 (d; 1). Let Ep = ff 2 L1 loc;t Lx = supt>0 t 2 2p kf kp < 1g and let Cp be the norm of the bilinear operator B on (Ep )d . Let ~u0 (B_ pd=p 1;1 (IRd ))d be ~ :~u0 = 0 and such that ket~u0 kEp < 4C1 . Let ~u be the mild solution such that r p of the Navier{Stokes equations with initial value ~u0 (with k~ukEp < 2C1 p ). Then the following assertions are equivalent: (A) ~u is asymptotically self-similar in the sense that there exists a function d 1 V~ 2 (Lp (IRd ))d so that limt!1 t 2 2p k~u p1t V~ ( pxt )kp = 0 (B) et~u0 is asymptotically self-similar in the sense that there exists a function 1 d U~ 2 (Lp (IRd))d so that limt!1 t 2 2p ket~u0 p1t U~ ( pxt )kp = 0 (C) There exists a distribution ~v0 in (B_ pd=p 1;1 (IRd ))d homogeneous with degree 1 so that limj! 1 2j(d=p 1) kj (~u0 ~v0 )kp = 0. In this case, we have U~ = e~v0 and the function p1t V~ ( pxt ) is the mild solution of the Navier{Stokes equations with initial value ~v0 . Proof: (A) implies (B) by Lemma 23.4: if f~(t; x) = p1t V~ ( pxt ), we have that ~ f~) = ~g(t; x); we then de ne et~u0 = ~u + B (~u; ~u) is asymptotic in Ep to f~ + B (f; ~U as ~g(1; x). p p (B) implies (C). We write ~z0;t (x) = t~u0 ( t x). (B) is equivalent to limt!+1 ke~z0;t U~ kp = 0. This gives for all j 2 ZZ that limt!+1 kSj ~z0;t
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244
Mild solutions
e Sj U~ kp = 0. Since the norms k~z0;tkB_ pd=p 1;1 remain bounded, we nd that ~z0;t is weakly convergent to a distribution ~v0 . This distribution is then homogeneous with degree 1. Moreover, we have U~ = et~v0 . Then, Theorem 23.2 gives that limj! 1 2j(d=p 1) kj (~u0 ~v0 )kp = 0. Finally, (C) implies (A): if (C) is satis ed, then ~v0 is the weak limit of 2j ~u0 (2j x) as j goes to 1; hence, ket~v0 kp lim inf j! 1 2j ket(2j ~u0 (2j :))kp t 2dp 12 ket~u0 kEp < 2C1 p . We then apply Theorem 23.2 and get that ~u is asymptotically self-similar, since the solution ~v(t; x) associated with ~v0 is selfsimilar.
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Part 5:
Decay and regularity results for weak and mild solutions
© 2002 by CRC Press LLC
Chapter 24 Solutions of the Navier{Stokes equations are space-analytical
In this chapter, we prove that an elementary modi cation of the proof of existence of mild solutions gives a proof of the spatial analyticity of those solutions. Spatial analyticity was rst discussed by Masuda [MAS 67] and time analyticity was considered by Foias and Temam [FOIT 89]. We consider only spatial analyticity. In order to avoid technical notations, we only consider global solutions in three simple cases: global solutions in B_ Pd M1;1 (Le Jan and Sznitman's solutions [LEJS 97]), global solutions in H_ d=2 1 and global solutions in Ld. For each case, we brie y recall the construction of global solutions based on the Picard contraction principle (see Chapter 15), then we prove spatial analyticity. First, we state the Picard contraction principle for global solutions:
Proposition 24.1: (The Picard contraction principle) Let E be a Banach space of functions Rde ned on (0; +1) IRd so that the ~ :(~u ~v) ds is bounded bilinear operator B de ned by B (~u; ~v) = 0t e(t s) IPr d d d from E E to E . Let E be the space de ned by f 2 E if and only if f 2 S 0 (IRd ) and (etf )t>0 2 E . Then, there exists a positive constant CE so that ~ :~u0 = 0 and k(et~u0)t>0 kE < CE , there exists a solution for all ~u0 2 E d with r d ~uR t 2 E for the Navier{Stokes equations with initial value ~u0 : ~u = etu~0 (t s) IPr ~ :(~u ~u) ds. 0e 1. The Le Jan and Sznitman solutions
Le Jan and Sznitman [LEJS 97] considered, as a very simple space convenient to the study of Navier{Stokes equations, the space E of tempered distributions f 2 S 0 (IRd ) so that f^( ) is a locally integrable function on IRd and sup2IRd j jd 1 jf^( )j < 1. This space may be de ned as a Besov space based on the spaces P M of pseudomeasures (i.e. P M is the space of the Fourier transforms of essentially bounded functions: P M = F L1 ). More precisely, E = B_ Pd M1;1 . 247 © 2002 by CRC Press LLC
248
Regularity results
Proposition 24.2: (Existence of global solutions) i) f 2 B_ Pd M1;1 if and only if etf 2 E , where E is de ned by g 2 E if and only if the spatial Fourier transform g^(t; ) is a locally integrable function on (0; 1) IRd and supt>0 sup2IRd j jd 1 jg^(t; )j < 1. ii) The bilinear operator B is bounded from E d E d to E d . iii) There exists a positive constant C0 so that for all ~u0 2 (B_ Pd M1;1 )d with r~ :~u0 = 0 and k(et~u0 )t>0 kB_ Pd M1;1 < C0 , there exists a solution ~u 2 E d for the Navier{Stokes equations with initial value ~u0. Proof: We have only to prove point ii). The proof is based on the following elementary lemmas: Lemma 24.1: If w~ = B (~u; ~v), then we have
jw^~ (t; )j C1 j j
Z t
0
2 e (t s)jj j~u^(s; )j j~v^(s; )j ds
R Lemma 24.2: For all 2 IRd , IRd j 1jd
1
1
jjd
1
d C2 jjd1 2 .
R 2 Lemma 24.3: For all 2 IRd , 0t e (t s)jj ds j1j2 .
This obviously gives the boundedness of B . We now slightly modify the proof to get analyticity.
Theorem 24.1: (Analyticity of global solutions) i) f 2 B_ Pd M1;1 if and only if et f 2 F , where F is de ned by g 2 F if and only if the spatial Fourier transform p g^(t; ) is a locally integrable function on (0; 1) IRd and supt>0 sup2IRd e t jj j jd 1 jg^(t; )j < 1. ii) The bilinear operator B is bounded from F d F d to F d. iii) There exists a positive constant C3 such that for all ~u0 2 (B_ Pd M1;1 )d ~ :~u0 = 0 and k(et~u0 )t>0 kB_ d 1;1 < C3 , there exists a solution ~u 2 F d for with r PM the Navier{Stokes equations with initial value ~u0. p This solution is space-analytic on the domain f(x + iy) 2 CI d = jyj < tg. We just have to prove ii) by transforming Lemma 24.1 into the following lemma: p t Lemma U~ and ~v (t; :) = p t 24.4: If w~ = pB (t~u; ~v), and if ~u(t; :) = e ~ ~ e V , then w~ (t; :) = e W with
jW~^ (t; )j C4 j j
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Z t
0
2 e (t s)=2 jj jU~^ (s; )j jV~^ (s; )j ds:
Fourier analysis
249
Proof: To prove Lemma 24.4, we prove, for 0 < s < t and ; following inequality: p p p 2 2 1 (24:1) e (t s)jj e s j j e s jj e2 e t jj e 2 (t s)jj
2 IRd , the
Such an inequality was already considered by Foias and Temam [FOIT 89]. We rst notice that, due to the triangle inequality, we have
ps j j ps jj ps j j;
p p 2 2 p 2 1 1 hence e (t s)jj e s j j e s jj e 2 (t ps)jj p sjj e 2 (t s)jj . Thus, we s)j j 2. We write I = want prove thatpI =p 21 (t s)pj j2 + ( t p to p s)j j 1 12 ( t + ps)j j . If tj j 2, then we have I 0 < 2, whereas ( pt if tj j < 2, we have I tj j < 2. Theorem 24.1 is then easily proved by using Lemmas 24.4, 24.2, and 24.3. 2. Analyticity of solutions in
H_ d=2 1
Foias and Temam [FOIT 89] proved spatial analyticity for solutions in Sobolev spaces of periodical functions in an elementary way. We adapt this proof to the case of the Sobolev space H_ d=2 1 de ned on the whole space and follow the same line as for the Le Jan and Sznitman solutions. We start with an existence theorem of global solutions in H_ d=2 1 (with a di erent proof than in Chapters 15 and 19):
Proposition 24.3: (Existence of global solutions) i) f 2 H_ d=2 1 if and only if etf 2 E , where E is de ned by g 2 E if and only if the spatialR Fourier transform g^(t; )is a locally integrable function on (0; 1) IRd and IRd j jd 2 supt>0 jg^(t; )j 2 d < 1. ii) The bilinear operator B is bounded from E d E d to E d . iii) There exists a positive constant C0 such that for all ~u0 2 (H_ d=2 1 )d ~ :~u0 = 0 and k(et~u0)t>0 kH_ d=2 1 < C0 , there exists a solution ~u 2 E d for with r the Navier{Stokes equations with initial value ~u0. We have only to prove point ii). Again using Lemmas 24.1 and 24.3, we replace Lemma 24.2 with the following elementary lemma:
Lemma 24.5: Let d 3. There exists a constant C5 so that for all measurable functions U and V on IRd we have Z
j jd
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Z
Z
4 jU V j2 d C5 ( j jd 2 jU ( )j2 d )1=2 ( j jd 2 jV ( )j2 d )1=2
250
Regularity results
Proof: Lemma 24.5 is equivalent to the fact that the pointwise product is a bounded operator from H_ d=2 1 H_ d=2 1 to H_ d=2 2 . Using the decomposition of the product into paraproducts, fg = _ (f; g)+ _ (g; f )+ _(f; g) with _ (f; g) = P j 2ZZ Sj 2 f j g , we nd that
k_ (f; g)kH_ d= C kf kB_ 1 ;1 kgkH_ d= k _ (g; f )kH_ d= C kgkB_ 1 ;1 kf kH_ d= : k_(f; g)kH_ d= C k_(f; g)kB_ d ; C 0 kf kH_ d= kgkH_ d= 8 <
2
2
2
2
1
2
1
2
2
1
2
1
1
2 1
2
1
2
1
Lemma 24.5 readily implies Proposition 24.2 when d 3. It remains to consider the case d = 2, in which case H_ d=2 1 = Ld. We prove analyticity in Ld in the next section. We now slightly modify the proof to achieve analyticity.
Theorem 24.2: (Analyticity of global solutions) i) f 2 H_ d=2 1 if and only if etf 2 F , where F is de ned by g 2 F if and only if the spatial Fourier transform p g^(t; ) is a locally integrable function on (0; 1) IRd and j jd=2 1 (supt>0 e t jj jg^(t; )j) 2 L2 (IRd ). ii) The bilinear operator B is bounded from F d F d to F d. iii) There exists a positive constant C6 so that for all ~u0 2 (H_ d=2 1 )d with r~ :~u0 = 0 and k(et~u0)t>0 kH_ d=2 1 < C6 , there exists a solution ~u 2 F d for the Navier{Stokes equations with initial value ~u0. p This solution is space-analytic on the domain f(x + iy) 2 CI d = jyj < tg. Proof: B is bounded from F d F d pto F d . We write j~u^(t; )j ptjj We check that d= 2 e U ( ) with j j 1 U 2 L2 and j~v^(t; )j e tjj V ( ) with j jd=2 1 V 2 2 L . Wep apply Lemmas 24.4 and 24.3 and get that w~ = B (~u; ~v ) satis es supt>0 e t jj jw^~ (t; )j 2C4 j j 1 U V ( ). To conclude the proof, we must prove R R R 2 that ( j jd 2 jUjVj2 j d )1=2 C5 ( j jd 2 jU ( )j2 d )1=2 ( j jd 2 jV ( )j2 d )1=2 , which is exactly the content of Lemma 24.5.
3. Analyticity of solutions in
Ld
Spatial analyticity for solutions in Lebesgue space was considered by Grujic and Kukavica [GRUK 98]. p Their proof was based on the study of the equation satis ed by ~u(t; (1 + i t)x) for small 's. We give here a di erent proof (Lemarie-Rieusset [LEM 00]) based on multilinear singular integrals.
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Proposition 24.4: (Existence of global solutions) i) f 2 Ld if and only if et f 2 E1 \ E2 , where E1 = Cb ([0; 1); Ld(IRd )) and E2 is de ned by: g 2 E2 , t1=8 g(t; x) 2 L1 ((0; 1); L4d=3(IRd)) and tlim t1=8 kg(t; x)k4d=3 = 0: !0 ii) The bilinear operator B is bounded from E2d E2d to E1d \ E2d . iii) There exists a positive constant C0 such that for all ~u0 2 (Ld )d with r~ :~u0 = 0 and k(et~u0 )t>0 kE2 < C0 , there exists a solution ~u 2 E1d \ E2d for the Navier{Stokes equations with initial value ~u0 .
Proof: Point i) is a simple consequence of the Bernstein inequalities (which give Ld B_ d0;1 B4d=1=34;1 ) and of the characterization of Besov spaces through the heat kernel. Point ii) is obvious: we just write for ~u and ~v in E2d that
kB (~u; ~v)k4d=3 C and
Z t
ds 1 s1=8 k~uk4d=3 s1=8 k~vk4d=3 1=4 7 = 8 j t s j s 0
Z t
1 ds s1=8 k~uk4d=3 s1=8 k~vk4d=3 1=4 : 3 = 4 s 0 jt sj This proves Proposition 24.4.
kB (~u; ~v )kd C
Once again, we slightly alter the proof and get analyticity of the solutions:
Theorem 24.3: (Analyticity of global solutions) P Let 1 be the operator de ned by the Fourier multiplier k k1 = dj=1 jj j: R 1 f (x) = (21)d eix: k k1f^( ) d . Then: i) f 2 Ld if and only if etf 2 F1 \ F2 , where p F1 = ff 2 Cb ([0; 1); Ld(IRd))= sup ke t1 f kd < 1g t>0 and
p F2 = fg 2 L1loc((0; 1); L d )= sup t ke t gk d < 1 and 4 3
t>0
1 8
1
4 3
lim t 8 kgk 43d = 0g:
t!0
1
ii) The bilinear operator B is bounded from F2d F2d to (F1 \ F2 )d . iii) There exists a positive constant C7 so that for all ~u0 2 (Ld )d with ~r:~u0 = 0 and k(et~u0 )t>0 kd < C7 , there exists a solution ~u 2 (F1 \ F2 )d for the Navier{Stokes equations with initial value ~u0. p This solution is space-analytic on the domain f(x + iy) 2 CI d = jyj < tg.
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Regularity results The proof relies on several basic lemmas.
Lemma 24.6: Let 1 p < 1. If Tj 2 L(Lp (IR); Lp (IR)) for 1 j d, then T1 : : : Td de ned by T1 : : : Td('1 : : : 'd ) = T1('1 ) : : : Td('d ) belongs to L(Lp (IRd); Lp (IRd)). Proof: We may assume that all the Tj but one are identity operators (then write T1 : : : Td = (T1 Id : : : Id)Æ(Id T2 : : : Id)Æ: : :Æ(Id : : : Id Td)). But we obviously see that Id : : : Id Tj Id : : : Id operates on Lp (IRd ) = Lp (dx1 : : : dxj 1 dxj+1 : : : dxd ; Lp (dxj )). Lemma 24.7: p p Let 1 < p < 1. The linear operators At;s de ned by At;s = e(t s)=2 e( t s) 1 for 0 s < t < 1 are equicontinuous on Lp (IRd ). Proof: We may write At;s = Tt;s : : : Tt;s where Tt;s is an operator on Lp (IR): Z t s 2 p p 1 eix f^( )e 2 +( t s)jj d Tt;s f (x) = 2 IR We may then use the Littlewood{Paley theory for functions in Lp and use more precisely the Marcinkiewicz multiplier theorem: a Fourier multiplier m(D) with m 2 C 1 1 (IR ) operates on Lp (IR) (1 < p < 1) if we have m 2 L1 (IR ) and dd m( ) 2 L1 (IR ) and we have km(D)kL(Lp;Lp) Cp (kmk1 + k ddk mk1 ) t s 2 p p (Stein [STE 70]). We p have just to estimate pt;s p= ke 2 +( t s)jj k1 and ps)j j)e t 2 s 2 +( t s)jj k . We know, from the t;s = k((t s) 2 ( t p p 1 proof of Lemma 24.4, that 41 (t s) j j2 + ( t s)j j 4. Thisp gives p that 2 2 t;s e4 and that t;s 5k 41 (t s) jp j2 e 41 (t s)jj k1 ke 14 (t s)jpj +(p t s)jj k1 ps)j j e 14 (t s) jj2 +( t s)jj k and 2 1 +ke 4 (t s) jj k1 k 14 (t s) j j2 + ( t 1 1=2 e 4 4 p nally t;s 5 2 e + 4e . Lemma 24.8: Letp2 < p < 1. The bilinear operators Bt de ned by Bt (f; g) = p p e t 1 (e t 1 f:e t 1 g) for t > 0 are equicontinuous from Lp Lp to Lp=2 . Proof: We have Bt (f; g) =
1 (2)d
Z Z
p eix:(+) e t(k+k1 kk1 kk1 ) f^( )^g() d d
We then split the domain of integration on subdomains, depending on the sign of j , of j andRof j + j . For this, we introduce the monodimensional R operators K1 f = 21 0+1 eix f^( ) d , Kp1f = 21 01 eix f^( ) d , Lt;1 ;2 f = R f if 1 2 = 1 and Lt;1;2 f = 21 eix e 2 tjjf^( )d if 12 = 1. We de ne for
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~ = ( 1 ; : : : ; d ) and ~ = ( 1 ; : : : ; d ) in f 1; 1gd and for t > 0 the operators Zt;~ ; ~ = K 1 Lt; 1; 1 : : : K d Lt; d ; d . Then, Bt (f; g) =
X
( ~ ; ~;~ )2f 1;1gd
K 1 : : : K d Zt;~ ; ~ f Zt;~ ;~ g
3
The operators Zt;~ ; ~ are equicontinuous on Lp (IRd ) for 1 < p < 1 (because the kernel of e jDj in dimension 1 is given by 1 1+1x2 2 L1 (IR) and since the operators K1 and K 1 are combinations of the identity operator and of the Hilbert transform), and the operators K 1 : : : K d are bounded on Lp=2 for 2 < p < 1. This nishes the proof.
Proof of Theorem 3: Point i) is a direct consequence of Lemma 24.7: we just write et f = e t1 At;0 et=2 f . Since et=2 f belongs to E1 \ E2 , et f belongs to F1 \ F2 . p In order ii), we write for f~ and ~g in (F2 )d : f~ = e t1 F~ ptto prove point 4d=3 d 1G ~ with t1=8 F~ and t1=8 G ~ in (L1 ~) = and ~g = e t Lx ) . We de ne Bt (F~ ; G (Bt (Fj ; Gk ))1j;kd and write ~ g) = e B (f;~
pt
1
Z t
0
~ :Bs (F~ ; G ~ ) ds: At;s e(t s)=2 IPr
~ ) in (L2d=3(IRd))dd . We then use Lemma 24.8 to estimate the norm of Bs (F~ ; G Of course, this proof works as well for local solutions, hence for Lebesgue spaces Lp with p > d. An easy result follows:
Theorem 24.4: ~ :~u0 = 0, there exists T > 0 For all p 2 [d; 1) et ~u0 2 (Lp )d such that r and a solution ~u 2 C ([0; TR ]; (Lp)d ) of the Navier{Stokes equations with pt initial t (t s) ~ t value ~u0 : ~u = e u~0 IPr:(~u ~u) ds so that ~u = e ~v with 0e ~v 2 L1 ([0; T ]; (Lp)d ).
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Chapter 25 Space localization and Navier{Stokes equations In this chapter, we describe recent results of Brandolese [BRA 01] on spatial localization of solutions to the Navier{Stokes equations. Well-localized solutions were described by Furioli and Terraneo [FURT 01] and by Miyakawa [MIY 00]. Limitations on the localization were rst described by Dobrokhotov and Shafarevich [DOBS 94] and extended by Brandolese, who gives some additional interesting constructions of well-localized solutions. 1. The molecules of Furioli and Terraneo
We begin with the molecules described by Furioli and Terraneo [FURT 01] and by Brandolese [BRA 01]. Those molecules were introduced with a double prospect. On one hand, Furioli and Terraneo began their thesis by studying mild solutions in (C ([0; T ); L3(IR3 )))3 (a study which led to the theorem on uniqueness described in Chapter 27); a special case was the subspace of solutions which belong (C ([0; T ); F_ 12;2))3 ; the Triebel-Lizorkin space F_12;2 is the space of distributions vanishing at in nity and such that their Laplacian belongs to the Hardy space H1 ; this space appeared likely to provide a better insight into the solutions to the Navier{Stokes equations: if ~u 2 (H1 (IR3 ))3 , then ~ ~u 2 (L3=2 )33 , so that, by the div-curl theorem (Theorem ~u 2 (L3 )3 and r ~ 12.1), (~u:r)~u 2 (H1 (IR3 ))3 and (since the Riesz transforms operate on H1 ), r~ p 2 (H1 (IR3 ))3 ; thus, the three terms which contribute to @t~u have the same regularity. On the other hand, the Hardy space has a very simple structure, according to the theory of atomic decompositions; an atomic decomposition could then be seen as a model for decomposing a Navier{Stokes ow into simple structures. In terms of the wavelet theory, instead of trying to decompose ~u on a wavelet basis (as discussed in Chapter 12, and as studied by Federbush [FED 93], Cannone [CAN 95], and Meyer [MEY 99] ), we could try to decompose it as a vaguelettes series and see how the vaguelettes evolved. The rst step looks at the evolution of a single vaguelette, before looking at the interaction of vaguelettes due to the nonlinearity. Because of the nonlocal character of the equation (because of the pressure), Furioli and Terraneo did not consider atoms (compactly supported elementary functions in H1 ), but molecules in the 255 © 2002 by CRC Press LLC
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sense of Coifman and Weiss [COIW 77]. They considered an initial value ~u0 R 0 3 ~ so that ~u0 2 (S0 ) , r:~u0 = 0 and IR3 (1 + jxj)2Æ j~u0 j2 dx < 1. They showed that this localization and regularity were preserved on an interval [0; T (~u0)], provided that Æ 2 (3=2; 9=2) and Æ 1=2 2= IN. Brandolese [BRA 01] simpli ed this proof by using the localization of homogeneous Sobolev spaces introduced by Bourdaud [BOU 88a] [BOU 88b].
Theorem 25.1: Let 2 IN so that > d=2. Let Æ d=2; + d=2 + 1). Let E ;Æ be the space
2 IR so that Æ > d=2 and Æ 2 (
E ;Æ = ff 2 L1 (IRd )= jxj f (x) 2 H Æ and @ (jxj f (x))jx=0 = 0 for j j < Æ
d g 2
normed by kf kE ;Æ = k jxj f kH Æ and let F ;Æ = ff 2 C0 (IRd ) = f^( ) 2 E ;Æ g. Then: a) F ;Æ ff 2 C0 (IRd ) = (1 + jxj)Æ+d=2 f 2 L1 g d b) For every T > 0, the bilinear p B is continuous on (C ([0; T ]; F ;Æ )) , p operator with an operator norm O T (1 + T ) ~ :~u0 = 0, then there exists a positive T and a solution c) If ~u0 2 (F ;Æ )d and r d ~u 2 (C ([0; T ]; F ;Æ )) to the Navier{Stokes equations with initial value ~u0.
The theorem can be proved in the Fourier variable by a series of lemmas on the space E ;Æ and on the realization of the homogeneous Sobolev space H_ Æ . If we look at the de nition of the homogeneous Besov space B_2Æ;2(IRd ) (Chapter 3), we have already seen two de nitions. More precisely, de ne for N 2 IN and P @ f (0)x (so that f 2 C 1 the polynomial PN (@ )f as PN (@ )f = j j
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De nition 25.1: (Realization of the homogeneous Sobolev spaces) Let Æ 2 IR so that Æ d=2 2= IN. Let N (Æ ) be the smallest integer N so that N > Æ d=2. Then the realization H_ Æ of the homogeneous Sobolev spacePB_ 2Æ;2 (IRd ) is de ned as the space of distributions f 2 B_ 2Æ;2 (IRd ) so that f = j2ZZ j f PN (Æ) (@ )j f . For Æ < d=2, we have H_ Æ = B_ 2Æ;2 . For Æ > d=2, we have H_ Æ = ff 2 B_ 2Æ;2 = d @ _Æ dx f (0) = 0 for j j < Æ 2 g. H is thus a space of distributions: if f = 0 in H_ Æ , f = 0 in S 0 . We recall the fundamental result of Bourdaud [BOU 88b]:
Lemma 25.1: (Pointwise multipliers of H_ Æ ) Let Æ 0 so that Æ d=2 2= IN. Let m be a function of class C N on d IR nf0g with N d + 1 + max(0; Æ d=2) so that, for all 2 INd with j j @ m 2 L1 . Then the pointwise multiplier operator M de ned by N , jxjj j dx m Mm f (x) = m(x)f (x) is bounded on H_ Æ . Proof: When Æ < d=2, we notice that the Fourier multiplier m(D) (de ned by F (m(D)f )( ) = m( )f^( )) is a Calderon{Zygmund operator and that the weight jxj2Æ belongs to the Muckenhoupt class A2 ; thus, m(D) is bounded on L2 (jxj2Æ dx) (Stein [STE 93]). This proves that Mm is bounded on H_ Æ for Æ < d=2. We now prove the result for Æ 2 (K + d=2; K + 1 + d=2), by induction on K 2 IN. We rst notice that f 2 H_ Æ for Æ > d=2 if and only if f (0) = 0 and r~ f 2 (H_ Æ 1 )d . Thus, we try to check that @j (mf ) belongs to H_ Æ 1 ; the case of m@j f is dealt with by the induction hypothesis. We write (@j m)f = (jxj@j m) jfxj and we conclude by showing that jfxj belongs to H_ Æ 1 ; indeed, since f (0) = 0, we R P write jfxj = dk=1 jxxkj 01 @k f (tx) dt; by induction hypothesis, jxxkj is a pointwise multiplier of H_ Æ 1 ; moreover, by homogeneity, @k f (tx ) belongs to H_ Æ 1 with R1 Æ d= 2 1 norm t k@k f kH_ Æ 1 ; since Æ > d=2, we nd that 0 k@k f (tx)kH_ Æ 1 dt < 1. We now are able to begin the proof for Theorem 25.1. We rst comment on the requirement \f 2 L1" in the de nition of E ;Æ . This requirement expresses only the fact that f is de ned by its restriction on IRd nf0g:
Lemma 25.2: Let Æ 2 IR so that Æ > d=2 and Æ d=2 2= IN. Let h 2 H Æ (IRd ) be such that @ h(0) = 0 for j j < Æ d2 . Let 2 IR so that d=2 < < Æ + d=2. Then the function H (x) = hjx(jx ) belongs to L1 (IRd ). Proof: The hypothesis on h may be written as h 2 L2 \ H_ Æ . For jxj > 1, h is square-integrable and jxj is square-integrable (since > d=2), so that their product H is integrable. For jxj < 1, we write that h 2 H_ Æ ; hence,
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Regularity results P
h = j2ZZ j h PN (Æ) (@ )j h. To estimate j (x) = j h PN (Æ) (@ )j h, we may estimate the size of each term in the sum and get (according to the Bernstein inequality)
jj (x)j kj hk1 +
1 k@ j hk1 jx j C 2jd=2 (1+2j jxj)N (Æ) ! j j
1 k hk j 2
or we may estimate the size of the remainder in the Taylor integral expansion and get
jj (x)j C
X
j j=N (Æ)
k@ j hk1 jx j C 2jd=2 (2j jxj)N (Æ) kj hk2;
we now use the fact that kj hk2 C khkH_ Æ 2 jÆ and write
jh(x)j C khkH_ Æ jxjN (Æ)
X
2j jxj<1
2j(d=2 Æ+N (Æ) + jxjN (Æ)
1
X
2j jxj1
2j(d=2 Æ+N (Æ)
1) ;
since we have N (Æ) 1 < Æ d=2 < N (Æ) this gives jh(x)j C khkH_ Æ jxjÆ d=2 . Since Æ d=2 > d, this gives that H is integrable in the neighborhood of 0.
Lemma 25.3: Under the assumptions of Theorem 25.1 on and Æ, if f and g belong to E ;Æ and if 2 INd with j j = , then x (f g) belongs to H Æ . Proof: Choose 2 D(IR) so that Supp [ 3=4; 3=4], (x) = ( x) 2 P and k2ZZ (x k) = 1. De ne !(x; y) = ( jxj2jx+jjyj2 ) so that !(x y; y) + !(y; x y) = 1. We then write f = jxFj and g = jxGj ; thus, f g(x) = R F (x y) G(y) R F (x y) G(y) R F (x y) G(y) jyj dy = jx yj jyj ! (y; x y ) dy + jx yj jyj ! (x y; y ) dy = j x y j R R !(y; x y) Fjx(x yjy ) Gjy(jy ) dy + !(y; x y) Gjx(x yjy ) Fjy(jy ) dy. We want to prove R that H (x) = x !(y; x y) Fjx(x yjy ) Gjy(jy ) dy belongs to H Æ . We have kH kH Æ R jG(y)j F (x y) Æ jyj dy supy2IRd k (x y + y) !(y; x y) jx yj kH Æ (dx); since H is a shiftinvariant Banach space of distributions, we may rewrite this inequality into R kH kH Æ jGjy(jy )j dy supy2IRd k (x + y) !(y; x) Fjx(jx ) kH Æ (dx) and try to estimate (x; y)F (x) in L2 (dx) and in H_ Æ (dx), where (x; y) = !(y; x) (xj+xjy ) . The function !(y; x) is smooth and homogeneous with degree 0 on IRddnf(0; 0)g and is p supported by f(x; y) = jyj 3jxjg. Thus, is itself smooth and homogeneous with degree 0 on IRddnf(0 ; 0)g. Moreover, its derivatives with respect to x will @ (x; y )j C j(x; y )j j j C jxj j j . Thus, Lemma satisfy, for all 2 INd, j @x
25.1 gives us control of the norm of (x; y)F (x) in H_ Æ uniformly with respect to y. Since we know that jxGj is integrable, this proves that H 2 H Æ .
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Lemma 25.4:
Under the assumptions of Theorem 25.1 on and Æ , if f and g belong to E ;Æ , if j , k and l belong to f1; : : : ; dg and if t > 0, then the function 2 h(x) = jxxjj jxxkj e tjxj xl f g belongs to E ;Æ with norm
p khkE ;Æ C p1 (1 + t)kf kE ;Æ kgkE ;Æ : t
Proof: We have to prove that jxj h(x) 2 L2 \ H_ Æ . We write X jxj x e tjxj2 x x f g: P jxj h(x) = jxxjj jxxkj l
2 j j= j j= jx j
2 Using Lemma 25.1, we must prove that h (x) = e tjxj xl x f g belongs to 2 L2 \ H_ Æ . From Lemma 25.3, we get that kh k2 C ke tjxj xl k1 kx f gk2 0 1 = 2 C t kf kE ;Æ kgkE ;Æ . On the other hand, we know that x f g belongs to H Æ . We introduce a smooth function 2 D(IRd), which is equal to 1 in the neighborhood of 0. We know that (1 (x))x f g belongs to H Æ , hence to H_ Æ since 1 is equal identically to 0 in the neighborhood of 0. Similarly, we know that xl (x)x f g belongs to H Æ and we are going to check that its derivatives of order less than Æ d=2 are equal to 0: if Æ d=2 , we have f 2 L1 and g 2 L1 , thus f g 2 L1 and xl (x)x f g = O(jxj +1 ) with + 1 > Æ d=2; if Æ d=2 < , we write that, for y 2 IRd , we have jyj jxj=2 or jx yj jxj=2, hence jf (y)j C kf kE ;Æ jxjÆ d=2 or jg(x y)j C kgkE ;Æ jxjÆ d=2 , this gives that f g(x) = O(jxjÆ d=2 ) and thus that xl (x)x f g = O(jxjÆ d=2+1 ). Now, we write 2 2 h = (e tjxj xl ) (1 (x)) x f g + (e tjxj ) (x) xl x f g ;
Lemma 25.1 then gives that kh kH_ Æ
Ct
1=2 (1 + pt)
kf kE ;Æ kgkE ;Æ .
Proof of Theorem 25.1: Point b) is a direct consequence of Lemma 25.4, ~ :f~ ~g with f~(s) 2 since we may decompose the Fourier transform of e(t s)IPr 2 j d d t j j (F ;Æ ) and ~g(s) 2 (F ;Æ ) as a sum of terms jj jkj e l f^m g^n (with j , k, l, m and n in f1; : : : ; dg). Point c) is a consequence of point b), according to the Picard contraction principle. We now prove point a). We already know that F ;Æ C0 . We want to estimate the decay of f 2 F ;Æ at in nity. We write for f 2 F ;Æ h = ( ) =2 f and k = jxj d+ that f = c k h for some positive constant c . We choose a smooth function 2 D(IRd) equal to 1 in a neighborhood of 0 and supported by the ball B (0; 1=2), and we de ne K (x; y) = c !( xjxjy )k (x y) and H (x; y) = c (1 !( xjxjy ))k (x y); then we have: Z Z X @ y f (x) = h(y) K (x y) dy + h(y) (H (x; y) H (x; 0) ) dy: ! j jÆ d=2 @y
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We then write that (1 + jyj) Æ jK (x; y)j C (1 + jxj) Æ jx yj d+ and thus Z
j h(y) K (x y) dyj C kh^ kH Æ (1 + jxj)
Æ
Z
1=2 dy d + d 2 jx yjjxj=2 jx yj
which is O(jxj d=2 Æ ) since > d=2. Similarly, we have the estimate
@ y jyjN (Æ) 1 H ( x; 0) j C ! jxjN (Æ)+d j jÆ d=2 @y X
jH (x; y)
min(1;
1
2jyj jxj ) :
this is obvious by the Taylor integral formula when jyj < jxj=2; on the other hand, when jyj jxj=2, we control H (x; y) by jx yj d+ C jxj d+ and @ d+ +j j ; we then multiply the estimate O( jyjj j ) by @y H (x; 0) by jxj jxjd +j j ( 2jjxyjj )N (Æ) j j 1 , which is greater than 1. This gives R
j h(y) H (x; y) Pj jÆ d=2 @y@ H (x; 0) y ! dyj 1=2 R yj2N (Æ) 2 4jyj2 C kh^ kH Æ jxj N (Æ) d+ +1 j(1+ jyj)2Æ min(1; jxj2 ) dy which is O(jxj d=2 Æ ) since d=2 1 < Æ N (Æ) < d=2. 2. Spatial decay of velocities
Theorem 25.1 proves the existence of solutions that are decaying at in nity like jxj d 1+ for any > 0 (since we have the decay 0(jxj Æ d=2+ ) with the restriction Æ < + 1 + d=2). The decay jxj d 1 is easily obtained (Miyakawa [MIY 00], Brandolese [BRA 01]) as we shall see (in Proposition 25.1) and is optimal (Dobrokhotov and Shafarevich [DOBS 94], Brandolese [BRA 01], Theorem 25.2).
Proposition 25.1: For Æ 2 IR, let YÆ = ff 2 L1loc = (1 + jxj)Æ f 2 L1 g. Then: a) For Æ 2 [0; d + 1] and for every T > 0, the bilinear p operator p B is continuous on (Cb ((0; T ); YÆ ))d , with an operator norm O T (1 + T ) ~ :~u0 = 0, then there exists a positive T b) If ~u0 2 (YÆ )d with 0 Æ d + 1 and r d and a solution ~u 2 (Cb ((0; T ); YÆ )) to the Navier{Stokes equations with initial value ~u0 . Moreover, we may choose T min(1; k~ukYÆ2 ) for a positive constant
, which depends only on d and Æ. ~ :~u0 = 0 and let, for some positive T , a solution c) Let ~u0 2 (L1 )d and r 1 d ~u 2 (Cb ((0; T ); L ) to the Navier{Stokes equations with initial value ~u0. If ~u0 belongs more precisely to (YÆ )d with 0 Æ d + 1, then ~u belongs to (Cb ((0; T ); YÆ ))d .
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Proof: a) is easily seen just by looking at the size of the Oseen kernel; from Proposition 11.1, we know that we have ~ g)(t; x)j C jB (f;~
Z tZ
0
IRd
1
jx yj + pt s d+1
jf~(s; y)jj~g(s; y)j ds dy:
R
dy ~ g)(t; :)k1 p d+1 ptC s and get kB (f;~ j x yj+ t s p C t sup0<s
For Æ = 0, we write
~ g)(t; x) C (I (t; x) + J (t; x)) jB (f;~
sup kf (s; :)kYÆ sup0<s
0<s
where we de ne I and J as 8 > <
R R 1 1 I (t; x) = 0t jyjjxj=2 ds dy jx yj+pt s d+1 (1+jyj)Æ RtR 1 1 > ds dy: : J (t; x) = 0 jyj<jxj=2 jx yj+pt s d+1 (1+jyj)Æ p Æ R R C t 1 We have I (t; x) jx2jÆ 0t IRd d+1 ds dy jxjÆ . For jy j < jxj=2, we p jx yj+ t s have jx yj jxj=2; for Æ < d, we obtain
J (t; x)
Z tZ
0
jyj<jxj=2 jx
p pt s jxjÆ jy1jÆ ds dy C t jxj Æ :
2Æ
yjd Æ R
For d Æ d +1, we write that jyj<jxj=2 (1+dyjyj)Æ C (Æ > d) or C (1+ln jxj) 1 (Æ = d) and that (for jyj < jxj=2) jx2djd+1+1 ; hence, we get jx yj+pt s d+1 J (t; x) Ctjxj d 1 Ctjxj Æ for Æ > d and J (t; x) Ctjxj d 1 (1 + ln jxj) Ctjxj Æ for Æ = d. This gives the boundedness of B from (L1p((0; T ); YpÆ )d (L1 ((0; T ); L1)d to (L1 ((0; T ); YÆ )d with an operator norm O T (1 + T ) . b) is a direct consequence of a), through the Picard contraction principle. We must check that f0 7! (et f0)0
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Regularity results
\T1
k~u(t; :)kYÆ
t p ke ~u0kYÆ + kRBt(~up;1~u)kYÆ C (1 + T )(k~u0 kYÆ + M 0p t s k~u(s; :)kYÆ ds p R C (1 + T )k~u0kYÆ + MC (1 + T ) 0t pt1 s dsk~u0kYÆ p R R +C 2 (1 + T )2 M 2 0t 0s pt1 s ps1 k~u(; :)kYÆ d ds R A + B 0t k~u(; :)kYÆ d p p p with A = C (1 + T )(1 + 2M T )k~u0 kYÆ and B = C 2 (1 + T )2 M 2 . Gronwall lemma then gives that k~ukYÆ AeBt .
The
We now prove that this result cannot be true for Æ > d + 1, by using the inclusion, for Æ > d + 1, YÆ L2 (IRd ; (1 + jxj)d+2 dx):
Theorem 25.2: Let ~u0 be a divergence-free vector eld in (L2 (IRd ; (1+ jxj)d+2 dx))d and let, for some positive T , a solution ~u 2 (C ([0; T ]; L2(IRd ; (1 + jxj)d+2 dx))d to the Navier{Stokes equations u0. Then ~u satis es that, for every R with initial value ~ tR 2 [0; T ], the matrix ( IRd uj (x) uk (x) dx)1j;kd is a multiple of the identity: of ~u0 will not IRd uj (t; x) uk (t; x) dx = (t)Æj;k . In particular, the localization R persist unless ~u0 satis es the orthogonality properties IRd u0;j (x) u0;k (x) dx = (0)Æj;k . Before proving Theorem 25.2, we prove some useful lemmas:
Lemma 25.5:
a) (Restriction theorem) For s > 1=2, the mapping f (x; y ) 2 S (IR IRd 1 ) 7! f (0; y) 2 S (IRd 1 ) has a continuous extension from H s (IRd) to H s 1=2 (IRd 1). b) For s 1, the mapping f (x; y ) 2 H s (IR IRd 1 ) 7! f (x;y) x f (0;y) maps boundedly H s (IRd ) to H s 1 (IRd ).
Proof: a) is quite obvious; looking at the Fourier transforms, we have to prove an inequality such as Z
IRd 1
(1+jj2 )s 1=2 j
Z
IR
f (; ) d j2 d
Z
(1+jj2 +j j2 )s jf (; )j2 d d:
IRIRd 1
This is proved through the Cauchy{Schwarz inequality
j
Z
IR
since
f (; ) d j2
R
IR (1 + j j
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Z
IR
(1 + jj2 + j j2 )s jf (; )j2 d
2 + j j2 ) s
d = (1 + jj
2 )1=2 s R
Z
IR
(1 + jj2 + j j2 ) s d
IR (1 + X
2) s
dX .
Spatial localization
263
To prove b), we write (at least for f 2 S (IRd )) the identity f (x;y) x f (0;y) = @ s 1 . For 0 and 2]0; 1] we have 0 @x f (x; y ) d. Of course, @x f 2 H for g 2 H kg(x; y)kH C ()kgkH 1=2 . (This is obvious for 2 IN by computing the L2 norms of the derivatives up to order ; then extend the result to 2= IN by interpolation.) Thus, we get k f (x;y)x f (0;y) kH s 1 R1 @ 0 k @x f (x; y )kH s 1 d < 1. R1 @
Lemma 25.6: Let d 2. Let ! be a smooth function on IRd f0g which is homogeneous with degree 0. If ! belongs locally to H d=2 (i.e. if '! 2 H d=2 (IRd ) for all ' 2 D(IRd )), then ! is constant. Proof: We prove the result by demonstrating that ! is constant on every subspace of dimension 2. Since the hypotheses are invariant under rotations, we only have to check that ! is constant on the plane x3 = : : : = xd = 0. We use the restriction theorem on H d=2(IRd ) and nd that !(x1 ; x2 ; 0; : : : ; 0) belongs locally to H 1 (IR2 ), and thus that @1 !(x1 ; x2 ; 0; : : : ; 0) and @2 !(x1 ; x2 ; 0; : : : ; 0) belong locally to L2 (IR2 ); since they are homogeneous with degree 1, we nd that they must be equal to 0 (just integrate on the ball j(x1 ; x2 )j 1). Proof of Theorem 25.2: We start from the identity ~u(t) = et~u0
Z t
0
~ :~u ~u ds e(t s)r
Z t
0
~ p ds e(t s) r
R = t e(t s)@
and we de ne qj j p ds. We have @k qj = @j qk . 0 d ) and let ( ) = '( )^ Let ' 2 D (IR qj ( ). Since qj = et u0;j uj j R t (t s) r~ :(uj ~u) ds and ~u(t) is bounded in L2 ((1 + jxj)d+2 dx), we nd that 0e d+2;1 . Thus, j q^j is the sum of a function in H (d+2)=2 and a function in B1 d= 2+1 belongs to H , hence is continuous. Since k j = j k , j is equal to 0 on the hyperplane j = 0. We then apply point b) in Lemma 25.5 and get that j = j gj with gj 2 H d=2. But gj does not depend on j since we have j k gj = j k gk ; hence, gj = gk a.e.. We thus write j = jP g. We use the fact that ~u is divergence free; hence, j2f1;:::;dg @j qj = R t (t s) P @j @k (uj uk ) ds. Taking the Fourier transforms and mulj;k2f1;:::dg 0 e tiplying by '( ), we obtain
ij j2 g( ) = '( ) R
X
Z t
j;k2f1;:::dg 0
e(t s) j k F (uj (s)uk (s))( ) ds:
Let 0t e(t s)F (uj (s)uk (s))( ) ds = Uj;k (t; ). The function Uj;k belongs to d+2;1 ). In particular, '( )(Uj;k (t; ) Uj;k (t; 0)) belongs to L2 \ L1 ((0; T ); B1 ( d +1) = 2 H_ for every t, and so does jjj2k '( )(Uj;k (t; ) Uj;k (t; 0)) (according to Lemma 25.1). Since g 2 H d=2 and since L2 \ H_ (d+1)=2 H d=2, we get
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Regularity results P
j k d=2 for every t. Lemma 25.5 that 1j;kd jj2 Uj;k (t; 0) '( ) belongs to H P j gives that 1j;kd jj2k Uj;k (t; 0) = C (t), thus Uj;k (t; 0) = C (t)Æj;k . Since R R Uj;k (t; 0) = 0t uj (s; y)uk (s; y) ds dy, Theorem 25.2 is proved.
Brandolese has shown how to construct solutions to the Navier{Stokes equations with a better localization than allowed by Theorem 25.2. Those localised solutions are symmetric in the following sense: let be the cycle in IRd (x1 ; x2 ; : : : ; xd ) 7! (x2 ; x3 ; : : : x1 ); then a vector eld ~a is symmetric in the sense of Brandolese if Æ ~a = ~a Æ and if ~aj is an even function with respect to the j th variable xj and an odd function with respect to the kth variable xk for k 2 f1; : : : ; dg, k 6= j . Then Brandolese shows that when ~u0 is a divergence-free symmetric vector eld, which belongs to (YÆ )d with d + 1 < Æ < d + 2, the mild solution associated with ~u0 remains in (YÆ )d [BRA 01]. 3. Vorticities are well localized
Brandolese has shown that, in contrast to the case of velocities (Theorem 25.2), the vorticity may keep a very rapid decay at in nity [BRA 01]. We begin by introducing the vorticity matrix associated to the velocity ~u:
Lemma 25.7: ~ be the operators ~u 7! ROT (~u) = (@j uk @k uj )1j;kd Let ROT and K ~ (M ) = ( P1kd @k mj;k )1jd . and M = (mj;k )1j;kd 7! K ~ u 2 (Ld;1 )d g. Then ROT boundedly maps E d to Let E = fu 2 C0 (IRd ) = r ~ boundedly maps (Ld;1 )dd to E d. Moreover, K ~ Æ ROT = Id (Ld;1)dd and K d on E . Proof: This is obvious since @k is a convolution operator with a kernel that is homogeneous with degree d + 1 and smooth outside f0g and thus belongs to d L d 1 ;1 . De nition 25.2: (Vorticity matrix) If ~u is a solution to the Navier{Stokes equations, its vorticity matrix is de ned by = ROT (~u). The Navier{Stokes equations may be expressed in terms of the vorticity matrix:
Proposition 25.2: (Navier{Stokes equations and vorticity matrix) ~ u 2 (Ld;1)d g. Let ~u0 2 E d with r ~ :~u0 = 0 and let Let E = fu 2 C0 (IRd ) = r
0 = ROT (~u0). Then: (A) There exists a positive T and a solution ~u 2 (C ([0; T ); E )d to the Navier{ Stokes equations with initial value ~u0 .
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Spatial localization
265
(B) The following assertions are equivalent for ~u 2 (C ([0; T ); E )d : (B1) ~u is a solution to the Navier{Stokes equations with initial value ~u0 : there exists p 2 D0 ((0; T ) IRd )d so that 8 < @t ~ u= :
~ :~u ~u ~u r r~ :~u = 0 ~u(0; :) = ~u0
r~ p
(B2) there exists p 2 (D0 (0; T ) IRd )d so that 8 u= < @t ~ :
~ (p + 21 j~uj2 ) ~u + ROT (~u)~u r r~ :~u = 0 ~u(0; :) = ~u0
~ ( ) and (B3) There exists 2 (C ([0; T ); Ld;1)dd so that ~u = K
~ ( )) @t = + ROT ( K
(0; :) = 0
Proof: E is a Banach algebra for the pointwise product: if kukE = kuk1 + P k 1j d @j f kLd;1 , then kuv kE kukE kv kE . We thus obtain the inequality ( ke t s)IPr~ :~u ~vkE C pt1 s k~ukE k~vkE . This gives (A). The equivalence between (B1) and (B2) is a direct consequence of the identities r~ :~u ~u =P(~u:r~ )~u + (r~ :~u)~uPand (~u:r~ )~u = r~ ( 12 j~uj2 ) ROT (~u)~u (i.e. P 1kd uk @k uj = 1kd uk @j uk 1kd uk (@j uk @k uj )). We obviously have (B2))(B3). The converse implication is proved by the following remark. We have local uniqueness for the solutions of the problems (B2) and (B3); thus, if is a solution of (B3) in (C ([0; T ); L1;d))dd and ~u is a solution of (B2) in (C ([0; T~u); E ))d , then we may assume T T~u and
= ROT (~u) on (0; T~u ). Moreover, if T > T~u, we nd that k~ukE remains bounded as t goes to T~u ; since the existence of a solution of (B2) on [t0 ; t0 + ] is granted for = O(k~u(t0 )kE2 ), we see that ~u may be extended up to T . Theorem 25.3: R R Let X = ff 2 L1 (IRd ) = ejxjjf (x)j dx < 1 and ejj jf^( )j dx < 1g, with norm kf kX = kejxjf k1 + kejj f^k1 . Then: A) i) X S . ii) For u 2 X , v 2 X and 1 k d, ku @k v kX C kukX kv kX . iii) For u 2 X , t 2 (0; 1] and 1 k d, ket @k ukX C p1t kukX . ~ u 2 (Ld;1 )d g. Let ~u0 2 E d with r ~ :~u0 = 0 and let B) Let E = fu 2 C0 (IRd ) = r d d
0 = ROT (~u0). If 0 2 X , then there exists a positive T and a solution ~u 2 (C ([0; T ); E )d to the Navier{Stokes equations with initial value ~u0 so that ROT (~u) 2 (C ([0; T ); X )dd.
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Regularity results
Proof: The Schwartz class S may be de ned by the conditions f 2 S if and only if, for all and in INd, x @ f 2 L2 . Integration by parts and Leibnitz rule for di erentiation then gives us that f 2 S if and only if, for all and in INd , x f 2 L1 and @ f 2 C0 . Thus, X S . We have f 2 X ) f^ 2 L1 \ L1 ) j1j f^ 2 L1 . Thus, we get kejxju @k vk1 1 1 jxj we may estimate the Fourier (2)d ke uk1 k jj v^k1 C kukX kv kX . Moreover, R @ @ 1 k k transform of u v by jF (u v)j (2)d ju^()j j 1 j jv^( )j d; we write ejj ejj ej j to obtain Z
@ 1 ejjjF (u k v)j d (2)d
Z
ejjju^( )j d
1 jj since ejj v^ 2 L1 \ L1 loc , we have jj e v^ C kukX kvkX . To control et u for t 1, we write
2 L1
Z
Z
ejj
1 j j jv^( )j d ;
and thus kejjF (u @k v)k1
j@k etuj ju(y)j (4t1)d=2 jx 2t yj e
jx yj2 4t dy
jx yj and ejxj ejyje pt and, thus, we get Z
C ejxjj@k et uj dx p t
Z
R
the control of the Fourier transform is simpler: 2 and thus, since kk e tjj k1 = p12e , Z
jxj2
ejxjjxje 4 dx ejxjjuj dx with C = ; 2(4)d=2
1 ejj jF (@k et u)j d p 2e
Z
F (@k et u) = ik e
tjj2 u^( ),
ejj ju^j d:
This constitutes proof for (A). If 0 belongs to X dd, thenRet 0 2 C ([0; 1]); X dd). The bilinear op~ (!)) ds is bounded on erator de ned by ( ; !) = 0t e(t s) ROT ( K p d d C ([0; T ]; X ) for every T 1 with an operator norm O( T ). Thus, if T is small enough, we may apply the Picard contraction principle to the equation (B3) = et 0 + ( ; ). Hence, (B) is proved.
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Chapter 26 Time decay for the solutions to the Navier{ Stokes equations In this chapter, we describe some classical and some recent results on time decay of solutions to the Navier{Stokes equations. The problem of time decay of the solutions was described by Wiegner [WIE 87] and Schonbek [SCHO 85]. A discussion of limitations on the decay was recently given by Mikayawa and Schonbek [MIYS 01] and Brandolese [BRA 01] who gives some additional interesting constructions of rapidly decaying solutions. 1. Wiegner's fundamental lemma and Schonbek's Fourier splitting device
The Fourier splitting device introduced by Schonbek [SCHO 85] is based on the following idea: if we look at the heat kernel et , it may be viewed as a low-frequency p lter that selects the frequencies so that tj j2 1; thus, the frequency 1= t is the limit over which high frequencies are to be dumped by the kernel. Then, we estimate the L2 norm of the solution ~u of the molli ed Navier{Stokes equations
~u
= et~u
0
Z t
0
~ :[(~u ! ) ~u] ds e(t s) IPr R
(with ! (x) = d !(x=), ! 2 D(IRd ), ! 0 and R ! dx = 1) by splitting the norm into two pieces (2)d k~u(t; x)k2L2 (dx) = jjf (t) ju^ (t; )j2 d + R 2 1=2 ). jj>f (t) ju^(t; )j d with f (t) = O(t We prove more precisely the following lemma of Wiegner [WIE 87]:
Lemma 26.1: ~ :~u0 = 0 and let, for > 0, ~u be the solution of the Let ~u0 2 (L2 )d with r R ~ :[(~u ! ) ~u] ds. molli ed Navier{Stokes equations ~u = et ~u0 0t e(t s) IPr Then there exists a constant d , which does not depend on ~u0 nor R on so that for any positive continuous function ' on [0; 1) and for (t) = 0t '(s) ds, we have the inequality, for all t 2 [0; 1): e(t) k~u
2 (t; :)k2
k~u0k22 + d
Z t
0
d+2 e(s) '(s) kes~u0 k22 + '(s) 2 R (s) ds
267 © 2002 by CRC Press LLC
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Regularity results Rs
with R (s) = ( 0 k~u (; :)k22 d )2 .
Proof: We have d ~ ~uk22 ); (e k~uk22 ) = e ('k~uk22 + 2h~u j@t~ui) = e ('k~uk22 2kr dt hence 0 '(t)
d 2 dt (e k~u k2 )
(t) R
= (2e )d '(t) 2j j2 jF ~uj2 d . Let 2 D(IRd) with and ( ) = 1 q for j j 1 and ( ) = 0 for j j 2. We obviously have 2j j2 '(t)( '2(t) ), and we thus get
s Z t 2 d 2 ( t ) t 2 ~ :[(~u !) ~u]k2 ds)2 (e k~uk2 ) 2e '(t) ke ~u0 k2 +( k( D )r dt '(t) 0
We now write, for j = 1; : : : ; d and for > 0, k(D)@j f k2 C (d+2)=2 kf k1 and we arrive at Z t d 2 t 2 ( d +2) = 2 (e k~uk2 ) 'e 2ke ~u0k2 + C' ( k~uk22 ds)2 : dt 0
2. Decay rates for the
L2
norm
We may now introduce Wiegner's theorem [WIE 87] on the decay rate of restricted Leray weak solutions of the Navier{Stokes equations:
Theorem 26.1: (Wiegner's theorem) ~ :~u0 = 0 and let ~u be a retricted Leray weak Let ~u0 2 (L2 (IRd ))d with r solution for the Navier{Stokes equations with initial value ~u0 . Then: (A) limt!1 k~u(t; :)k2 = 0. (B) If ~u0 belongs more precisely to (L2 \B_ 2 ;1 )d with 0 < (d+2)=2 (so that ket~u0 k2 C (1 + t) =2 ) then supt>0 (1 + t) =2 k~u(t; :)k2 < 1. Moreover, if < (d + 2)=2, limt!0 t =2 k~u et ~u0 k2 = 0. Proof: Step 1: Decay of the L2 norm 1 (A) is a direct consequence of Lemma 26.1, with '(t) = (t+e) ln( t+e) , (t) = ln ln(t + e) and e = ln(t + e). Since k~uk2 k~u0 k2, we have R(s) s2 k~u0k42 ; thus, Lemma 26.1 then states that
k~u(t; :)k22
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R
t (s) s k2 + '(s)(d+2)=2 s2 k~u0 k4 ds 2 2 2 ln(k~ut 0+k2e) + d 0 e '(s) ke R ~ut 0( : s) 0
e
'(s) ds
Time decay
269
The same inequality holds with ~u instead of ~u; we know from the study of the limiting process for the molli ed equations (Chapter 13) that ~u is the strong limit of a sequence ~u in (L2loc ((0; 1) IRd ))d ; as a consequence, we get that there is a subsequence ~u such that for almost every t ~u(t; :) converges to ~u in (L2loc (IRd ))d . Since the L2 norm of ~u is bounded uniformly with respect to , this subsequence converges weakly to ~u in L2 , so that for almost every t we have k~u(t; :)k2 lim sup!0 k~u(t; :)k2 . Thus, for almost every t, we have:
k~u(t; :)k22
R
t (s) s k2 + '(s)(d+2)=2 s2 k~u0 k4 ds k ~u0 k22 2 2 0 : ln(t + e) + d e '(s) ke R ~ut 0( s) 0
e
'(s) ds
Moreover, t 7! ~u(t; :) is weakly continuous from [0; 1) to (L2 )d ; since the majorant of k~u(t; :)k22 is a Rcontinuous function, we get that the inequality is valid for all t > 0. Since 01 e ' ds = +1, we conclude by checking that d+2 2 d d+2 lims!+1 kes~u0 k2 = 0 and lims!+1 '(s) 2 s2 = lims!+1 s 2 (ln s) 2 = 0 (for d 3, the choice of '(t) = 1=(t +1) would have been suÆcient; for d = 2, we have to deal with a function ' decaying to 0 at 1 more rapidly than 1=t).
Step 2: Control of t =2 k~u(t; :)k2 To prove (B), we choose '(t) = 1+ t ; hence, e = (t+1) with > (d+2)=2. Thus, Lemma 26.1 gives (for ket~u0 k2 C (~u0 )(1 + t) =2 ) (t+1) k~u(t; :)k22 k~u0 k22 +CC (~u0 )2 (t+1) +C
Z t
0
(s+1)
1 (d+2)=2 R
(s) ds:
where C depends only on d, and . If we assume that we have the estimate k~uk2 D( ; ~u0 )(1 + t) for some 0, we get for 6= 1 Z t
0
(s +1)
(s) ds C (d; ; )D( ; ~u0 )
1 (d+2)=2 R
2 (1+ t) (d+2)=2+max(0;2 4 )
and thus
D( ; ~u0 )2 k22 + C C (~u0 )2 + C k~u(t; :)k22 (tk~u+01)
(t + 1) (t + 1)(d+2)=2 max(0;2
4 )
and nally k~u(t; :)k2 D(Æ; ~u0)(1 + t) Æ with Æ = min( =2; d 4 2 + min(1; 2 )). For d 3, we start with = 0 and we conclude after nitely many iterations that k~u(t; :)k2 D( =2; ~u0)(1+ t) =2 ; since the constant D( =2; ~u0 ) does not depend on , we conclude that the same estimate is valid for k~u(t; :)k2 .
Step 3: The case d = 2 The same proof is valid for d = 2 provided that we can start with > 0, since we nd in that case Æ = min( =2; min(1; 2 )); if = 0, Æ = 0 and iterating
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Regularity results
the estimate does not give anything. Up to now, we only have an estimate that was proved in (A):
k~u0k22
k~u(t; :)k22 ln(t + e) + 2
R
s 2 s u0 k2 + 0 k~u k2 d 2 2 (s+e) ln(s+e) 0 s+e ke ~
Rt 1
ln(t + e)
ds
:
We then useR the estimates ket~u0 k2 C (~u0 )(t+e) =2 with > 0, k~u(t; :)k2 dt k~u0k2 and 01 (t+e)(ln( t+e))2 = 1 and we obtain 2 2
2 k~u0 k42 : k~u(t; :)k22 ln(k~ut 0+k2e) + 2(Ct +(~ue0)) + ln( t + e)
D(~u0 ) where the constant D(~u ) does not depend Thus, we have k~u(t; :)k22 ln( 0 t+e) 2 on . We again use Lemma 26.1, with ' = (t+e) ln( and e = (ln(t + e))2 . t+e) We obtain Rs R t 2 ln(s+e) k~u k22 d 2 s 2 0 2 k e ~ u k + 2 ds 0 2 (s+e) ln(s+e) : k~u(t; :)k22 (ln(kt~u+0ke2))2 + d 0 s+e 2 (ln(t + e)) R R e+ s R s d We then estimate 0s k~uk22 d by 0s ln(d +e) = [ ln(+e) ]0 + 0 (ln(+e))2 = R e+s + o( s d ) C e+s ; hence, ln(s+e) ln(s+e) 0 ln(+e)
k~u(t; :)k22 C
k~u0k22
C (~u0 )2 D(~u0 )2 E (~u0 ) + + 2 2 (ln(t + e)) t ln(t + e) (ln(t + e)) (ln(t + e))2
where the constant E (~u0 ) does not depend on . As a consequence, we have Rs 2 d C E (~u )2 (s + e)(ln(s + e)) 2 . Thus, using Lemma 26.1 with k ~ u k 0 2 0 ' = (t + 1) 1 and e = (t + 1) with > , we get R s ku~ k22 d 2 1 s 2 ke ~u0 k2 + 0 (s+1) ds 0 k22 + d 0 (kt~u+1) (tR+1) R s t (1+s) 2 ku~ 0 k22 + C (u~ 0 )2 + CE (~u )2 0 (ln(s+e))2 [ 0 k~u k22 d] ds : d (1+t) 0 (t+1) (t+1) Rt
k~u(t; :)k22
(1+s)
Now, we de ne N; = maxN tN +1 k~u(t; :)k22 . We have 0; k~u0 k22 and, for 1 N,
N; F (~u0 )N + G(~u0 )N P
X
0kN
(1 + k) 2 (ln(k + e))
2
X
0pk
p; :
For > 1, we have 0kN (1 + k) 2 (ln(k + e)) 2 C ( )N 1 (ln N ) thus, X 1 N; F (~u0 )N + C ( ) G(~u0 ) : 2 N (ln N ) 0kN k;
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2
and
Time decay
271
P
If N0 is chosen so that 2C ( )G(~u0 ) N N0 N (ln1N )2 N X k=0
k; N0 k~u0k22 + F (~u0 )
1, we obtain for N N0 N
N X
1X k + 2 k=0 k; k=N0
and we get that for some exponent Æ 2 [0; 1) and some constant H (~u0 ) we have Rt PN Æ H ( ~ u ) N . We get that u(t; :)k22 ds CH (~u0 )(1 + t)Æ , giving 0 k=0 k; 0 k~ us Rt
3 R s k~u k2 d 2 ds 1 2 0 (1 + s) 0 H (~u0 )2 (1 + t) 2+2Æ
(t + 1) 2 2Æ and we get that k~u(; :)k22 I (~u0 )(1 + t) with = min( ; 2 2Æ) > 0.
Step 4: Control of the uctuation We now prove that, when < (d + 2)=2, the tendency et~u0 prevails over the uctuation w~ = B (~u; ~u) when t goes to +1: when ket~u0 k2 = O(t =2 ), k~u et~u0 k2 = o(t =2 ). Let us de ne w~ = ~u et~u0 . We have ~ :[(~u ! ) ~u ]; @t w~ = w~ IPr hence, d ~ (t; :)k2 2 dt kw
~ w (t; :)k22 + 2hr ~ w~ j(~u ! ) ~ui = 2kr 2 ~ w (t; :)k2 2hr ~ et~u0 j(~u ! ) ~u i = 2kr
We now work for t 1. We again consider a positive continuous function ' on R [1; 1) and we de ne (t) = 1t '(s) ds. Then, d ~ k2 ) 2 dt (e kw
Since w~ =
2 2kr ~ w~ k22 2hr ~ et~u0j(~u !) ~ui) = e ('kw~ kq 2 e ('k( '2(t) D)w~ k22 2hr~ et~u0 j(~u !) ~u i):
R t (t s) e IP ~ :[(~u
0
r
! ) ~u ]
ds, we again nd that
s
Z t
k( '(2t) D)w~ k22 C'(d+2)=2 ( k~uk22 ds)2 : 0 ~ et~u0k1 On the other hand, we have kr B_ 2 ;1 B_ 1 d=2;1 . Hence,
jhr~ et~u0 j(~u !) ~uij Ct
Ct
+1+d=2
2
+1+d=2
2
k~u0kB_ 2 ;1
since
k~u0kB_ 2 ;1 k~uk22 :
Thus, we have the inequality Z t +1+d=2 d 2 ( d +2) = 2 2 k~u0 kB_ ;1 k~u k22 : (e kw~ k2 ) C'e ' ( k~uk22ds)2 +' 1 t 2 dt 0
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Regularity results
We know that k~u(t; :)k2 C (~u0 )(1 + t) =2 , so that we have for t 1 and Rt 6= 1 0 k~uk22 ds D(~u0 )tmax(1 ;0) , where the constant D(~u0 ) does not 3 d 2 depend on . We then take ' = t 1 with > max( d+2 2 ; 2 + 4 ). Thus,
e = t and we have
t kw~ (t; :)k22 kw~ (1; :)k22 + CD(~u0 )2 t 0 + CC (~u0 )2 k~u0kB_ 2 ;1 t 1 with
max(2 2 ; 0) 0 = d+2 2 1 = 32 + d 4 2 If 2 (0; (d + 2)=2) or if = 0 and d 3, , 0 and 1 are greater than , so we get the result (notice that kw~ (1; :)k2 k~u(1; :)k2 + ke~u0 k2 2k~u0k2 and that there exist a subsequence w~ n so that, for almost every t, w~ n (t; :) in weakly convergent to w~ (t; :) Rin (L2 )d ). If = 0 and d = 2, we must use the ~ et~u0 k1 = 0. estimates limt!+1 sup>0 1t 0t k~uk22 ds = 0 and limt!+1 tkr 3. Optimal decay rate for the
L2
norm
We now prove that the results in Theorem 26.1 are optimal, in the sense that, generically, we cannot achieve a better decay. We begin with the description of the asymptotic behavior of the solutions as t goes to +1 (a result of Carpio [CAR 96] and of Fujigaki and Miyakawa [FUJM 01]).
Theorem 26.2: ~ :~u0 = 0. Let ~u be a retricted Leray Let ~u0 2 (L2 (IRd ) \ B_ 2 (d+2)=2;1 )d with r weak solution for the Navier{Stokes equations with initial value ~u0 . Then: d+2 (A) ~u satis es the inequality supt>0 (1 + t) 4 k~u(t; :)k2 < 1. R1 ~ 0 = ~u0 IPr ~ : 0 ~u ~u ds. Then, limt!+1 t d+2 ~ 0 k2 = 0. 4 k~u et U (B) Let U Proof: (A) is a consequence of Theorem 26.1. To prove (B), we introduce the vector ~z de ned by ~z We have
R ~ :[(~u !) ~u ] ds = 0t (et e(t s) )IPr Rt t ~ : 0 [(~u !) ~u] ds + ~u et~u0 : = e IPr
~ :[(~u !) ~u]; @t~z = ~z (Id et )IPr
hence, d 2 dt k~z (t; :)k2
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~ z (t; :)k22 + 2hr ~ (Id et)~z j(~u ! ) ~ui = 2kr 2 ~ w (t; :)k2 2hr ~ etZ~ j(~u !) ~ui = 2kr
Time decay
273
Rt
~ : 0 [(~u ! ) ~u] ds. We now work for t 1. with Z~ = ~z + ~u0 IPr We again consider a positive continuous function ' on [1; 1) and we de ne Rt (t) = 1 '(s) ds, arriving at 2 2kr ~ ~zk22 2hr ~ et Z~ j(~u ! ) ~u i) = e ('k~zkq 2 e ('k( '2(t) D)~z k22 2hr~ etZ~ j(~u !) ~u i):
d 2 dt (e k~z k2 )
~ :[(~u !) ~u ] = IPr ~ :(et e(t s))[(~u !) ~u] ds, Since (et e(t s))IPr we get that q
k(
2 '(t)
D)~z k22
R
C'(d+2)=2 ( 0t k(et e(t s) )[(~u ! ) ~u]k1 ds)2 R C'(d+2)=2 ( 0t st k~uk22 ds)2 :
~ et~u0 k1 Ct (d+2)=2 k~u0k _ (d+2)=2;1 since On the other hand, we have kr B2 _B2 (d+2)=2;1 B_ 1d 1;1 . We know that
k~u(t; :)k2 C (~u0 )(1 + t)
(d+2)=4
and this gives
kr~ etIPr~ :
Rt
u ! ) ~u] 0 [(~
dsk1
Ct
(d+1)=2 k R t [(~ 0 u ! ) ~u ] C 0 C (~u0 )2 t (d+1)=2 :
dsk1
~ et~z k1 kr ~ et~u0 k1 + kr ~ et~u k1 + kr ~
Finally, Rwe have kr t t t d= 4 1 = 2 ~ : 0 [(~u ! ) ~u] dsk1 and we have kr ~ e ~uk1 Ct e IPr k~uk2 ( d +2) = 2 CC (~u0 )t . Hence, for t 1,
jhr~ etZ~ j(~u !) ~u ij D(~u0 )t
d+1
2
k~uk22
where the constant D(~u0 ) does not depend on . Thus, we have the inequality Z t d+1 d 2 s (e k~zk2 ) C'e '(d+2)=2 ( k ~uk22ds)2 + D(~u0 )' 1 t 2 k~uk22 : dt 0 t p
According to the case d = 2, it is desirable to replace st by st . We then again R p write k~u(t; :)k2 C (~u0 )(1+t) (d+2)=4 and thus 0t st k~uk22 ds CC (~u0 )2 t 1=2. 2d+1 We then take ' = t 1 with > max( d+4 2 ; 2 ) and we get
t k~z(t; :)k22 k~z(1; :)k22 + CC (~u0 )4 t 0 + CD(~u0 )C (~u0 )2 t 1 with
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0 = d+2 2 +1 1 = d 2 1 + d+2 2
274
Regularity results R
Since k~z(1; :)k2 C 01 k~uk22 ds + k~u(1; :)k2 + ke~u0 k2 C k~u0k22 + 2k~u0k1 , we nd that for some constant E (~u0 ) that does not depend on we have, for t 1, k~zk2 E (~u0 )t (d+2)=4 t 1=4 . Moreover, we have R R ketIPr~ : t1 [(~u !) ~u] dsk2 Ct (d+2)=4 t1 kuk22 ds C 0 C (~u0 )2 t (d+2)=4 t d=2: This gives for a constant F (~u0 ) which does not depend on and for t 1 Z 1 ~: t(d+2)=4k~u et~u0 + et IPr [(~u !) ~u] dsk2 F (~u0 )t 1=4 : 0
If we take the Rsequence ~u which strongly converges to ~u in (L2loc ((0R; 1) IRd ))d , ~ : 01 [(~u !) ~u ] ds weakly converges to etIPr ~ : 01 [~u ~u] ds then et IPr in (L2 (IRd ))d . This gives Z 1 ~: t(d+2)=4k~u et~u0 + etIPr [~u ~u] dsk2 F (~u0 )t 1=4 : 0
(B) has been proved. In order to describe the consequences of the asymptotic formula given in Theorem 26.2, we prove two useful lemmas:
Lemma 26.2: ~ :~u0 = 0 and R (1 + jxj)j~u0 (x)j dx < 1. Then: Let ~uR0 2 (L2 (IRd ))d with r (A) ~u0 dx = ~0. (B) ~u0 2 (B_ 2 (d+2)=2;1 )d . R (C) If ~u0 2 (B_ 2 ;1 )d for some > d+2 u0 (x) dx = ~0 for every 2 , then xj ~ j 2 f1; : : : ; dg. Proof: (A) is easy: the Fourier transforms u^j are continuous and satisfy Pd u ^ ^j (0; : : : ; 0; j ; 0; : : : ; 0) = 0 for j 6= 0; by continuj j ( ) = 0. RThus, on u j =1 ity, u^j (0) = 0 and uj (x) dx = R0. (B) is equivalent to supj2ZZ 1=22j jj1 jF (~u0 )j2 jjdd+2 < 1. Since we have R (1 + jxj)j~u0 (x)j dx < 1, the Fourier transform of ~u0 has bounded derivatives, hence is Lipschitzian. SinceR F (~u0 )(0) = 0, we have jF (~u0 )( )j C j j. We now prove (C). If xk uj 6= 0, then we de ne the cone X = f = k@l u^j k1 jl j 21 j@k u^j (0)jjk jg l= 6 k We then have Z j@ u (0)j2 Z lim inf 2j(d+2) jF (~u0 )j2 d k j jk j2 d > 0 j !+1 4 j 2 ;1=2jj1 1=22 jj1
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Time decay
275
and thus ~u0 cannot belong to (B_ 2 ;1 )d for > (d + 2)=2.
Lemma 26.3: p Let g 2 H 1 (IRd ) and let gt (x) = t d=2 g (x= t)R. Then: (A) For f 2 L1 (IRd ), limt!+R1 td=4 kf gt ( f dx)gt k2 = 0R. (B) For f 2 L1 (IRd ) with (1 + jxj)jf (x)j dx < 1 and f dx = 0, R P limt!+1 t(d+2)=4 kf gt + dj=1 ( xj f dx)@j gt k2 = 0.
p
Proof: (A) is anR easy estimate p on rescaling. Let f[t] (x) =ptd=2 f ( t x). Then d= 2 fR gt(x) = t f (y)g((xR y)= t) dy =Rt d=2 (fp[t] g)(x= t) and td=4 kf gt (R f dx)gt k2 = kf[t] g ( f dx)gk2 = k td=2 f ( t y)(g(x y) g(x)) dyk2 jf (y)j kg(x pyt ) g(x)kL2 (dx) dy. R 1 d We R now prove (B). Let ! 2 D(IR) with IR ! dx = 1. For f 2 L (IR ) with (1 + jxj)jf (x)jRdx < 1 and for 0 k d, we de ne fk by f0 = f and, for k 1, fk (x) = IRk fR(y1; : : : ; yk ; xk+1 ; : : : ; xd ) dy1 : : : dyk !(x1 ) : : : !(xk ). ThenR fk 1 fk satis es IR (fk 1 fk )(x1 ; : : : ; xk 1 ; y; xk+1 ; : : : ; xd ) dy = 0 and IR j(fk 1 fk )(x1 ; : : : ; xk 1 ; y; xk+1 ; : : : ; xd )j jyj dy 2 L1(IRd 1); thus, the function Fk , de ned by Fk (x)
R k = x1 (fk 1 fk )(x1 ; : : : ; xk 1 ; y; xk+1 ; : : : ; xd ) dy R +1 = xk (fk 1 fk )(x1 ; : : : ; xk 1 ; y; xk+1 ; : : : ; xd ) dy; P
satis es Fk 2 LR1 (IRd ) and @k Fk = fk 1 fk ; thus, we get f = fn + dk=1 @k Fk (with fRn = 0 if f dx = 0). P P If f dx = 0, we get f gt = 1kd (@k Fk )gt =Rt 1=2 1kd Fk (@k g)t . P (A) gives then that limt!1 t(d+2)=4 kf gt 1kdR( Fk dx)@k gt k2 = 0. Now, P @ ^ ^ ^ we Rhave f = 1kd ik Fk ; hence, @k f (0) = i Fk dx; since @@k f^(0) = i xk f (x) dx, (B) is proved. We may now state the result of Miyakawa and Schonbek [MIYS 01]:
Theorem 26.2: ~ :~u0 = 0 and R (1 + jxj)j~u0 (x)j dx < 1. Let Let ~u0 2 (L2 (IRd ))d with r ~u be a retricted Leray weak solution for the Navier{Stokes equations with iniR tial value ~u0 . De ne, for 1 j; k d, j;k = xk u0;j (x) dx and j;k = R 1R 1 jj2 ). We then have 0 uj (x; s)uk (x; s) dx ds and let W = F (e d+2 4 k~uk2 = kV~ k2 lim t t!+1
where, for 1 j d,
Vj =
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X
( j;l + j;l )@l W
1ld
@@ @ k;l l k j W: 1ld 1kd X
X
276
Regularity results
d+2 In particular, limt!+1 t 4 k~uk2 = 0 if and only if the two following conditions are R full lled: (A) xj ~u0 (x) dx R=R~0 for every j 2 f1; : : : ; dg. (B) The matrix ( uj (t; x)uk (t; x) dt dx)1j;kd is a scalar multiple of the identity. d+2 Proof: The formula limt!+1 t 4 k~uk2 = kV~ k2 is a direct consequence of Theorem 26.2 and Lemma 26.3. d+2 In particular, limt!+1 t 4 k~uk2 = 0 if and only if V~ = O. We have 2 P P j j V^j ( ) = i e jj2 Pj ( ) with Pj ( ) = 1ld 1kd ( j;l + j;l )l k2 k;l j k l . V~ = 0 if and only if Pj = 0 for 1 j d. For j 6= k, j 6= l and k 6= l, we have @j @k @l Pj = k;l l;k = 2 k;l . Thus, if V~ = 0, we get k;l = 0 for k 6= l and P 2 2 2 . For l 6= j , we get @ 3 P = ; 0 = Pj = j;j j j j + 1ld j;l l j jP l;l j l j;l l j thus, j;l = 0 for j 6= l and Pj = j 1ld ( j;j + j;j l;l )l2 . Pj = 0 then gives for l = j j;j = 0 and for l 6= j l;l = j;j + j;j = j;j .
The condition (B) in Theorem 26.3 is not easily checked, since it depends on the solution ~u. However, Brandolese could give examples of solutions with a decay of the L2 norm better than t (d+2)=4 , by using symmetric divergence-free vector elds as initial data [BRA 01].
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Chapter 27 Uniqueness of Ld solutions In this chapter, we brie y address the uniqueness problems for mild solutions in C ([0; T ); (Ld )d ). Given ~u0 2 (Ld )d , we may not directly construct a solution in C ([0; T ); (Ld )d ), since B is not continuous on C ([0; T ); (Ld )d ) (Oru [ORU 98]; see Chapter 28). Solutions are always constructed in a smaller space (see Kato [KAT 84], Giga [GIG 86], Cannone [CAN 95], or Planchon [PLA 96]) and, thus, uniqueness was rst granted only in the subspaces of C ([0; T ); (Ld)d ) where the iteration algorithm was convergent. In 1997, Furioli, Lemarie-Rieusset and Terraneo [FURLT 00] proved uniqueness in C ([0; T ); (Ld)d ):
Theorem 27.1: (Uniqueness) If ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) d IR so that ~u and ~v belong to C ([0; T ); (Ld (IRd ))d ) and have the same initial value, then ~u = ~v . This theorem was later reproved by many authors through various methods. In this chapter, we detail the proofs of Furioli, Lemarie-Rieusset and Terraneo [FURLT 00], Meyer [MEY 99], and Monniaux [MON 99]. The proof of Lions and Masmoudi [LIOM 98] will be described in the next chapter. This theorem was extended to the case of Morrey-Campanato spaces by Furioli, LemarieRieusset and Terraneo [FURLT 00] and May [MAY 02], as we shall see in the nal section. 1. The uniqueness problem
We take a more general approach to the problem of uniqueness.
De nition 27.1: (Regular space)
A regular space is a Banach space X such that we have the continuous embeddings D(IRd ) X L2loc (IRd ) and such that moreover: (a) for all x0 2 IRd and for all f 2 X , f (x x0 ) 2 X and kf kX = kf (x x0 )kX . (b) for all 2 (0; 1) and for all f 2 X , f (x) 2 X and kf (x)kX C kf kX for a constant C which depends neither on nor on f . (c) D(IRd ) is dense in X .
277 © 2002 by CRC Press LLC
278
Regularity results We have the obvious result:
Lemma 27.1: (Regular spaces and Morrey{Campanato spaces). Let X be a regular space so that X is continuously embedded into Lploc (IRd ) for some 1 p 1. Then X is continuously embedded in mp;d , the closure of the smooth compactly supported functions in the Morrey-Campanato space M p;d . In particular, every regular space is continuously embedded in m2;d . We then consider the problem of uniqueness in C ([0; T ); X d): Uniqueness problem: Let X be a regular space. If ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) IRd so that ~u and ~v belong to C ([0; T ); X d ) and have the same initial value, then do we have ~u = ~v ? Of course, we do not know whether the uniqueness problem has a positive answer in the maximal regular limit space m2;d. We rst easily check that local uniqueness implies global uniqueness:
Lemma 27.2: Let X be a regular space. Assume that we have local uniqueness: if T > 0 and if ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) IRd so that ~u and ~v belong to C ([0; T ); X d ) and have the same initial value, then there exists a positive so that we have ~u = ~v on [0; ] IRd. Then we have global uniqueness: ~u = ~v on [0; T ).
Proof: Let = maxfT > 0 ~u = ~v on [0; T ]g. If < T , we have by continuity that ~u( ) = ~v( ). Moreover, ~u(t + ) and ~v(t + ) are solutions of the Navier{ Stokes equations on (0; T ) IRd ; hence, by local uniqueness ~u( + t) = ~v( + t) for 0 t , which constitutes a contradiction. As a consequence, we may obviously check the following result: For every > 0, we have uniqueness in m2;d+ . In particular, we have uniqueness in every regular space X so that for some 2 [0; 1) and some constant C , we have, for all f 2 X , sup0<<1 kf (x)kX C kf kX .
Proposition 27.1:
Proof: We even have uniqueness for solutions in the larger space (L1 ((0; T ); M 2;d+))d . It is enough to check that we have local uniqueness, since the mapping t 7! ~u(t) is continuous from [0; T ) to (M 2;d+)d , endowed with the -weak topology. Hence, Lemma 27.2 may be modi ed easily to t to the case of (L1 ((0; T ); M 2;d+))d . Local uniqueness is then obvious, since we have, for two solutions ~u and w~ with the same initial value, w~ = ~u ~v = B (w; ~ ~v) B (~u; w~ ). Theorem 17.1 gives that kw~ kM 2;d+ kB (w; ~ ~v )kM 2;d+ + kB (; ~u; w~ )kM 2;d+
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Uniqueness in Ld
CT 2(d+)
279
sup kw~ (s)kM 2;d+ ( sup k~u(s)kM 2;d+ + sup k~v(s)kM 2;d+ ):
0<s
0<s
2(d+)
This then gives ~u = ~v on [0; min(T0 ; T )) with CT0 sup0<s
0<s
(sup0<s
Thus, the problem of uniqueness is open only for limit spaces (for which kf (x)k kf kX ). According to Lemma 27.2, we focus on local uniqueness. The basic idea in Furioli, Lemarie-Rieusset and Terraneo [FURLT 00] is to split the solutions in tendency and uctuation, and to use di erent estimates on each term. More precisely, we consider two mild solutions ~u = et~u0 B (~u; ~u) = et~u0 w~ 1 and ~v = et~u0 B (~v ; ~v) = et~u0 w~ 2 in C ([0; T ); X d ) and write w~ = ~u ~v = w~ 2 w~ 1 = B (w; ~ ~v) B (~u; w~ ); nally,
w~ = B (w~ 1 ; w~ ) + B (w; ~ w~ 2 ) B (et~u0 ; w~ ) B (w; ~ et~u0 ): Thus the role of the uctuations is clearly seen: they control the behavior of w~ . We use the regularization properties of the heat kernel for the term et~u0 p (mainly, that limt!0 tket~u0 k1 = 0), while we use the fact that, for i = 1 or 2, we have lim!0 sup0t kw~ i (t)kX = 0; thus we may assume that the norm of w~ i is very small. 2. Uniqueness in
Ld
In this section, we give three proofs of Theorem 27.1. According to Lemma 27.2, we consider only the problem of local uniqueness.
Proof No. 1: The Besov space approach The construction of solutions in C ([0; T ); (Ld(IRd))d ) by Cannone and Planchon aimed at proving that the uctuation ~u et~u0 was in a smaller subspace (such as C ([0; T ); (Y )d ) with Y a Besov space (Y = B_ d0;1 in Cannone [CAN 95] and Furioli, Lemarie-Rieusset and Terraneo [FURLT 00], or Y = B_ d0;2 in Planchon [PLA 96]) or,pas one may easily check for a mild solution ~u 2 (L1 ((0; T ); Ld))d so that t~u 2 (L1 ((0; T ) IRd ))d , the Lorentz space Y = Ld;1). The idea in [FURLT 00] was that the problem of uniqueness could be treated in a greater space, since one did not bother to get an estimate especially in the Ld norm, the solutions already having been given. The example of Le Jan and Sznitman [LEJS 97] is particularly interesting. They consider the limit space X = B_ Pd M1;1(IRd) de ned by: f 2 B_ Pd M1;1 (IRd ) i f^ 2 L1loc(IRd ) and j jd 1 f^ 2 L1 , checking in a very simple way that the operator B is well behaved in this space. As a matter of fact, since the Riesz transforms are obviously bounded on p this space, we may just look at the scalar R bilinear operator A(u; v) = 0t e(t s) (uv)ds operating on L1(X ). We do R p not try to prove that 0t ke(t s) (uv)kX ds can be controlled; this is too
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280
Regularity results
coarse an estimate. The estimate we use is more direct: p 1 (uv) 2 X since (taking the Fourier transform) it is enough to notice that jjd1 1 jjd1 1 jjCd 2 (an obvious inequality due to radial invariance and homogeneity); then:
jFfA(uv)(t; :)g( )j jjCd 1 sup0<s
These simple estimates suggested that certain limit spaces could be handled, as far as their norm can be computed frequency by frequency or at least by frequency packets. This is the case of the Besov l1 spaces:
Lemma 27.3: (Besov spaces and heat kernel) ;1 Let X be a shift-invariant Banach space of distributions so that X B_ 1 s; 1 _ for some 2 IR. Let the Besov space BX be P de ned for s + < 0 as the space of tempered distributions f so that f = j 2ZZ j f in S 0 and so that 2js kj f kX 2 l1 (ZZ). Then Z 1 8f 2 L1((0; 1); B_ Xs;1 ) e f ( ; :) d 2 B_ Xs;1 : 0
Proof: According to the Bernstein inequalities, ( ) maps B_ Xs;1 to B_ Xs 2 ;1 for s + 2 < 0, while k( )1 et gkB_ Xs 2 ;1 C t (1 )kgkB_ Xs 2 ;1 if R
1. Thus, we split the integral F = 01 e f ( ; :) d into GA + HA , with
R
GA = 0A ( )1 e ( ) f ( ; :) d R HA = A+1 ( )1+ e ( ) f ( ; :) d s;1 and with 0 < < min(1; s ). We obtain kGA kB_ Xs 2 ;1 C A kf kL1 t B_ X
kHA kB_ Xs+2 ;1 C A kf kL1t B_ Xs;1 . Since such splitting may be done for any positive A, we get that F 2 [B_ Xs 2 ;1 ; B_ Xs+2 ;1 ]1=2;1 = B_ Xs;1 . In order to use Lemma 27.3, we must apply some conditions for a pointwise product to belong to a Besov space:
Lemma 27.4: (Pointwise product in Besov spaces) Let 2 p < d and let q 2 [p; ddpp ). Then the pointwise product boundedly maps B_ pd=p 1;1 B_ q0;1 to B_ pd=p 1 d=q;1 and boundedly maps B_ pd=p 1;1 B_ pd=p 1;1 to B_ pd=p 2;1 . Proof: We use the paradi erential calculus (seePChapter P 3), writing uv = _ (u; v) +P_ (u; vP ) + _(u; v) with _ (u; v) P= Pj2ZZ ( kj 3 k u)j v, _ (u; v) = j2ZZ ( k<j 3 k vj u and _ = j2ZZ jk jj2 k uj v.
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Uniqueness in Ld
281
We assume that u 2 B_ pd=p 1;1 and that v 2 Lq . The term _ (v; u) is easy to estimate: X
k(
kj 3
k v)j ukp C 2jd=q kvkB_ 1d=q;1 kj ukp
and this gives k_ (v; u)kB_ pd=p 1 d=q;1 C kukB_ pd=p 1;1 kvkB_ 1d=q;1 . On the other hand, we may estimate _ (u; v) by
k_ (u; v)kB_ pd=p
1 d=q;1
C sup 2j(d=p j 2ZZ
X
1 d=q) k(
kj 3
k u)j vkp
and by writing 1=p = 1=q +1=r with r > d. According to the Bernstein inequalities, we have that kk ukr C 2k(d=p d=r) kk ukp C 2k(1 d=r)kukB_ pd=p 1;1 ; thus, X
k(
kj 3
k u)j vkp (
X
kj 3
kk ukr kj vkq C 2j(1
d=r) kuk d=p 1;1 kv k 0;1 ; B_ q B_ p
this gives k_ (u; v)kB_ pd=p 1 d=q;1 C kukB_ pd=p 1;1 kvkB_ q0;1 . Moreover, the Bernstein inequalities give that X X kk ukpkj vkq kl _(u; v)kp C 2ld=q kl _(u; v)k ppq+q C 0 2ld=q j l 4 jk j j2 and this gives k_(u; v)kB_ pd=p 1 d=q;1 C kukB_ pd=p 1;1 kvkB_ q0;1 since d=p 1 > 0. Thus, the pointwise product is bounded from B_ pd=p 1;1 B_ q0;1 to d=p B_ p 1 d=q;1 ; just by using the embedding B_ q0;1 B_ 1d=q;1 . Similarly, the pointwise product is bounded from B_ pd=p 1;1 B_ pd=p 1;1 to B_ pd=p 2;1 : embedding B_ pd=p 1;1 B_ d 1;1 allows the control of (v; u) (and thus of _ (u; v)), while embedding B_ pd=p 1;1 B_ d0;1 gives control of _(u; v). We now prove uniqueness in Ld. We begin by proving uniqueness in the smaller space H_ d=2 1 (d 3). This result is a direct consequence of Lemmas 27.3 and 27.4. We just write that H_ d=2 1 B_ 2d=2 1;1 B_ qd=q 1;1 with d < q < 2d=(d 2). The proof of uniqueness in H_ d=2 1 is then obvious. We write
w~ = B (w~ 1 ; w~ ) + B (w; ~ w~ 2 ) B (et~u0 ; w~ ) B (w; ~ et~u0 ) where we have w~ , w~ 1 , w~ 2 in L1((0; T ); (B_ 2d=2 1;1 )d ) and t(1 d=q)=2 et~u0 in L1 ((0; T ); (Lq )d ). In order to estimate B (w~ 1 ; w~ ), we write B (w~ 1 ; w~ ) = R t (t s) ~ :w~ 1 w~ 2 (L1 (0; T ); (B_ 2d=2 1;1 )d ) [accord~z(s) ds with ~z = 1 IPr 0e ing to Lemma 27.4] and we apply Lemma 27.3. We proceed in the same way for B (w; ~ w~ 2 ). To estimate B (et~u0; w~ ), we use Lemma 27.4 and we write kB (et~u0 ; w~ )kB_ 2d=2 1;1 R 0t (t s)(1+C d=q)=2 kw~ (s)kB_ d=2 1;1 s(1 d=q)=2 kes~u0kq s(1 dsd=q)=2 : 2
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282
Regularity results
Thus, we nd
kw~ (t)kB_ 2d=2
1;1
C
sup kw~ (s)kB_ d=2 1;1 sup A(s) 2
0<s
0<s
with
A(s) = kw~ 1 (s)kB_ d=2 1;1 + kw~ 2 (s)kB_ d=2 1;1 + s(1 d=q)=2 kes~u0kq : 2 2 Since limt!0 A(t) = 0, we get local uniqueness in H_ d=2 1 . This proof does not work directly for the space Ld; we should be operating with the Besov space B_ d0;1 , which is not embedded in L2loc . We then use the fact that the uctuation has a better regularity than the tendency, in that it has a bounded norm in a Besov space with a positive regularity exponent:
Lemma 27.5:
The operator B is bounded from L1 ((0; T ); (Ld )d ) L1 ((0; T ); (Ld )d ) to L1 ((0; T ); (B_ pd=p 1;1 )d ) for every p 2 [d=2; 1].
Proof: This is a direct consequence of Lemma 27.3. We just write B (~u; ~v) = R t (t s) ~ :~u ~v. We have, for f and g in Ld, fg 2 e ~z(s) ds with ~z = 1 IPr 0 d=p 2 ; 1 0;1 B_ Ld=2 B_ d= for all p d=2 according to the Bernstein inequalities. p 2 1 We nd that ~z 2 L ((0; T ); (B_ pd=p 1;1 )d ) and so does B (~u; ~v), according to Lemma 27.3. The proof of uniqueness in Ld is now obvious. We write
w~ = B (w~ 1 ; w~ ) + B (w; ~ w~ 2 ) B (et~u0 ; w~ ) B (w; ~ et~u0) d=p 1;1 )d ) for every p d=2 (for where we have w~ , w~ 1 , w~ 2 in L1 ((0; T ); (B_ 1 w~ 1 and w~ 2 , apply Lemma 27.5; for w~ , write that w~ = ~u ~v = w~ 2 w~ 1 ) and t(1 d=q)=2 et~u0 in L1 ((0; T ); (Lq )d ) for every q d. We then choose p 2 (max(2; d=2); d) and q 2 (d; dpdp ). Lemmas 27.3 and 27.4 give the control
kw~ (t)kB_ pd=p
1;1
C
sup kw~ (s)kB_ pd=p 1;1 sup A(s)
0<s
0<s
with
A(s) = kw~ 1 (s)kB_ pd=p 1;1 + kw~ 2 (s)kB_ pd=p 1;1 + s(1 d=q)=2 kes~u0 kq : Since limt!0 A(t) = 0, we get local uniqueness in Ld.
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Uniqueness in Ld
283
Proof No. 2: The Lorentz space approach For Ld, there is now a simpler way to achieve uniqueness: while the space _Bd0;1 is not good for proving uniqueness, it can be replaced by another l1 space: Ld;1 (Meyer [MEY 99]).
Lemma 27.6: (Lorentz spaces and heat kernel) Let p 2 (1; d) and let 1=q = 1=p 1=d. Then: Z 1 p 8f 2 L1 ((0; 1); Lp;1 ) e f ( ; :) d 0
2 Lq;1 :
Proof: According to the Sobolev inequalities, ( ) maps Lp;1 to Lr;1 for 0 < 2 < d=p and 1r = 1p 2d , while k( ) 21 + et gkLr;1 C t ( 12 ) kgkLr;1 . R p Thus, we split the integral F = 01 e f ( ; :) d into GA + HA , with
R
GA =R 0A ( )1 e ( ) 1=2 f ( ; :) d HA = A+1 ( )1+ e ( ) 1=2 f ( ; :) d
with 0 < < 21 ( p1 1d ). We obtain kGA k d dq2 q ;1 C A kf kL1 t Lp;1 and L kHA kL d+2dq q ;1 C A kf kL1t Lp;1 . Since such splitting may be accomplished dq dq for any positive A, we get that F 2 [L d 2 q ;1 ; L d+2 q ;1 ]1=2;1 = Lq;1 . The proof of uniqueness in Ld is now obvious. We write
w~ = B (w~ 1 ; w~ ) + B (w; ~ w~ 2 ) B (et~u0 ; w~ ) B (w; ~ et~u0 ) where we have w~ , w~ 1 , w~ 2 in L1 ((0; T ); (Ld;1)d ) (for w~ 1 , write w~ 1 = et~u0 ~u 2 L1 ((0; T ); (Ld )d ) and Ld Ld;1; for w~ , write that w~ = ~u ~v ) and t(1 d=q)=2 et~u0 in L1 ((0; T ); (Lq )d ) for every q d. Since the pointwise product maps Ld;1 Ld;1 to Ld=2;1 , Lemma 27.6 gives control of B (w~ 1 ; w~ ) and d of B (w; ~ w~ 2 ). Moreover, the operator (f; g) 7! ( ) 2q (uv) boundedly maps d; 1 q; 1 d; 1 L L to L , according to the Sobolev inequalities, giving
Rt
0 (t
kB (et~u0 ; w~ )kLd;1 ~ (s)kLd;1 s(1 d=q)=2 kes~u0 kq s(1 ds d=q)=2 : s)(1+d=q)=2 kw C
Thus, we nd
kw~ (t)kLd;1 C
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sup kw~ (s)kLd;1 sup A(s)
0<s
0<s
284
Regularity results
with
A(s) = kw~ 1 (s)kLd;1 + kw~ 2 (s)kLd;1 + s(1 d=q)=2 kes~u0 kq : Since limt!0 A(t) = 0, we get local uniqueness in Ld. Proof No. 3: The maximal regularity approach Monniaux [MON 99] has given simpler proof based on the maximal Lp (Lq ) regularity property for the heat kernel. We recall this theorem (see Chapter 7, Theorem 7.3): Theorem 27.3: (Maximal Lp (Lq ) regularity for the heat kernel.) R The operator A de ned by f (t; x) 7! Af (t; x) = 0t e(t s) f (s; :) ds is bounded from Lp ((0; T ); Lq (IRd )) to Lp ((0; T ); Lq (IRd )) for every T 2 (0; 1], 1 < p < 1 and 1 < q < 1. The proof of uniqueness in Ld is now obvious. We select p 2 (2; 1) and write w~ = B (w~ 1 ; w~ ) + B (w; ~ w~ 2 ) B (et~u0 ; w~ ) B (w; ~ et~u0) where, p t for T <1 T , w~ 1 , w~ 2d ind L1 ((0; T ); (Ld)d ), w~ in Lp ((0; T ); (Ld)d ) and te ~u0 in (LR ((0; T ) IR )) . In order to estimate B (w~ 1 ; w~ ), we write that ~ :w~ 1 w~ 2 (Lp (0; T ); (Ld)d ) B (w~ 1 ; w~ ) = 0t e(t s) ~z(s) ds with ~z = 1 IPr [according to the Sobolev embedding H_ d=12 Ld] and we apply Theorem 27.3. To accomplish the estimate of B (et~u0 ; w~ ), we write
kB (et~u
~ )k 0; w
Lpt Ldx
k
Z t
0
ptC s kw~ (s)kd pskes~u0k1 pdss kp :
We write t 7! kw~ (t)kd 2 Lp (0; T ) and t 7! t 1=2 2 L2;1((0; T )). Since p > 2, we may use the estimates on pointwise product and on convolution in Lorentz spaces Lp;q (IR) (Chapter 2) and get that pointwise product maps Lp L2;1 to L2p=(p+2);p and that convolution maps L2p=(p+2);p L2;1 to Lp . Thus, we nd for all T 2 (0; T ):
kw~ (t)kLptLdx ((0;T )IRd ) C kw~ kLptLdx ((0;T )IRd ) with
p
sup A(s)
0<s
A(s) = kw~ 1 (s)kLd + kw~ 2 (s)kLd + skes~u0k1 : Since limt!0 A(t) = 0, we get local uniqueness in Ld.
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Uniqueness in Ld
285
3. The case of Morrey{Campanato spaces
The proofs given in Section 2 work in a more general pattern. Furioli, Lemarie-Rieusset and Terraneo [FURLT 00] extended the proof through Besov spaces to the case of Morrey{Campanato spaces by using the Besov spaces over Morrey{Campanato spaces described by Kozono and Yamazaki [KOZY 97]:
Theorem 27.4: Let mp;d be the closure of the smooth, compactly supported functions in the Morrey{Campanato space M p;d . If p > 2 and ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) IRd so that ~u and ~v belong to C ([0; T ); (mp;d )d ) and have the same initial value, then ~u = ~v . Hence, we have uniqueness in C ([0; T ); (Y )d ) for all regular space Y in which the test functions are dense and which is embedded into Lploc for some p > 2. We do not prove Theorem 27.4 here, since May [MAY 02] proved a more general result by extending the approach of Monniaux (i.e., by using the maximal Lp Lq property of the heat kernel):
Theorem 27.5: Let X1(0) be the closure of the smooth compactly supported functions in the multiplier space X1 . If ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) IRd so that ~u and ~v belong to C ([0; T ); (X1(0))d ) and have the same initial value, then ~u = ~v. Remark: Let us recall that we have, for all p > 2, M p;d X 1 M 2;d (Chapter 21). Proof: We choose p 2 (2; 1). We consider for T < T the space ET = ff 2 R L2loc((0; T ) IRd ))= supx0 2IRd k jx x0 j<1 jf (x; t)j2 dx 1=2 kLp((0;T )) < 1g. We then write w~ = B (w~ 1 ; w~ ) + B (w; ~ w~ 2 ) B (et~u0 ; w~ ) B (w; ~ et~u0 )
p where w~ 1 , w~ 2 in L1 ((0; T ); (X1)d ), w~ in ETd and tet~u0 in (L1 ((0; T ) IRd ))d . We compute the norm in ET by usingPa partition of unity: we choose ' 2 D(IRd ) with Supp ' [ 1; 1]d and k2ZZd '(x k) = 1, and we de ne 'k (x) = '(x k); then, the norm in ET is equivalent to the norm supk2ZZd k'kP f kLptL2x ((0;T )IRd ) . We de ne K = fk 2 ZZ = d(Supp '; Supp 'k ) 1g and = k2K 'k . We rst estimate ' B (w~ 1 ; w~ ). We write w~ = ~ + ~ where we have ~ = P k ~ (x k)w~ and = l=2K 'l (x k)w~ . Then: B (w~ 1 ; ~ ) =
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Z t
0
e(t s) (Id )~z(s) ds with ~z =
1 ~ :w~ 1 ~ IPr Id
286
Regularity results
We have ~z 2 (Lp (0; T ); L2(IRd )))d [since we have 2 (Lp (0; T ); L2(IRd)))d ; hence, w~ 1 ~ 2 (Lp (0; T )L2(IRd )))dd ]. According to Theorem 27.3, Z t
0
e(t s) ~z(s) ds 2 (Lp (0; T ); L2(IRd )))d :
On the other hand, we have
k
Z t
0
e(t s) ~z(s) dskL2 (dx) Ct1 1=p k~zkLpL2 ((0;T )IRd ) :
In order to estimate 'k B (w~ 1 ; ~ ), we write
k'k B (w~ 1 ; ~ )kLptL2
X
l=2K
k'k B (w~ 1 ; 'k+l w~ )kLptL2
We know (Proposition 11.1) that the operators 1 @i @j @k e(t s) have integrable kernels (t s)(1d+1)=2 Ki;j;k ( ptx s ) (1 i; j; k d), which can be controlled by 1 C px p C (t s)(d+1)=2 jKi;j;k ( t s )j ( t s+jxj)d+1 jxjd+1 . Thus, we may control, for l 2= K , 'k B (w~ 1 ; 'k+l w~ ) by
j'k (x)B (w~ 1 ; 'k+l w~ )(t; x)j R C jlj (d+1) 0t k (x k l)w~ 1 (s; x)kL2 (Rdx)k'(x k l)w~ (s; x)kL2 (dx) ds C 0 jlj (d+1) sup0<s
We now estimate 'k B (et~u0; w~ ). We split again w~ in w~ = ~ + ~ . We rst write (since p > 2): R kB (et~u0 ; ~ )kLptL2 k 0t ptC s k ~ (s)kL2 pskpes~u0 k1 pdss kp C 0 k ~ kLpL2 sup0<s
In order to estimate 'k B (w~ 1 ; ~ ), we write
k'k B (et~u0; ~ )kLptL2
X
l=2K
k'k B (et~u0 ; 'k+l w~ )kLptL2
which gives
j'k (x)B (et~u0 ; 'k+l w~ )(t; x)j Rt ( d +1) s u0 k 2 C jlj LR(dx) k'(x k 0 k (x kp l)e ~ 0 ( d +1) s C jlj sup0<s
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l)w~ (s; x)kL2 (dx) ds l)w~ (s; x)kL2 (dx) pdss k l)w~ (s; x)kLptL2x
Uniqueness in Ld
287
and nally that
k'k B (et~u0; ~ )kLpL2 CT 1+1=2 Thus, we nd for all T
1=p
0<s
pskes~u k kw~ k : 0 1 ET
2 (0; T )
kw~ (t)kET C kw~ kET with
sup
sup A(s)
0<s
p
A(s) = kw~ 1 (s)kX1 + kw~ 2 (s)kX1 + skes~u0 k1 :
Since limt!0 A(t) = 0, we achieve local uniqueness in X1(0).
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Chapter 28 Further results on uniqueness of mild solutions In this chapter, we prove some further results on uniqueness. We rst prove that uniqueness in (C ([0; T ]; Ld(IRd))d could not be directly provided by the boundedness of the bilinear operator B : Oru [ORU 98] proved that B is not continuous on (Cb ([0; T ); Ld)d (nor in the subspace of divergence-free vector elds). We then prove two additional uniqueness results: uniqueness in (L1 ([0; T ]; Ld(IRd ))d when d 4 (Lions and Masmoudi [LIOM 98]) and uniqueness in (C ([0; T ]; B 1)d \ (L1 ([0; T ]; L2))d \ (L2 ([0; T ]; H 1))d , when ~u0 2 (B p \ L2 )d (p < 1) (Chemin [CHE 99], May [MAY 02]), where B p is the closure of the test functions in Bpd=p 1;1 . 1.
Nonboundedness
Cb ([0; T ); (Ld)d )
of
the
bilinear
operator
B
on
Oru [ORU 98] proved the following result on the bilinear operator B :
Theorem 28.1: (Nonboundedness on C ([0; T ]; (Ld)d )) Let d 3. Let B be the bilinear operator B (~u; ~v)(t; :) = and, for T
2 (0; +1], let ET
Z t
0
~ :~u ~v ds e(t s) IPr
be the space
ET = ff~ 2 (C ([0; T ); IRd ))d = Then B is not bounded from ET
ET
~ :f~ = 0g: sup kf~kd < 1 and r
0
to ET .
Proof: If B is continuous on E1 , it is continuous on ET for all T : if ~u 2 ET and ~v 2 ET , we may de ne ~u 2 E1 by ~u(t; x) = ~u(min(t; T ); x) and the same for ~v ; then, on [0; T ], we have B (~u; ~v) = B (~u ; ~v ). Letting go to 0 gives control of B on ET . Moreover, if B is continuous on ET0 for one given T0 , then it is continuous on ET for all T : de ne for ~u 2 ET and > 0, X~u(t; x) = 289 © 2002 by CRC Press LLC
290
p
Regularity results
p
~u(t; x); then we have X TT ~u 2 ET0 and B (~u; ~v) = X TT0 B (X TT ~u; X TT ~v ). 0 0 0 Thus, we may assume T = 1. We then consider 2 S (IRd) so that is radial and its Fourier transorm ^ is supported by the ball f = j j 1=2g. We de ne f~(t; x) = (f1 ; f2 ; 0; : : : ; 0) with f1 (t; x) = 1x2t ( p1x t ) and f2 (t; x) = 1x1t ( p1x t ). We have f~ 2 E1 and ~ f~) 2= E1 . Let us write ~g = B (f; ~ f~). We have we are going to prove that B (f; 1 ~ ~ ~ @t~g = ~g + IPr:f f ; if ~g 2 E1 , we have @t~g 2 (L ((0; 1); B_ 13;1 ))d , thus ~g(t; :) converges weakly to a limit ~h(x) in (S 0 (IRd ))d as t ! 1. Since ~g(t; :) is bounded in (Ld )d and since Ld is a dual space, we nd that ~h 2 (Ld)d . But we R 1 (1 s) ~ ~ :f~ f~ ds. Let us compute hd, the dth component of ~h: have h = 0 e IPr Z 1
hd(x) =
0
@ e(1 s) d (@12 f12 + 2@1 @2 (f1 f2 ) + @22 f22 ) ds:
The Fourier transform of hd is given by
h^ d ( ) =
i (2)d
Z 1
0
e
(1 s)jj2 d ( 2 f^ f^ + 2 f^ f^ + 2 f^ f^ ) j j2 1 1 1 1 2 1 2 2 2 2
ds:
We call !j;k = @@j ^ @@k ^ and we de ne
( ) = d2 (12 !2;2( ) 21 2 !1;2( ) + 22 !1;1( )): j j We then may write ^hd( ) =
i (2)d
Z 1
0
e
s)(d
3)=2 (p1
e (d
2)=2 (p
(1 s)jj2 (1
s ) ds
and, nally, writing = (1 s)j j2 ,
h^ d ( ) =
i 1 (2)d j jd
Z jj2
1
0
j j ) ds:
The support of is contained in the ball B (0; 1); thus, h^ d is equal for j j 1 to m( ) where
m( ) =
i 1 d (2) j jd
Z 1
1
0
e (d
2)=2 (p
j j ) ds:
If M is the inverse Fourier transform of m, we nd that hd = M + where is a smooth function (since its Fourier transform is compactly supported). But M is a homogeneous distribution with degree 1 (M (x) = 1 M (x) for > 0); hence, it cannot belong to Ld in the neighborhood of 0 unless it is identically
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Further results on uniqueness
291
equal to 0. Thus, we must prove that m is not a null function. This is easily proved, when we choose ^( ) = (j j2 ) with nonincreasing on [0; 1). Indeed, we compute m(0; 2 ; : : : ; d ) = m(0; 0 ) as Z 1 p d 22 0 i e (d 2)=2 !1;1 ( (0; 0 )) ds; m(0; 0 ) = d 0 d +2 (2) j j j j 0 hence, as
m( ) =
4i d 22 (2)d j 0 jd+2
Z 1
0
p 0 12 0 (jj2 ) 0 (12 + j j j IRd
Z
d 2 e 2 [
0 j2 ) d] ds:
The integral does not vanish; hence, m is not identically equal to 0, and thus hd 2= Ld.
L1 (Ld) (d 4)
2. Uniqueness in
The uniqueness theorem in (C ([0; T ]; Ld(IRd ))d (Furioli, Lemarie-Rieusset and Terraneo [FURLT 00]) described in Chapter 27 was extended in dimension d 4 by Lions and Masmoudi [LIOM 98]:
Theorem 28.2: (uniqueness in (L1 ([0; T ]; Ld(IRd ))d ) If ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) d IR so that ~u and ~v belong to L1([0; T ); (Ld (IRd ))d ) and have the same initial value, then ~u = ~v . Proof: Once again, we just have to prove local uniqueness. If ~u and ~v are equal on [0; T ), then we use the fact that they are weakly continuous in (S 0 (IRd))d to get that the equality is still valid for t = T ; moreover, since Ld is a dual space and since ~u and ~v are bounded in (Ld )d , we nd that we have ~u(T ) = ~v(T ) in (Ld )d . If we can prove local uniqueness, we are able to get global uniqueness. The proof then follows a theorem of Von Wahl [WAH 85] for solutions in d 1 2 2 1d (L1 L t x \ Lt Lx \ Lt Hx ) . In order to prove that ~u = ~v on some interval [0; ), we may suppose that ~u is the Kato solution in C [0; ]; (Ld)d ; if we consider two bounded solutions in (Ld )d and if we prove that each bounded solution is equal to the continuous solution in (Ld)d for some interval, then the bounded solutions are equal on the smallest interval. The proof of Von Wahl relies on an energy equality on w~ = ~u ~v. First, we write that w~ 2 (L2 ((0; T ); H 1 )d and ~u and ~v are in (L1 ((0; T ); Ld ))d ; ~ (~u ~u ~v ~v) = w~ IPr ~ (w~ ~u + ~v w~ ); then, we write @t w = w~ IPr the Sobolev embeddings (when d 3) give that H 1 H_ 1 L2d=(d 2) and thus the pointwise product maps L1Ld L2H 1 to L2 L2 ; thus we obtain that @t w~ 2 (L2 ((0; T ); H 1 ))d . Thus, we may write that @t jw~ j2 = 2w:@ ~ t w~ and integration on x gives: (28:1)
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@t kw~ k22 + 2
Z
IRd
jr~ w~ j2 dx = 2
Z
IRd
~ ]~u + [~v:r ~ ]w~ ) dx: w: ~ ([w: ~r
292
Regularity results R
R
R
~ ]w~ dx ~ ]w~ dx and IRd w~ :[~v :r ~ ]~u dx = IRd ~u:[w: ~r ~r We use the equalities IRd w~ :[w: = 0. Then, we use the continuity of ~u and write ~u = et~u0 w~ 1 where limt!0 kw~ 1 kd = 0, yielding Z
1 (28:2) @t kw~ k22 + 2
Z
Z
~ ]w~ dxj+j et~u0 :[w: ~ ]w~ ) dxj: jr~ w~ j2 dx j dw~ 1 :[w: ~r ~r d d
IR
IR
IR
Pointwise product maps Ld H_ 1 L2 to L1; hence, Z
~ ]w~ dxj C1 kw~ 1 kd kr ~ w~ k22 : j dw~ 1 :[w: ~r IR
On the other hand we have w~ 2 (L1 L2 )d . We then write L1 L2 \ L2H_ 1 L1 L2 \ L2L2d=(d 2) Ld(1 2=p) Lp for 2 < p < 2d=(d 2). We have Ld B_ d0;d Bq d(1=d 1=q);d for d q. We select q = d2 =(d 2) and p = 2q=(q 2). The pointwise product maps Lq Lp L2 to L1 , giving R
~ ]w~ dxj j IRdet~u0 :[w: ~r
C ket~u0 kq kw~ kp kr~ w~ k2 d(1=2 C2 ket~u0 kq kw~ k12 d(1=2 1=p)kr~ w~ k1+ 2
1=p)
:
We have d(1=2 1=p) = d=q, 1 d=q = 2=d and 1 + d=q = 2 2=d, which then gives R
1=d) ~ ]w~ dxj C ket~u0 kq kw~ k22=dkr ~ w~ k2(1 j IRdet~u0 :[w: ~r 2 d Cd2 ket~u0 kdqkw~ k22 + d d 1 kr~ w~ k22 : On [0; ) with small enough, we have C1 kw~ 1 kd 1=d; thus, (28.2) nally
yields
@t kw~ k22 2
(28:3)
C2d t d 2 ke ~u0kq kw~ k2 : d
Since ~u0 2 (B_ q 2=d;d )d , we have et~u0 2 (Ld((0; 1); Lq ))d . The Gronwall lemma 2C2d R t s d gives us that kw~ (t)k22 kw~ (0)k22 e d 0 ke ~u0 kq ds = 0. Now, if we want to prove the theorem of Lions and Masmoudi, we can prove that, if ~u and ~v are two solutions of the Navier{Stokes equations in (L1 ((0; T ); Ld ))d with the same initial value ~u0 , then w~ = ~u ~v satis es w~ 2 (L1 L2 \ L2H 1 )d (at least, when T < 1, which we may assume, since we are interested in local uniqueness). But it is enough to write that w~ = B (w; ~ ~u) B (~v ; w~ ): if we know that w~ 2 (L1 ((0; T ); Lq ))d for some q 1, we may write that w~ ~u 2 (L1 ((0; T ); Lr ))dd with r = dq=(d + q) if q d=(d 1) and r = 1 if q d=(d 1) (since we then would have by interpolation between Ld and Lq that w~ 2 (L1 ((0; T ); Ld=(d 1)))d ), which gives
p kw~ (t)kr C t
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sup kw~ (s)kmax(q;d=(d
0<s
1))
sup k~v(s)kd + k~u(s)kd :
0<s
Further results on uniqueness
293
Therefore, w~ 2 (L1 Lq )d for all q 2 [1; d]. Moreover, we may write Z t
1 ~ ~ ~u + ~v w~ ] ds: 1=2 IPr:[w ( ) 0 For d=(d 1) < q d, we have w~ 2 (L1 Lq )d (L2 Lq )d (since we assume ~ :[w~ ~u + ~v w~ ] 2 (L2 Ldq=(d+q))d . The maximal T < 1), and thus ( )1 1=2 IPr Lp Lq regularity theorem for the heat kernel (Theorem 7.3) gives then that w~ 2 (L2 Lr )d with r = dq=(d + q) 2 (1; d=2]. Provided that r = 2 d=2, i.e., d 4, the proof is complete. (
)1=2 w~
=
e(t s)
3. A uniqueness result in
B_ 11;1
We end the chapter with a result described in Chemin [CHE 99]:
Theorem 28.3: (Uniqueness in (C ([0; T ]; B ( 1))d ) Let B p be the closure of the test functions in Bpd=p 1;1 , and let ~u0 2 ~ :~u0 = 0. (B p \ L2 )d for some p < 1, with r If ~u and ~v are two weak solutions of the Navier{Stokes equations on (0; T ) d IR such that ~u and ~v belong to the space (C ([0; T ); B 1 )d \ (L1 ([0; T ); L2 ))d \(L2 ([0; T ); H 1 ))d and have the same initial value ~u0 , then ~u = ~v. Remark: Chemin proved the theorem for d = 3 using energy estimates proved by Chemin and Lerner [CHEL 95]. May [MAY 02] demonstrated that the proof was easily extended to greater values of d by using the maximal LpLq regularity of the heat kernel instead of energy estimates. We always assume p > d and T < 1, with no loss of generality since B q for q > p. The proof is based on the paradi erential calculus of Bony [BON 81]. More P precisely, let and be the operators (f; g) = Pj2IN Sj 2 fP j g and (f; g) = S0 fS0 g + 2 f 0 g + 1 f 0 g + 1 f 1 g + j2IN j f ( j 2kj+2 k g), so that fg = (f; g) + (g; f ) + (f; g). We de ne the paradi erential decompositions of the bilinear B , B , B and B , by the formulas
Bp
8 > <
~ g) =R0te(t s)IPr ~ g) ds with M (f;~ ~ g) = ((fj ; gk ))1j;kd ~ :M (f;~ B (f;~ R t (t s) ~ ~ ~ g) = ((gk ; fj ))1j;kd ~ B (f;~g) =R0e IPr:M (f;~g) ds with M (f;~ > : t ~ g) = 0e(t s)IPr ~ g) ds with M (f;~ ~ g) = ((fj ; gk ))1j;kd ~ :M (f;~ B (f;~ ~ g) = B (f;~ ~ g) + B (f;~ ~ g) + B (f;~ ~ g). so that we have B (f;~ The proof begins by describing more precisely the regularity of s solution ~u of the Navier{Stokes equations in (L1 ([0; T ); L2))d \ (L2 ([0; T ); H 1 ))d :
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Regularity results
Lemma 28.1: i) If f~ and ~g belong to (L1 ([0; T ); L2 ))d \ (L2 ([0; T ); H 1 ))d , then ~h = 1+r ))d for 0 < < r 1. ~ B (f;~g) belongs to (L2=r ([0; T ); Hd= (d ) 1 ; 1 ii) For f 2 B1 , the operator g 7! (f; g ) is bounded from Hps to Hps 1 for every p 2 (1; 1) and s 2 IR and g 7! (f; g ) is bounded from Hps to Hps 1 for every p 2 (1; 1) and s > 1. ~ g) is bounded iii) For f~ 2 L1 ((0; T ); (B11;1 )d ), the operator ~g 7! B (f;~ q s d on L ((0; T ); (Hp ) ) for every p and q in (1; 1) and every s 2 IR and the ~ g) is bounded on Lq ((0; T ); (Hps )d ) for every p and q in operator ~g 7! B (f;~ (1; 1) and every s > 1. ~ :~u0 = 0. If ~u is a weak solution of iv) Let ~u0 2 (B11;1 \ L2 )d , with r the Navier{Stokes equations on (0; T ) IRd so that ~u belongs to the space (L1 ([0; T ); B11;1 ))d \ (L1 ([0; T ); L2 ))d \ (L2 ([0; T ); H 1 ))d and has initial value ~u0 , then w ~ = et u~0 ~u is a solution of the xed point-problem in q d L ((0; T ); (Hp ) ) (with q 2 (2; 1), = 1 + 1=q and p = d=(d 1=q)) w~ = 2B (~u; w~ ) B (~u; w~ ) + w~ 0 with
w~ 0 = 2B (~u; et~u0 ) + B (~u; et~u0 ) 2 Lq ((0; T ); (Hp )d ):
Proof: We rst study the pointwise product between two functions f and g in H r with 0 < r 1. We write fg = (f; g) + (g; f ) + (f; g). Now, for 2 (0; r), we de ne p > 2 by 1=p = 1=2 =d and we write that L2 Bp ;2 (due to the Bernstein inequalities) and this gives that (2 j kj f kp )j2IN 2 l2 and (2 j kSj f kp )j2IN 2 l2 . Since (2jr kj gk2)j2IN 2 l2 and since we have r ;1 C kf k2 kg kH r and k(f; g )kB r ;1 r > 0, we nd that k(f; g)kBd= (d ) d=(d ) r , or more accurately kfg k r C kf k2kgkH r . Thus, we nd that fg 2 Hd= H d (d )
C (kf k2kgkH r + kf kH r kgk2). 2=r r 2 2 1 Since L1 t Lx \ Lt Hx Lt Hx for 0 < r 1, we write Z t p 1+r ~ (t s) (
p
)
B (f;~g) =
0
e
(d )
~z ds
~ :f~ ~g. We have ~z 2 (L2=r Ld=(d ))d . The maxwith ~z = ( )r 1 IPr ~ g) 2 imal LpLq regularity property of the heat kernelR gives then that B (f;~ t (t s) 1+ r 2 =r d (1+ r ) =2 ~z ds; ~ _ (L ((0; T ); Hd=(d ))) . Moreover, B (f;~g)= 0 e ( ) Rt C ~ g)kd=(d ) 0 hence, kB (f;~ zkd=(d ) ds. Since T < 1 and (t s)(1+ r)=2 k~ ~ g) 2 (L2=r Ld=(d ))d . thus s 7! s(1+ 1 r)=2 2 L1((0; T )), we nd that B (f;~ 1+r ))d . ~ g) 2 (L2=r ((0; T ); Hd= Thus, we proved that B (f;~ (d )
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Further results on uniqueness
295
Point ii) is a classical consequence of the following characterization of the space Hp : f 2 Hp (1 < p < 1 and 2 IR) if and only if f 2 S 0 , S0P f 2 Lp and ( )=2 (Id S0 )f 2 Lp ; moreover, for 1 < q < 1, kf kq k( j2ZZ 4 j jj ( )=2 f j2 )1=2 kq . Now, if T = Pj2IN Tj with Supp T^j f = 2j 3 j j 2j+3 g and if f 2 S , we write
jh(
)=2 T jf ij
X
X
j 2IN jj kj5
jhTj jk (
)=2 f ij;
P
hence, k( )=2 T kp C k( j2IN 4j jTj (x)j2 )1=2 kp. This gives the control of (f; g) in Hps 1 when f 2 B11;1 (hence kSj f k1 C 2j kf kB11;1 for j 2 IN) and g 2 Hps . P Similarly, when > 0 and T = j2IN Tj with Supp T^j f = j j 2j+3 g, P P we get kT kHp C k( j5 4j kj 5 jTk (x)j 2 )1=2 kp . Since the convolution with (2 j 1jP5 )j2ZZ is a bounded mapping on l2(ZZ), we nd that we have kT kHp C 0 k( j2IN 4j jTj (x)j2 )1=2 kp . This gives the control of (f; g) in Hps 1 when f 2 B11;1 , g 2 Hps and s > 1. Point iii) is a direct consequence of point ii) and of the maximal Lp Lq ~ g) = ~ :M (f;~ regularity property of the heat kernel. We just have to write IPr 1 q s d ~ ~ ~ ~ (Id )Z with Z = (Id ) IPr:M (f;~g) 2 (L ((0; T ); Hp )) . Now, if ~u is a weak solution of the Navier{Stokes equations on (0; T ) IRd so that ~u belongs to the space (L1 ([0; T ); B11;1 ))d \ (L1 ([0; T ); L2 ))d \ (L2 ([0; T ); H 1 ))d and has initial value ~u0 , and if q 2 (2; 1), we apply point i) with r = 2=q and = 1=q, to get that w~ = B (~u; ~u) 2 (Lq ((0; T ); Hp )d with = 1 + 1=q and p = d=(d 1=q). Moreover, we have w~ = B (~u; ~u) = 2B (~u; ~u) + B (~u; ~u) = 2B (~u; et~u0 w~ ) + B (~u; et~u0 w~ ). Point iii) gives then that 2B (~u; w~ ) + B (~u; w~ ) 2 (Lq ((0; T ); Hp )d (since 1 < ), and so does w~ 0 = w~ + 2B (~u; w~ ) + B (~u; w~ ). The next step is to study the xed point equation in another space:
Lemma 28.2: i) For f 2 B11;1 , the operator g 7! (f; g ) is bounded from Bps;1 to Bps 1;1 for every p 2 [1; 1] and s 2 IR and g 7! (f; g) is bounded from Bps;1 to Bps 1;1 for every p 2 [1; +1] and s > 1. ii) For p 2 (d; 1), 2 (1; 1 + d=p) and T > 0, we de ne ET ;p as the space
ET ;p
= ff 2 Cb
([0; T ); B p)
=
t(1+ d=p)=2 f
2 L1 ((0; T ); Bp ;1 ) and
limt!0 t(1+ d=p)=2 kf kBp ;1 = 0
g:
~ g) and ~g 7! Then, for f~ 2 L1 ((0; T ); (B11;1 )d ), the operators ~g 7! B (f;~ ;p d ~ B (f;~g) are bounded on (ET ) for every p 2 (d; 1) and every 2 (1; 1+d=p).
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296
Regularity results
Proof: Point i) is easily checked. Before proving point ii), we give an exact description of B p for p < 1: a distribution f belongs to B p if and only if f 2 S00 , S0 f 2 Lp, for all j 2 IN j f 2 Lp and limj!+1 2j(1 d=p) kj f kp = 0. Thus, we have, for > 0, Bpd=p 1+;1 B p . If f~ 2 L1 ((0; T ); (B11;1)d ) and t(1+ d=p)=2~g 2 (L1 ((0; T ); Bp ;1 ))d , (1+ d=p) ~ ~ g) and t (1+ 2 d=p) IPr ~ g) be~ :M (f;~ with > 1, we nd that t 2 IPr :M (f;~ 1 2 ; 1 d
; 1 d ~ long to (L ((0; T ); Bp )) . The norm of B (f;~g) in (Bp ) for 2 ~ g)kB 2;1 + k( )( +2)=2 B (f;~ ~ g)kB 2;1 . We may be estimated as kB (f;~ p p have ~ g)kB 2;1 k( ) ( 2 +2) B (f;~ p with
(
C
Z t
0
(t
ds
~ t)N2 (~g; t) ( +2) (1+ d=p) N1 (f; s) 2 s 2
~ t) = sup0<s
For 2 [ 2; ), we get that
t
(d=p 1) ~ g)kB 2;1 2 k( ) ( 2 +2) B (f;~ p
C N1 (f;~ t)N2 (~g; t):
With the values = 2 and = d=p 1+ ( 2 (0; +1 d=p)), we nd that ~ g) belongs to (B p )d , then with the value = d=p 1, we nd that B (f;~ ~ g) B (f;~ ~ g)kBp C (1 + t)N1(f; ~ t)N2 (~g; t). belongs to L1((0; T ); (B p )d) and that kB (f;~ ~ Since limt!0 N2 (~g; t) = 0, we nd that limt!0 kB (f;~g)kBp = 0. Thus, t 7! ~ g) 2 (B p )d is continuous at O+ . We now prove the continuity of the map B (f;~ on (0; T ). For 0 < t1 < t2 , we write, as we have done before, the identity for R 0 < s < t1 : e(t2 s) e(t1 s) = 0t2 t1 e d e(t1 s) , which gives
k(e(t2 and
k(
s)
)
and we obtain
t t e(t1 s) )kL((Bp 2;1 )d;(Bp 2;1 )d) C min(1; 2 1 ) t1 s
(d=p+1 ) (t2 s) 2 (e
e(t1 s))kL((Bp 2;1 )d;(Bp 2;1 )d ) t2 t1 C min( (d=p+1 ) ; (d=p+3 ) ) 2 (t1 s) jt1 sj 2 1
(1+ d=p) ~ g)(t1 ) B (f;~ ~ g)(t2 )kBp CT t2 t1 2 N1 (f; ~ t2 )N2 (~g; t2 ): kB (f;~ t2
~ g)kBp ;1 . Now we deal with the value = , i.e., to estimate kB (f;~ R t We want more accurately to estimate t(1+ d=p)=2 k 0 e(t s)~z(s) dskBp 2;1
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Further results on uniqueness when sup0<s
0
ke(t
s)~z(s)k 2;1 Bp
ds C
1.
Z t=2
0
297
We split the integral in two 1
t s
k~z(s)kBp
2;1 ds;
for t=2 < s < t, we write that 1[t=2;t](s)~z 2 (L1 ((O; t); Bp 2;1 )d with, more (1+ d=p) 2 precisely, supt=2<s
k
Z t
0
e(t s) f dskBp 2;1 CT sup kf (s)kBp 2;1 0<s
for every t < T and every f 2 L1 ((0; T ); Bp 2;1). We proved a similar result in Chapter 27 (Lemma 27.3) for homogeneous Besov spaces. This gives the R R control for (Id S0 ) 0t e(t s)f ds = 0t e(t s)(Id S0 )f ds; the remainder R R is easily controlled by k 0t e(t s)S0 f dskBp 2;1 C 0t kf kBp 2;1 ds. The following point is to replace the estimate f~ 2 (L1 (B11;1 ))d by f~ 2 1 (L ((0; T ) IRd ))d :
Lemma 28.3: A) i) For f 2 L1 , the operator g 7! (f; g ) is bounded from Hps to Hps for every p 2 (1; 1) and s 2 IR and g 7! (f; g) is bounded from Hps to Hps for every p 2 (1; 1) and s > 0. ~ g) ii) For T > 0, if f~ 2 (L1 ((0; T ) IRd ))d ), the operator ~g 7! B (f;~ q s d is bounded on L ((0; T ); (Hp ) ) for every p and q in (1; 1) and every s 2 IR p with an operator norm bounded by C T sup0
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298
Regularity results
~ g)(t)kHps C R0t p 1 kM (f;~ ~ g)kHps d . Since we Now, we write kB (f;~ t p have k p1t kL1 (0;T ) = 2 T , we nd that ~ g)kLq ((0;T );(H s )d) C pT kB (f;~ p
sup
0<s
kf~k1k~gkLq ((0;T );(Hps )d) :
Similarly, we write R
~ g)kBp ;1 ~ g)(t)kBp ;1 C 0t p 1 kM (f;~ kB (f;~ t 0 1 = 2 (1 d=p ) = 2 Ct t sup0<s
~ g)(t)k d=p kB (f;~ Bp
C 0 t1=2
1;1
C
Z t
0
1
jt j (d=p2 )
~ g)kBp ;1 kM (f;~
+1 d=p d=p 1 ( +1 d=p)=2 k~gk ;1 +d=p 1 : sup kf~k1 sup k~gkB +d=p 1 ;1 s B p p 0<s
~ g) belongs more precisely to (B p )d and We do not have to check that B (f;~ ~ that the map t 7! B (f;~g) is continuous, since L1 B11;1 , so that we may use the results of Lemma 28.2. Lemmas 28.1 to 28.3 then give the following fundamental regularity result:
Lemma 28.4: ~ :~u0 = 0. If ~u is a weak Let ~u0 2 (B p \ L2 )d for some p 2 (d; 1), with r solution of the Navier{Stokes equations on (0; T ) IRd so that ~u belongs to the space (C ([0; T ); B 1 )d \ (L1 ([0; T ); L2 ))d \(L2 ([0; T ); H 1 ))d and has the initial value ~u0 , then ~u belongs to the space (ET ;p )d for every T < T and every 2 (1; 1 + d=p). Proof: We know that ~u belongs to Lq ((0; T ); (Hr )d ) (with q 2 (2; 1), = 1 + 1=q and r = d=(d 1=q) and that w~ is a xed point of the mapping ~g 7! 2B (~u;~g) B (~u;~g) + w~ 0 with w~ 0 = 2B (~u; 2t~u0 ) + B (~u; et~u0 ). ~ g) 7! 2B (f;~ ~ g) + B (f;~ ~ g) = Af~(~g) is conWe know that the operator (f;~ 1 1 ; 1 d q d tinuous from L ((0; Æ); (B1 ) )) L ((0; Æ); (Hr ) ) to Lq ((0; Æ); (Hr )d ) and that we have for a constant C0 , which does not depend on Æ 2 (0; T ):
kAf~(~g)kLq ((0;Æ);(Hr )d ) C0
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sup kf~kB11;1 k~gkLq ((0;Æ);(Hr )d )
0<s<Æ
Further results on uniqueness
299
~ g) 7! Af~(~g) is continuous from L1 ((0; Æ); (L1 )d )) Moreover, the operator (f;~ q d q L ((0; Æ); (Hr ) ) to L ((0; Æ); (Hr )d ) and we have for a constant C1 which does not depend on Æ 2 (0; T ):
p kAf~(~g)kLq ((0;Æ);(Hr )d ) C1 Æ
sup kf~k1 k~gkLq ((0;Æ);(Hr )d )
0<s<Æ
~ g) 7! Af~(~g) is continuous from Similarly, we know that the operator (f;~ ;p ;p 1 1 ; 1 d d d L ((0; Æ); (B1 ) ) (EÆ ) to (EÆ ) and we have for a constant C2 which does not depend on Æ 2 (0; T ):
kAf~(~g)k(EÆ ;p)d C2
sup kf~kB11;1 k~gk(EÆ ;p)d
0<s<Æ
~ g) 7! Af~(~g) is continuous from L1 ((0; Æ); (L1 )d )) Finally, the operator (f;~ (EÆ ;p )d to (EÆ ;p )d and we have for a constant C3 which does not depend on Æ 2 (0; T ): p kAf~(~g)k(EÆ ;p)d C3 Æ sup kf~k1 k~gk(EÆ ;p)d 0<s<Æ We then x T < T . Since the test functions are dense in B 1 and since [0; T ] is compact, L1((0; T ); L1) is dense in C ([0; T ]; B 1. Hence, we may split ~u in ~u = X~ + Y~ with sup0<s
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300
Regularity results
Proposition 28.1: ~ :~u0 = 0. If ~u and ~v are two Let ~u0 2 (B p )d for some p 2 (d; 1), with r weak solutions of the Navier{Stokes equations on (0; T ) IRd so that ~u and ~v belong to the space (ET ;p )d with 2 (1; 1 + d=p) and have the same initial value ~u0 , then ~u = ~v. Proof: We see easily that, one more time, it is enough to prove that we have local uniqueness: if ~u is a solution of the Navier{Stokes equations on (0; T ) IRd and if ~u 2 (ET ;p )d , then for t0 2 (0; T ) ~u(t + t0 ) is a solution of the Navier{ Stokes equations on (0; T t0 ) IRd with initial value ~u(t0 ) 2 (B p )d and ~u(t + t0 ) 2 (ET ;pt0 )d . Thus, local uniqueness would imply global uniqueness. If w~ = ~u ~v , w~ 1 = B (~u; ~u) and w~ 2 = B (~v; ~v ), we have w~ = B (w; ~ ~u) B (~v ; w~ ) = B (w; ~ w~ 1 ) B (w~ 2 ; w~ ) B (w; ~ et~u0) B (et~u0 ; w~ ): We are going to estimate
A(t; w~ ) = kw~ (t)kBpd=p 1;1 + t(1+ d=p)=2 kw~ (t)kBp ;1 with the help of the normsp sup0<s
A(t; B (w; ~ w~ 1 )) C sup kw~ (s)kB11;1 sup s(1+ d=p)=2 kw~ 1 (s)kBp ;1 0<s
0<s
A(t; B (w; ~ w~ 1 )) C sup s(1+ d=p)=2 kw~ (s)kBp ;1 sup kw~ 1 (s)kB11;1 0<s
0<s
0<s
0<s
A(t; B (w; ~ w~ 1 )) C sup
s(1+ d=p)=2 kw~ (s)kBp ;1
sup kw~ 1 (s)kB11;1
and
A(t; B (w~ 2 ; w~ )) C sup s(1+ d=p)=2 kw~ (s)kBp ;1 sup kw~ 2 (s)kB11;1 0<s
0<s
A(t; B (w~ 2 ; w~ )) C sup kw~ (s)kB11;1 sup s(1+ d=p)=2 kw~ 2 (s)kBp ;1 0<s
A(t; B (w~ 2 ; w~ )) C sup
0<s
0<s
sup kw~ 1 (s)kB11;1
0<s
Similarly, Lemma 28.3 gives
A(t; B (w; ~ et~u0 )) C sup s(1+ d=p)=2 kw~ (s)kBp ;1 sup
pskes~u k
A(t; B (w; ~ et~u0)) C sup s(1+ d=p)=2 kw~ (s)kBp ;1 sup
pskes~u k
0<s
0<s
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0<s
0<s
1
0
0
1
Further results on uniqueness
301
and
A(t; B (et~u0 ; w~ )) C sup s(1+ d=p)=2 kw~ (s)kBp ;1 sup
pskes~u k
A(t; B (et~u0 ; w~ )) C sup s(1+ d=p)=2 kw~ (s)kBp ;1 sup
pskes~u k
0<s
0<s
0<s
0<s
1
0
1
0
To control the remaining terms, we use the boundedness of the paraproduct from L1 Bps;1 to Bps;1 for p 2 [1; 1] and s < 0. This gives
A(t; B (w; ~ et~u0 )) C sup kw~ (s)kBpd=p 1;1 sup
pskes~u k
A(t; B (et~u0 ; w~ )) C sup kw~ (s)kBpd=p 1;1 sup
pskes~u k
0<s
0<s
and 0<s
0<s
0
0
Thus, we get that for some constant C0 we have
A(t; w~ ) C sup A(s; w~ ) sup A(s; w~ 1 ) + A(s; w~ 2 ) + sup 0<s
0<s
0<s
1
1
pskes~u k 0
1
Since ~u0 2 B p (i.e., since p it may be approximated through smooth functions), we have limt!0 tket~u0 k1 = 0; moreover, since > d=p 1, we have limt!0 t(1+ d=p)=2 ket~u0 kBp ;1 = 0. Since ~u 2 (ET ;p )d , we nd that w~ 1 = et~u0 ~u satis es limt!0 t(1+ d=p)=2 kw~ 1 kBp ;1 = 0. Finally, since ~u is continuous in B p norm and limt!0 ket~u0 ~u0kBpd=p 1;1 = 0, we nd that limt!0 kw~ 1 kBpd=p 1;1 = 0. Of course, w~ 2 satis es similar estimates and, for some positive , we have sup A(s; w~ 1 ) + A(s; w~ 2 ) + sup
0<s<
0<s
pskes~u k 0
1
2C1 ; 0
which gives w~ = 0 on [0; ]. Thus, Proposition 28.1 and Theorem 28.3 are proved.
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Chapter 29 Stability and Lyapunov functionals A Lyapunov functional for the Navier{Stokes equations is a norm k~u(t; :)kE which is decreasing for t 2 [0; 1). For instance, the energy norm k~u(t; :)k2 is a Lyapunov functional for the set of Leray solutions (at least, for the Leray solutions that satisfy Serrin's criterion of uniqueness, for which we have the ~ ~uk22 = 0). Under the assumption that ~u0 2 energy equality @t k~uk22 + 2kr d p d d (L \ L (IR )) with a small norm in Ld, Kato [KAT 90] has proved that the Lp norm was a Lyapunov functional for the solutions ~u 2 (C ([0; 1); Lp \ Ld))d (1 < p < 1). Kato [KAT 90] has also studied the norm in the potential spaces Hps for p 2 and s > 0 when the initial data has a small norm in (Ld)d . More recently, Cannone and Planchon [CANP 00] discussed the case of Besov norms as Lyapunov functionals, under a condition of size of ~u in (L1 ([0; 1); B_ 11;1 ))d . We adapt their proof, considering the data in (B_ ps;q \ BMO 1 )d with no scaling condition on s instead of considering only data in (B_ ps;q )d with the scaling condition s = d=p 1 (which ensures that Bps;q BMO 1 ); this adaptation has been made in a joint work with E. Zahrouni [ZAH 02].
1. Stability in Lebesgue norms We rst prove the stability theorem of Kato:
Theorem 29.1: (Stability in Lebesgue norms) For 1 < p < 1 there exists a positive p depending only on d and p so that: ~ :~u0 = 0 and k~u0 kd < p , (A) When ~u0 2 (Lp (IRd) \ Ld (IRd))d with r then the solution ~u 2 (C ([0; 1); Lp \ Ld))d for the Navier{Stokes equations with initial data ~u0 has a norm in (Lp (IRd ))d , which decreases on [0; 1) and limt!1 k~ukp = 0. (B) When p 2 we have a more general decay result: p may be chosen so ~ :~u0 = 0 and k~u0kd < p and when that, when ~u0 2 (Lp (IRd ) \ Ld (IRd ))d with r d d p d d ~ ~v0 2 (L (IR ) \ L (IR )) with r:~v0 = 0 and k~v0 kd < p , then the solutions ~u and ~v in (C ([0; 1); Lp \ Ld ))d for the Navier{Stokes equations with initial data ~u0 and ~v0 are getting closer in Lp norm as t increases: t 7! k~u ~v kp is decreasing on [0; 1). 303 © 2002 by CRC Press LLC
304
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Proof: Step 1: Regularity of the solutions We know (Theorem 15.3) that there exists (d) > 0 so that, when ~u0 2 d ~ :~u0 = 0 and k~u0kd < (d), then the Navier{Stokes equa(L (IRd))d with r tions with initial data ~u0 have a global solution ~u 2 (C ([0; 1); Ld))d with supt>0 k~u(t; :)kd 2k~u0kd. If moreover ~u0 2 (Lp )d for some p 2 [1; 1), then the persistency theorem (Theorem 18.3) gives that ~u 2 (C ([0; 1); Lp ))d . This mild solution is smooth. We know (Proposition 15.1) that for all 0 < T1 < T2 and all 2 INd we have supT1
d k~u ~vkpp = p d (@t~u @t~v):j~u ~v jp 2 (~u ~v ) dx: (29:1) dt IR P 2 Indeed, let = e jxj + dk=1 juk (x) vk (x)j2 1=2 . Then Z
d k kp = p d (@t~u @t~v):p 2 (~u ~v) dx: dt p IR 2 Since, for 0 < < 1, we have j~u ~vj e jxj =2 + j~u ~v j, we nd by the dominated convergence theorem that Z
d (@t ~u @t~v):j~u ~vjp 2 (~u ~v) dx: lim k kp = p !0 dt p IRd kkpp ke jxj2=2 kp + k~u ~v kp , and the majorant is a continuous function of t; hence, we nd that inequality (29.1) is valid at least in the sense of distributions on (0; 1); since the right-hand term in equality (29.1) is a continuous function of t, we nd that k~u ~vkpp is C 1 and that (29.1) is valid as an equality between continuous functions. Equation (29.1) may be rewritten as R d k~u ~v kp = p IRd (~u ~v ):j~u ~vjp 2 (~u ~v) dx p dt R (29:2) ~ :(~u ~u ~v ~v) :j~u ~vjp 2 (~u ~v) dx p IRd IPr
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305
R
Step 3: The term IRd (~u ~v):j~u ~v jp 2 (~u ~v ) dx equality (29.2), we de ne w~ = ~u ~v and look at the properties R To deal with of IRd w: ~ jw~ jp 2 w~ dx. 2 Recall that we have de ned = (jw~ j2 + e jxj )1=2 . is a C 2 positive function, and so is for every 2 IR. We have, for j 2 f1; : : : ; dg, @j ( ) = @j 1 . We then write R
2 ~ p 2 w~ dxj = jxj jR @j w:
followed by @j = (@j w~ :w~
R @j w: ~ p 2 w~ xxjj =+ = R R j@j w~ j2 p 2 dxj R jxj jR (p 2) jxj jR (@j w~ :w~ ) @j p 3 dxj ;
2 xj e jxj ) 1 . We get
R
R 2 p 2 2 p 4 jxj jR j@j w~ j dx j + (p 2) jxj jR j@j w~ :w~ j dxj = x =+ R @j w~ :p 2 w~ xjj = R (29:3) R R 2 +(p 2) jxj jR (@j w: ~ w~ ) xj e jxj p 4 dxj jxj jR @j2 w~ :p 2 w~ dxj :
We have j@j2 w~ :p 2 w~ j j@j2 w~ jp 1 2 L1 (IRd ) (note that @j2 w~ 2 (Lp )d since w~ 2 (Lp )d and since the Riesz transform operates boundedly on Lp ) and 2 2 j(@j w: ~ w~ ) xj e jxj p 4 j j@j w~ jp 1 jxj j =2 e jxj =2 2 L1 (IRd ) for 0 < p d p < p 1 (we have @j w~ 2 (L ) since (Id )w~ 2 (L )d and since the kernel of the convolution operator Id@j belongs to L1 (IRd )). We may then take the limit of (29.3) as R goes to +1, and then integrate with respect to the other variables xk 's, obtaining (29:4)
R
j@jRw~ j2 p 2 dx + (p
R
2) j@j w~ :w~Rj2 p 4 dx = 2 (p 2) (@j w: ~ w~ ) xj e jxj p 4 dx @j2 w~ :p 2 w~ dx:
We now take the limit as goes to 0. For p 2 we write j@j w: ~ w~ j2 p 4 p 2 j x j = 2 p 2 2 p 2 2 ) and we apply the dominated converj@j w~ j j@j w~ j (jw~ j + e gence theorem; for p < 2, we use the monotone convergence theorem since 1 R R 1 2 p 2 2 increases to Rjw~ j . Thus, we have lim ~ j dx = j@j w~ j jw~ jp 2 dx jw R !0 j@ 2 p 4 2 and lim!0 j@Rj w: ~ w~ j dx = j@j w~ :w~ j jw~ jp 4 dx. On the other hand, we 2 have seen that (@j w~ :w~ ) xj e jxj p 4 dx = O( =2 ) for 0 < < p 1 and that j@j2 w: ~ p 2 w~ j Rj@j2 w~ jp 1 , so thatRwe may apply the dominated convergence and get lim!0 @j2 w: ~ p 2 w~ dx = @j2 w: ~ jw~ jp 2 w~ dx. Thus R
(29:5)
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(~u R ~v ):j~u ~vjp 2 (~u ~v) dx = 2 p 2 dx j =1 j@j (~u ~v )j j~u ~v j Pd R (p 2) j=1 j@j (~u ~v):(~u ~v)j2 j~u ~vjp 4 dx: Pd
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Regularity results
Step 4: The function j~u ~v jp=2 If we approximate again j~u ~vj by , we have 2 = (p=2)(@j (~u ~v):(~u ~v) xj e jxj2 )p=2 2 @j p= and we have for 0 < < p=2 and w~ = ~u ~v 2 2 j j @j p= p
2 if p < 2 j@j w~ jjw~ j(p 2)=2 + =2 jxj je jxj =2 p= 2 if p 2 j@j w~ j(p 2)=2 + =2 jxj je jxj =2 p= 2
2
2 remains bounded in L2 as goes to 0. Taking the limit in D0 , Thus @j p= we nd that @j (j~u ~vjp=2 ) 2 L2 and that
kr~ (j~u ~vjp=2 )k2 C
d X j =1
kj~u ~vjp=2 1 @j (~u ~v)k2 C 0 k(~u ~v)k1p=2 k ~u ~vkpp 1=2 :
P Let us de ne Qp (f~) = dj=1 k jf~jp=2 1 @j f~k2 . Up to now we have proved the following results: 2d i) j~u ~vjp=2 2 H 1 L d 2 and kj~u ~vjp=2 k d2d2 C kr~ (j~u ~vjp=2 )k2 C 0 Qp (~u ~v)
ii) for some positive constant Ap
Z
d k~u ~vkpp Ap Qp (~u ~v)2 p IPr~ (~u ~u ~v ~v):j~u ~vjp 2 (~u ~v ) dx dt R ~ :(~u ~u ~v ~v):j~u ~v jp 2 (~u ~v ) dx for p 2 Step 5: Control of IPr Let = 2=p 2 (0; 1]. We rst write that j~u ~u ~v ~vj = j~u (~u ~v) + (~u ~v) ~vj (j~u ~vjp=2 ) (j~uj + j~v j): On the other hand, we have
j@j j~u ~vjp 2 (~u ~v) j C j@j (~u ~v)j j~u ~vjp 2= C j@j (~u ~v)j j~u ~vj p 1(j~u ~vj p )1 2
2
Since j~uj + j~vj 2 Ld, (j~u ~vjp=2 ) 2 L (d 2) and 0 < d1 + ( 12 d1 ) < 1, we may forget the operator IP (the Rieszp transforms operate boundedly onp Lr for 1 < r < 1); since j@j (~u ~v)j j~u ~v j 2 1 2 L2 and (for p > 2) (j~u ~v j 2 )1 2 2d L (d 2)(1 ) , we use the Holder inequality (since d1 + ( 12 d1 )+ 12 +(1 )( 12 d1 ) = 1), we nd that R j IPr~ (~u ~u ~v ~v):j~u ~vjp 2 (~u ~v) dxj C (k~ukd + k~vkd)k~u ~vk (d 2d2) Pd ( j=1 kj~u ~vjp=2 1 @j (~u ~v)k2 ) k j~u ~vjp=2 1 k (d 2)(1 2d ) 0 1 C (k~ukd + k~vkd)Qp (~u ~v) Qp (~u ~v)Qp (~u ~v) : d
2
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Lyapunov functionals
307
We obtain for two positive constants Ap and Bp : d k~u ~vkpp Ap 1 Bp (k~ukd + k~vkd ) Qp (~u ~v)2 dt
and dtd k~u ~vkpp 0 if k~ukd + k~vkd 1=Bp. R ~ :~u ~u):j~ujp 2~u dx for p < 2 Step 6: Control of (IPr 2 ~ :~u = 0, we write r ~ :~u ~u = (~u:r ~ )~u and Let = p 1 2 (0; 1). Since r thus
d
d
j =1
j =1
X X jr~ :~u ~uj C j~ujj~uj1 p=2 j@j ~ujj~ujp=2 1 = C j~uj(j~ujp=2 ) j@j ~ujj~ujp=2 1 : d
We then write j~uj 2 Ld , (j~ujp=2 ) 2 L (d 2) and dj=1 j@j ~ujj~ujp=2 1 2 L2; since 0 < 1d + 12 + ( 12 1d ) < 1, we may again forget the operator IP. Moreover, j~ujp 2~u 2 (L (1 2)(dd 2) )d since p 1 = p2 (1 ). Now, we obtain for two positive constants Ap and Bp : 2
P
d p k ~ukp Ap 1 Bp k~ukd Qp (~u)2 dt
and dtd k~ukpp 0 if k~ukd 1=Bp.
Remark: The theorem remains valid when we replace the Ld norm with the weaker Ld;1 norm. Starting from the statement that j~u ~vjp=2 2 H 1 , we use the sharp Sobolev inequality H 1 L2d=(d 2);2. Thus, we have only to check that for 2 (0; 1) and q 2 (1; 1) and r 2 [1; 1], we have for f 2 Lq;r jf j 2 Lq= ;r= . But Theorem 2.3 states that f may be decomposed as f = P this is very easy: j (1 1 =q) kfj k1 )j 2ZZ 2 lr (ZZ) and (2j=q kfj k1 )j 2ZZ 2 lr (ZZ) and f with (2 j 2ZZ j P where the fj 's have disjoint measurable supports. Then, jf j = j2ZZ jfj j with (2 j (1 1=q) kjfj j k 1 )j2ZZ 2 lr= (ZZ) and (2j =q kjfj j k1)j2ZZ 2 lr= (ZZ), which gives jf j 2 [L1= ; L1 ][1 1=q;r= ] = Lq= ;r= . This gives d k~u ~vkpp Ap 1 Bp0 (k~ukLd;1 + k~vkLd;1 ) Qp (~u ~v)2 for 2 p < 1 dt and d p k~uk Ap 1 Bp0 k~ukLd;1 Qp (~u)2 for 1 < p < 2: dt p
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2. A new Bernstein inequality Before turning our attention to the case of Besov norms, we consider a new kind of Bernstein inequality proved recently by Planchon [PLA 00] (generalizing a former result of Danchin [DAN 97]):
Proposition 29.1: Let p 2 (2; 1). Then there exists two positive constants cp and Cp so that for every f 2 S 0 (IRd ) and every j 2 ZZ we have ~ (jj f jp=2 )k22=p Cp 22j=p kj f kp cp 22j=p kj f kp kr where j f is the j -th dyadic block in the Littlewood-Paley decomposition of f .
Proof: By homogeneity, it is enough to prove the inequality for j = 0. Since p=2 > 1 and since 0 f is smooth, we nd that @ @ (j0 f jp=2 ) = p=2 sgn(0 f )j0 f jp=2 1 f; @xk @xk 0 hence, we may write
kr~ (j0 f j p )k2p = 2p kj0 f j p 1 r~ 0 f k2p p2 k0 f k1p p kr~ 0 f kpp Cp k0 f kp 2
2
2
2
2
2
according to the Bernstein inequality. For the converse part, we just de ne P gk = @k 0 f , so that 0 f = 1kd @x@ k gk . We then write R
j0 f jp dx
= =
R P
( 1kRd @x@ k gk )0 fj0 f jp 2 dx @ 1kd gk @xk (0 fj0 f jp 2 ) dx P @ 1kd (p 1)kgk kp k j0pf jp 2 @xk 0 f kp=(p 1) P p=2 1 @ 1kd (p 1)kgk kp k j0 f j 2p 1 @xk 0 f k2k0 f kp 2 C k0 f kp= p k j0 f j 2 1 @x@ k 0 f k2 P
where the last inequality follows from the Bernstein inequality k @k 0 f kp C k0 f kp .
If we look at a vector-valued function, we have a similar statement:
Proposition 29.2: Let p 2 (2; 1). Then there exists two positive constants cp and Cp so that for every (real-valued) f~ 2 (S 0 (IRd ))d and every j 2 ZZ we have cp
P p p @ @ ~ j f~k2 1kd k jj f~j 1 @xk j f~k2 + 1kd k jj f~j 2 @xk j f: Cp 2 2j kj f~kp= p
P
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2
2
Lyapunov functionals
309
where j f~ is the j -th dyadic block in the Littlewood-Paley decomposition of f~.
Proof: By homogeneity, it is enough to prove the inequality for j = 0. The majoration by Cp is a direct consequence of the Bernstein inequalities. For the P converse, we de ne ~gk = @k 0 f~, so that j0 f~j2 = 1kd @x@ k ~gk :0 f~, and then write R R P j0 f~jp dx = ( 1kd @x@ k ~gk ):0 f~j0 f~jp 2 dx R P = ~gk : @x@ k (0 f~j0 f~jp 2 ) dx 1 k d P p 2 @ 0 f~k p=(p 1) 1kP d (p 1)k~gk kp k j0 f~j @xk ~ 0 f~kp=(p 1) +(p 2) 1kd k~gk kp k j0 f~jp 3 @x@ k 0 f: P ~ ~ p=2 1 ~p 1 @ 1 kd k~gk kp k j0 f j 2 @xk 0 f k2 k0 f kp P p 2 1 ~ 0 f~k2 k0 f~kp= +(p 2) 1kd k~gk kp k j0 f~j 2 2 @x@ k 0 f: p P p 2 @ C k0 f~kp= p 1kd k j0 f~p j 2 1 @xk 0 f~k2 P 2 2 @ ~ 0 f~k2 : +C k0 f~kp= p 1kd k j0 f~j 2 @xk 0 f:
3. Stability and Besov norms We may now state the theorem on the Besov norms as Lyapunov functionals. We de ne the norm of ~u in (B_ ps;q )d as
k~ukB_ ps;q =
X
j 2ZZ
Z X d
2jsq ( (
k=1
jj uk j2 )p=2 dx)q=p 1=q :
Theorem 29.2: Let p 2 [2; 1), q 2 [1; 1), s > 1 so that s + 2q > 0. Then there exists a positive constant = (p; q; s) so that whenever ~u0 belongs to (CMO 1 )d (where CMO 1 is the closure of the test functions in BMO 1 ) and to (B_ ps;q )d ~ :~u0 = 0 and k~u0 kBMO 1 < , then the Koch-Tataru solution ~u 2 with r (C ([0; 1); CMO 1 ))d for the Navier{Stokes equations with initial data ~u0 belongs to (C ([0; 1); B_ ps;q ))d ; moreover, its norm in (B_ ps;q (IRd ))d decreases on [0; 1) and limt!1 k~ukB_ ps;q = 0. Proof: Step 1: Regularity of the solutions According to the Koch and Tataru theorem, we know that there exists ~ :~u0 = 0 and k~u0kd < (d), (d) > 0 so that, when ~u0 2 (BMO 1 (IRd ))d with r then the Navier{Stokes equations with initial data ~u0 have a global solution;
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Regularity results
this solution satis es supt>0 tk~uk1 C k~u0 kBMO 1 and we easily check that supt>0 k~ukBMO 1 C k~u0kBMO 1 (see Theorem 16.2 and its corollary). If moreover ~u0 2 (B_ ps;q )d for some p 2 [1; 1), q 2 [1; 1) and s > 1, then we have some information on the behavior of ~u due to the various persistency theorems already proved (see Theorem 19.3 for s > 0, Theorem 22.3 for s = 0 and Theorem 20.3 for s < 0). More precisely, we proved: i) for s > 0: ~u 2 (Cb ([0; 1); B_ ps;q ))d . ii) for s = 0: supt>0 t1=4 k~uk2p < 1. iii) for s < 0: t s=2 k~ukp 2 (Lq ((0; 1); dtt ))d . In all cases, we nd that ~u 2 (Cb ([0; 1); B_ ps;q ))d : we easily see that ~u et~u0 2 (L1 ([0; 1); B_ ps;1 ))d . Indeed, it is simple to estimate the size of kB (~u; ~u)kB1s+;1 for 2 (max( 1; 1 s); 1): i) for s > 0, we write that k~u ~ukB_ ps;q C k~ukB_ ps;q k~uk1 C (~u0 )t 1=2 and thus
kB (~u; ~u)kB1s ;1 +
R
~ jk1C (~u0 ) pd )=2 e(t ) IPr C 0 (t ) (1+)=2 C (~u0 ) pd = C0 t =2 C (~u0 ):
0t kj( Rt
ii) for s = 0, we write that k~u ~ukp C k~uk22p C (~u0 )t 1=2 and thus kB (~u; ~u)kB1s+;1 C t =2 C (~u0 ). iii) for s 2 ( 1; 0), we write that k~u ~ukp C k~ukpk~uk1 C (~u0 )t(s 1)=2 and thus R kB (~u; ~u)kB1s+;1 R 0t kj( )(s+)=2 e(t ) IPr~ jk1 C (~u0 ) (1ds)=2 C 0t (t ) (1+s+)=2 C (~u0 ) (1ds)=2 = C0 t =2 C (~u0 ): q
Since kB (~u; ~u)kB_ ps;1 C kB (~u; ~u)kB_ ps+;1 kB (~u; ~u)kB_ ps ;1 , we proved that B (~u; ~u) 2 (L1 ((0; 1); B_ ps;1 ))d . Thus, ~u 2 (L1 ((0; 1); B_ ps;q ))d . This gives that, for all [T1 ; T2 ] (0; 1) and all > s ~u 2 (L1 ((T1 ; T2 ); B_ p;q ))d ; indeed, if T0 2 (0; T1), ~u 2 (L1 ((T0 ; T2 ); B_ p;q ))d and 2 (0; 1), then we use the fact that ~u and all its derivatives are bounded on [T0 ; T2 ] (according to the regularity criterion Proposition 15.1) and we write that k~u ~ukB_ p;q C k~ukB_ p;q k~ukB1 u0 ; T0; T2 ) and we nd that on [T0 ; T2 ] we have 1+jj;1 C (~ ( t T 0 ) k~u e ~u(T0 )kB_ p+;q C (T2 T0)(1 )=2 C (~u0 ; T0 ; T2), while on [T1 ; T2 ] we have ke(t T0)~u(T0 )kB_ p+;q C (T1 T0 ) =2 C (~u0 ; T0 ; T2 ). ~ :~u ~u (hence, @t~u) belong to In particular, we nd that ~u and IPr 1 ;q d _ (L ((T1 ; T2 ); Bp )) for all s and all [T1 ; T2] (0; 1); hence, we get ~u 2 (C ((0; 1); B_ p;q ))d . For = s, the continuity is still valid for t = 0: we have limt!0 ket~u0 ~u0 kB_ ps;q = 0 by density of the smooth functions in B_ ps;q , while limt!0 kB (~u; ~u)kB_ ps;q = 0 is a consequence of limt!0 t1=2 k~uk1 = 0. We thus have proved the following estimates, that will be useful in the proof of Theorem 29.2:
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Lyapunov functionals
311
j) ~u 2 (C ([0; 1); B_ ps;q ))d . jj) ~u 2 (C ((0; 1); B_ ps+2=q;q ))d . jjj)For some < s, ~u et~u0 2 (L1 ((T1 ; T2 ); B_ p;q ))d for all [T1 ; T2] (0; 1). Step 2: Derivative of the B_ ps;q norm Recall that we have
k~ukqB_ ps;q =
X
j 2ZZ
Z X d
2jsq ( (
k=1
jj uk j2 )p=2 dx)q=p :
The series converges uniformly on all [T1 ; T2 ] (0; 1), since kj ~ukp kj ~u0 kp +kj B (~u; ~u)kp and since B (~u; ~u) is bounded in (B_ ps+2=q;q )d \ (B_ p;q )d on [T1 ; T2 ] with < s. Thus, we may write (at least in the sense of distributions): d X X d q d k ~ukB_ ps;q = 2jsq ( ( jj uk j2 )p=2 dx)q=p : dt dt k=1 j 2ZZ Z
P 2 Let j; = e jxj + dk=1 jj uk (x)j2 1=2 . Then Z
d k kq = qkj;kqp p d (@t j ~u):pj; 2 j ~u dx: dt j; p IR We nd that j dtd kj; kqpj kj;kqp 1 kj @t~ukp . Since we know that @t~u is, locally in time, bounded in (B_ ps;q )d , we nd by the dominated convergence theorem that Z
d k ~ukq = qkj ~ukqp p d @t j ~u:jj ~ujp 2 j ~u dx: dt j p IR This equality may be rewritten as d q dt kj ~ukp
=
R
qkj ~ukqp p IRd (j ~u):jj ~ujp 2 j ~u dx R ~ :~u ~u):jj ~ujp 2 j ~u dx: qkj ~ukqp p IRd (j IPr
R Step 3: The term IRd (j ~u):jj ~ujp 2 j ~u dx Recall that we proved in Section 1 the identity R (j ~u):jj ~ujp 2 j ~u dx = Pd R j@k j ~uj2 jj ~ujp 2 dx Pkd=1 R (p 2) k=1 j@k j ~u:j ~uj2 jj ~ujp 4 dx: We now use Proposition 29.2 and get the estimate Z
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(j ~u):jj ~ujp 2 j ~u dx Ap 22j kj ~ukpp :
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Regularity results
for some positive constant Ap . R ~ :~u ~u):jj ~ujp 2 j ~u dx when q 2 Step 4: Control of (j IPr We use the paraproduct decomposition of the products and write
kj _ (uk ; ul )kp C kSj+2 ~uk1 and
kj _(uk ; ul )kp=2 C
2 X m=
X
mj
2
2
kj+m ~ukp
km~uk2p
and thus we get
C 2j C 0 22j k~ukB_ 1 ;1 1
R
~ :~u ~u):jj ~ujp 2 j ~u dxj (j IPr P2 kSj+2~uk1kj ~ukpp 1 P m= 2 kj +m ~ukp +kj ~uk1 kj ~ukpp 2 mj 2 km~uk2p 2 kj ~ukpp 1 P m= 2 kj +m ~ukp P +kj ~ukpp 2 mj 2 km~uk2p :
j
We now sum over j to get: P
jsq q kj ~ukq p R d (j ~u):jj ~ujp 2 j ~u p IR qAp Pj2ZZ 2jsq kj ~ukqp p 22j qkj ~ukpp = qAp k~ukqB_ s+2=q;q 1
j 2ZZ 2
and
P
dx
jsq q kj ~ukq p j R (j IPr ~ :~u ~u):jj ~ujp 2 j ~u dxj P p C P j2ZZ k~ukB_ 11;1 2j(sq+2) kj ~ukqp 2 P kj ~ukp 2m= 2 kj+m~ukp + mj 2 km~uk2p C 0 k~ukB_ 11;1 k~ukqB_ 1s+2=q;q
j 2ZZ 2
(since s + 2=q > 0); hence,
d q k~uk s;q dt B_ 1
qAp (1 Bp k~ukB_ 1 ;1 )k~ukqB_ 1s 1
=q;q
+2
for two positive constants Ap and Bp , so that dtd k~ukqB_ 1 ukB_ 11;1 s;q 0 when k~ 1=Bp. R ~ :~u ~u):jj ~ujp 2 j ~u dx when q < 2 Step 5: Control of (j IPr We use the paraproduct decomposition of the products and write
kj _ (uk ; ul )kp C kSj+2 ~uk1
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2 X m=
2
kj+m ~ukp
Lyapunov functionals and
kj _(uk ; ul)kp=q C
X
mj
2
313
km~ukqp km~uk21 q
and thus we get R
~ :~u ~u):jj ~ujp 2 j ~u dxj (j IPr kSj+2 ~uk1 kj ~uP kpp 1 P2m= 2 kj+m ~ukp +kj ~ukq1 1 kj ~ukpp q mj 2 km~ukqp km~uk21 q P C 0 2j k~ukB_ 11;1 2j kj ~ukpp 1 2m= 2 kj+m ~ukp P +2j(q 1) kj ~ukpp q mj 2 2k(2 q) km~ukqp :
j
C 2j
We now sum over j to get: P
jsq q kj ~ukq p R d (j ~u):jj ~ujp 2 j ~u p IR qAp Pj2ZZ 2jsq kj ~ukqp p 22j qkj ~ukpp = qAp k~ukqB_ s+2=q;q 1
j 2ZZ 2
and
dx
R jsq ~ :~u ~u):jj ~ujp 2 j ~u dxj j ~ukqp p j (j IPr j 2ZZ 2 q kP C k~ukB_ 11;1 Pj2ZZ 2j(sq+2) kPj ~ukqp 1 P2m= 2 kj+m ~ukp +C k~ukB_ 11;1 j2ZZ 2j(s+1)q mj 2 2k(2 q) km~ukqp C 0 k~ukB_ 11;1 k~ukqB_ 1s+2=q;q P
(since s + 1 > 0); hence,
d q k~uk s;q dt B_ 1
qAp (1 Bp k~ukB_ 1 ;1 )k~ukqB_ 1s 1
for two positive constants Ap and Bp , so that dtd k~ukqB_ 1 s;q 1=Bp.
=q;q
+2
0 when k~ukB_ 1 ;1 1
Remark: Danchin [DAN 01] recently extended the Bernstein-like inequality described in Proposition 29.1 to the range 1 < p < 1. This allows us to extend Theorem 29.2 to Besov spaces B_ ps;q with 1 < p < 2 (replacing the condition s + 2q > 0 by s + q2 > 2 p p ).
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Part 6:
Local energy inequalities for the Navier{Stokes equations on IR3
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Chapter 30 The Ca arelli, Kohn, and Nirenberg regularity criterion 1. Suitable solutions Leray's theorem is based on an energy inequality. To give an accurate description of the local regularity of a Leray solution, Sche er introduced a local version of this energy inequality [SCH 77]. Weak solutions satisfying the local energy inequality are termed suitable by Ca arelli, Kohn, and Nirenberg [CAFKN 82]. We begin by de ning a loose version of weak solutions:
De nition 30.1: (Weak solutions) Let ~u(t; x) be a weak solution for the Navier{Stokes equations on (0; T ) IR3 : i) ~u is locally square integrable on (0; T ) IR3 ~ :~u = 0 ii) r ~ :(~u ~u) r ~p iii) 9p 2 D0 ((0; T ) IR3 ) @t~u = ~u r Then we de ne suitable solutions:
De nition 30.1: (Suitable solutions) Let ~u(t; x) be a weak solution for the Navier{Stokes equations on (0; T ) IR3 . ~u is suitable on the cylinder Q = (a; b) B (x0 ; r0 ) (where x0 2 IR3 , r0 > 0 and 0 < a
ZZ
ZZ
ZZ
jr~ ~uj2 dx dt j~uj2 (@t + ) dx dt+ (j~uj2 +2p)(~u:r~ ) dx dt
~u is suitable on (0; T )IR3 if ~u is suitable on all cylinders Q (0; T )IR3 . Let us rst comment on hypothesis jjj). In the original de nition of Caffarelli, Kohn, and Nirenberg, the hypothesis for suitable solutions was p 2 317 © 2002 by CRC Press LLC
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Energy inequalities
L5=4 (Q). The reason for using this exponent is that it was at that time the best exponent one could prove when exhibiting suitable solutions for the Navier{ Stokes equations on a bounded domain associated with a square{integrable initial value ~u0 (for the whole space, it is easy to check that the restricted Leray solutions are suitable on (0; 1) IR3 and that the associated pressure may be chosen in L5loc=3 ((0; 1) IR3 ); see Proposition 30.1 below). See the recent paper of Ladyzhenskaya and Seregin [LADS 99]. But Lin [LIN 98] proved the existence of suitable solutions with p 2 L3loc=2 ((0; T ) ) for bounded domains , by using regularity estimates for the pressure obtained by Sohr and Von Wahl [SOHW 86]. The computations are easier with the hypothesis p 2 L3=2(Q) and we always assume this hypothesis in our de nitions of suitability. Inequality (30.1) may be restated in the following form: there exists a local nonnegative measure on (0; T ) IR3 so that the following equality holds in D0 ((0; T ) IR3 ):
~ ~uj2 = j~uj2 @t j~uj2 + 2jr
(30:2)
r~ :f(j~uj2 + 2p)~ug
Existence of suitable solutions on (0; 1) IR3 is provided by the following remarks:
Proposition 30.1: (Leray solution) (A) A restricted Leray solution on (0; T ) IR3 is suitable. (B) A suitable Leray solution satis es the strong energy inequality. Proof: We rst prove (A). Recall that a restricted Leray solution ~u associated ~ :~u0 = 0 is given as a limit of with an initial data ~u0 2 (L2 (IR3 ))3 with r solutions ~un for the molli ed equations (with limn!1 n = 0). We have ~u 2 2 3 2 _1 3 (L1 ((0; T ); L R )) 2 \ (L ((0; T ); H )) . Obviously, i) sup0
r~ :(j~uj2 (u~ !) + 2p ~u)
and we let n go to 0. We know that un converges strongly to ~u in (L2loc )3 : for all R R~ 3 2 D((0; T )IR ) we have limn j (t; x)(~un (t; x) ~u(t; x))j2 dt dx = 0. This gives strong convergence in ((L6t L2x)loc )3 (since the functions ~u are uniformly 23 3 3 bounded in (L1 t Lx ) ) and strong convergence in ((Lloc ) (since the functions ~u 2 6 3 are uniformly bounded in (Lt Lx) ). Moreover, the functions p are uniformly
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The Ca arelli, Kohn, and Nirenberg criterion
319
bounded in the space L2t L3x=2 ; hence, we may assume that the sequence pn converges weakly to p in L2t L3x=2 . Thus, we have the following convergences in D0 : j~un j2 ! j~uj2 , j~un j2 (u~n !n ) ! j~uj2~u and pn ~un ! p ~u. Hence, ~ ~un j2 converges in D0 ; since r ~ ~un is weakly convergent in (L2loc)33 , 2jr ~ ~un j2 ! 2jr ~ ~uj2 + where is a nonnegative locally nite we nd that 2jr measure. The proof of (B) follows the proof of the strong energy inequality for the restricted Leray solutions (Proposition 14.1). We choose 2 D(IR) with 0 R R and dt = 1 and we take for t1 > t0 > 0 and > 0 (t) = 1 t 1 (( s t0 ) ( s t1 )) ds; moreover, we take 2 D(IR3 ) with = 1 in a neighborhood of 0, 0, and we de ne R (x) = (x=R). We apply the local energy inequality to ' = (t) R (x). We let R go to +1 and obtain ZZ
ZZ
j~uj2 1 (( t t0 ) ( t t1 )) dx dt If t0 and t1 are Lebesgue points for the measurable function t 7! k~ukL (dx), we 2
jr~ ~uj2 (t) dx dt
2
get when goes to 0 that 2
Z t1 Z
t0
Z
jr~ ~uj2 (t) dx dt j~u(t0 ; x)j2 dx
Z
j~u(t1 ; x)j2 dx:
For a xed (Lebesgue point) t0 , this inequality is then extended to all t1 > t0 by weak continuity. There are many other ways of nding suitable solutions on the whole space; for instance, Beir~ao da Vega [BEI 85] uses the approximation to the Navier{ Stokes equations by adding a bi-Laplacian in the equations (@t~u = 2~u + ~ :~u ~u r ~ p ). ~u r The measure expresses the lack of regularity of the solution ~u. This may be viewed in several ways. For instance, if we look at the proof of Serrin's theorem on energy equality, we nd the following basic result:
Proposition 30.2: (Energy equality) Let ~u be a weak solution for the Navier{Stokes equations on (0; T ) IR3 and assume that ~u satis es hypotheses j), jj) and jjj) of De nition 30.2 on a cylinder Q (0; T ) IR3 . If ~u 2 (Lpt Lqx(Q))d for some d 2 [d; +1] and p1 = 12 2dq , then ~u satis es (30.2) on Q with = 0. We begin the proof by recalling an elementary inequality:
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Energy inequalities
Lemma 30.1: 3 , f 2 L2 (B ), and r ~ f 2 (L2 (B ))d . Then, for 0 < < Let B = B (0; 1) IR R 3 3 ~ f k3L=22(B) ). 1, f 2 L (B (0; )) and B(0;) jf j dx C (kf k3L2(B) + kf k3L=22(B) kr Proof: Let ' 2 D(IR3 ) with ' = 1 for jxj (1 + 2)=3 and ' = 0 for jxj (2 + )=3. Then r~ ('f ) 2 L2 (IR3 ); thus, the Sobolev inequality gives us that 'f 2 L6(IR3 ) and kf kL3(B(0;) kf'k12=2 kf'k16=2 C kf k12=2(kf k1L=22(B) + kr~ f k1L=22(B) ). d Proof of Proposition 2: We use R Ra smoothing function 2 D(IR ) which is supported in B (0; 1), and satis es dx dt = 1, and de ne, for 0 < < r0 =2, (x) = 1d ( x ). According to Lemma 30.1, ~u belongs to (L3 ((a; b) B (x0 ; r0 2)))3 and @t~u belongs to (L3=2 ((a; b) B (x0 ; r0 2)))3 ; hence, we may write @t j ~uj2 = 2@t( ~u) : ~u = ( @t~u): ~u. Then:
( @t ~u): ~u = r~ : ( [r~ ~u]):( ~u) j [r~ ~u]j2 r~ : ( [~u ~u]):( ~u) + ( [~u ~u]):( [r~ ~u]) r~ : ( p)( ~u) 2 3 2 3 0 Since ~u 2 (L1 t Lx (Q)) (L (Q)) , we have in D ((a; b) B (x0 ; r0 2)) that 2 2 ~ ~ lim!0 @t j ~uj = @t j~uj and lim!0 r: ( [r ~u]):( ~u) = lim!0 21 :j ~ ~u 2 (L2 (Q))dd, we have in D0 ((a; b) ~uj2 = 12 :j~uj2 . Similarly, since r ~ ~u]j2 = jr ~ ~uj2 . We know from B (x0 ; r0 2)) the equality lim!0 j [r Lemma 30.1 and from assumptions j) and jj) that ~u 2 (L3 ((a; b)B (x0 ; r0 )))3 and from assumption jjj) that p 2 L3=2 ((a; b) B (x0 ; r0 ))). Hence, we getthe ~ : ( [~u ~u]):( ~u) = equalities in D0 ((a; b) B (x0 ; r0 2)): lim!0 r ~r: j~uj2~u and lim!0 r ~ : ( p)( ~u) = r ~ : p~u). Thus, under the sole assumptions j), jj) and jjj), we get that
~ ~uj2 2r ~ : j~uj2~u 2r ~ : p~u) @t j~uj2 = j~uj2 2jr ~ ~u]) +2 lim!0 ( [~u ~u]):( [r Now, if we assume that ~u 2 (Lpt Lqx(Q))3 , we nd (from the proof of Lemma 30.1) that ~u 2 (Lrt Lsx((a; b) B (x0 ; r0 )))3 with 1=p +1=r = 1=q +1=s = 2, and ~ ~u]) = thus in D0 ((a; b) B (x0 ; r0 2)) we have lim!0 2( [~u ~u]):( [r ~ ~ ~ j 2[~u ~u]:[r ~u] = lim!0 [( ~u) ( ~u)]:( [r ~u]) = lim!0 r ~ j~uj2~u . ~uj2 ( ~u) = r Duchon and Robert [DUCR 99] proposed an impressive visualization of how the energy inequality expresses the lack of regularity of the solution:
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The Ca arelli, Kohn, and Nirenberg criterion
321
Proposition 30.3: Let ~u be a weak solution for the Navier{Stokes equations on (0; T ) IR3 and assume that ~u satis es hypotheses j), jj) and jjj) of De nition 30.2 on a cylinder Q (0; T ) IR3 . If moreover ~u 2 (Lpt Lqx(Q))d for some d 2 [d; +1] and p1 = 12 2dq . Then ~ ~uj2 + j~uj2 r ~ :f(j~uj2 + 2p)~ug (A) The distribution = @t j~uj2 2jr 0 may be computed as the limit in D (Q) of 2, where =
Z
r~ ' ( ): j~u(t; x ) ~u(t; x)j2 (~u(t; x ) ~u(t; x)) d
and
= (~u ' ~u):
d Z X k=1
@k ' ( )(uk (t; x ) uk (t; x)) (~u(t; x ) ~u(t; x)) d R
where ' = 13 '( x ), ' is a smooth function supported in B (0; 1) with ' dx = 1 and where is de ned in D0 ((a; b) B (x0 ; r0 )). (B) Let b13=3;1 is the closure of D(IR3 ) in B31=3;1 . If for all ! 2 D(Q) the vector distribution !~u belongs to (L3 ((a; b); b13=3;1 ))3 ), then = 0 on Q.
Proof: From the proof of Proposition 30.2, we already know that ~ ~uj2 2r ~ : j~uj2 ~u 2r ~ : p~u) @t j~uj2 = j~uj2 2jr ~ ~u]): +2 lim!0 (' [~u ~u]):(' [r ~ : j~uj2 ~u 2 lim!0 (' [~u ~u]):(' [r ~ ~u]). Hence = r On the other hand, we develop j~u(t; x ) ~u(t; x)j2 (~u(t; x ) ~u(t; x)) and we get: ~ :(j~uj2~u) + ~u:r ~ (' j~uj2 ) j~uj2 (' r ~ :~u) + (R r ~ ' dx):j~uj2~u ' r ~ ~ +2~u: ' (~u:r)~u 2~u:(~u:r)(' ~u) ~ :(j~uj2~u) + ~u:r ~ (' j~uj2 ) + 2~u: ' (~u:r ~ )~u 2~u:(~u:r ~ )(' ~u) = ' r
=
We have
~ :(j~uj2 ~u) ~u:r ~ (' j~uj2 ) = lim r ~ : ' (j~uj2~u) ~u(' j~uj2 ) = 0 lim ' r !0 !0 in D0 (Q). Thus, the only signi cant term in when goes to 0 will be 2~u: ' ~ )~u 2~u:(~u:r ~ )(' ~u). (~u:r Similarly, we develop (~uk (t; x ) uk (t; x))(~u(t; x ) ~u(t; x)) and we get: ~ )~u +(R r ~ ' dx):~u ~u (~u:r ~ )' ~u (' (r ~ :~u)~u =(~u ' ~u): ' (~u:r ~ )~u (~u:r ~ )' ~u = (~u ' ~u): ' (~u:r
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Energy inequalities
Thus, we have ~ )~u (~u:r ~ )' ~u lim!0 2 = 2 lim!0 ' ~u: ' (~u:r ~ : ' ~u:' (~u ~u) 2(' [~u ~u]):(' [r ~ ~u]) r ~ :(j' ~uj2~u) = lim!0 2r 2 ~ :(j~uj ~u) 2 lim!0 (' [~u ~u]):(' [r ~ ~u]) = : =r
To prove (B), we x 0 > 0 and we consider < 0 . We consider ! 2 D(Q) equal to 1 on (a + 0 ; b 0 ) B (x0 ; r0 0 ) and de ne ~v = !~u. On Q0 = (a + 0 ; b 0) B (x0 ; r0 20 ), we have Z
Q0
j (t; x)j dx
Z
C b 0 sup k~v(t; x ) ~v(t; x)k33 dt a0 +0 jj< R
and we have the same estimate for Q0 j(t; x)j dx. We have the estimate supjj< k~v(t; x ) ~v (t; x)k33 Ck~v(t; :)k3B1=3;1 2 L1 ((a0 + 0 ; b0 0 )) and 3 we know that for almost every t lim!0 1 supjj< k~v(t; x ) ~v(t; x)k33 = 0. We then may conclude by dominated convergence.
2. A fundamental inequality Sche er [SCH 77] described how to choose special test functions in order to use the local energy inequality:
Lemma 30.2: (Sche er's test functions) Let ~u be a weak solution for the Navier{Stokes equations on (0; T ) IR3 , so that ~u is a suitable solution on the cylinder (0; 1) B (0; 1). Let '(x) 2 D(IR3 ) be a nonnegative function such that ' 1 on B (0; 1=4) and Supp ' B (0; 1=2) and we de ne 'x (y) = '(y x). Let W (t; y) be the fundamental solution of the heat equation: W (t; y) = which is the solution of @t W
1 3=2 jy4jt2 e 4t
W = Æ. The function x;s;k is then de ned as
x;s;k (; y ) = W (4
k +s
; y x):
We then have the inequality for x 2 B (0; 1=2), k 2 and 3=4 4 k < s < 1 RR R ~ ~u(; y)j2 dy d 23k jx yj< 1k j~u(s; y)j2 dy + 23k s 1k <<s; jx yj< 1k jr 2 4 2 RR D0 0<<1;jyj<1 j~u(; y)j3 d dy 2=3 + RR s ~ g('x (y) x;s;k (; y)) dy d e1=4 (8)3=2 0 (j~u(; y)j2 +2p(; y))f~u(; y):r
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The Ca arelli, Kohn, and Nirenberg criterion where D0 depends neither on ~u, nor on x; s; k.
323
Proof: Since ~u is locally L3 and p is locally L3=2 , we nd that @t ~u is locally L1t (Hx 3=2 ), which gives that ~u is weakly continuous from (0; 1) into D0 (B (0; 1)); 2 3 since ~u 2 (L1 t Lx ((0; 1) B (0; 1))) , we get that ~u is weakly continuous from 2 3 (0; 1)R into (L (B (0; 1))) . Now, we x 2 D(IR), Supp R[0; 1], 0 and dt = 1 and we introduce for > 0 the function !s;() = 1 2(2 ) 1 ( s ) d . If s > 1 and if is small enough ( < s 1 ), !s; is nonnegative and 2 2 bounded by 1. If for some X 2 (s; 1), we have (; y) 2 C 1 ((0; X ) B (0; 1)) and if is nonnegative, the local energy inequality gives for jxj < 1=2 and 0 < s < 1: RR jr~ ~u(; y)j2 !s;()'x (y) (; y) dy d RR j~u(; y)j2 f@ + y g(!s;()'x (y) (; y)) dy d RR ~ g(!s;()'x (y) (; y)) dy d + (j~u(; y)j2 + 2p(; y))f~u(; y):r R We let go to 0; then, for all Lebesgue point s of s 7! jyj<1 j~u(s; y)j2 dy, we get that R RR j~u(s; y)j2 'x (y) (s; y) dy + 12s jr~ ~u(; y)j2 'x (y) (; y) dy d RR 1 2 2 2 k k 1 0 2 j~u(; y)j 'x (y) (; y) dy d+ RRs 0 j~u(; y)j f@ + y g('x (y) (; y)) dy d RR ~ g('x (y) (; y)) dy d + 0s (j~u(; y)j2 + 2p(; y))f~u(; y):r Since ~u is weakly continuous from (0; 1) into (L2 ((B (0; 1))3 , we nd that this equality is valid for all s 2 (0; 1). We now let = x;s;k . For < s, we have f@ +y g x;s;k = 0. Moreover, if s 4 k < < s and jx yj < 2 k , then x;s;k (; y) (8) 3=2 23k e 1=4 . Thus, we get: RR R ~ ~u(; y)j2 dy d 23k jx yj< 1k j~u(s; y)j2 dy + 23k s 1k <<s; jx yj< 1k jr 2 4 2 1=4 (8)3=2 2kk1 R R 12 j~u(; y)j2 'x (y) x;s;k (; y) dy d+ e 0 RRs 2 ~ x;s;k (; y):r ~ 'x (y)) dy d 0R jR~us(; y)j ( x;s;k (; y) 'x (y) + 2r 2 ~ + 0 (j~u(; y)j + 2p(; y))f~u(; y):rg('x (y) x;s;k (; y)) dy d 3 3=2 If 0 < < 1=2 and s > 3=4 4 k, or if jx yj > 1=4, then x;s;k (; y) 2e (check that W (t; y) (4) 3=2 max(t 3=2 ; (6=e)3=2 jyj 3 )) and, for 1 j d, j@j x;s;k (; y)j (22)325=2 e2 (check that j2ytj W (t; y) (41)3=2 max( p21t2 ; e2217jy=2j4 )). This gives us that: 1=4 (8)3=2 2kk1 R R 12 j~u(; y)j2 'x (y) x;s;k (; y) dy d+ e 0 RRs 2 ( x;s;k (; y) 'x (y) + 2r ~ x;s;k (; y):r ~ 'x (y)) dy dg j ~ u ( ; y ) j RR 01 RR ~ 'x (y)j)dy dg C 02 j~u(; y)j2 j'x (y)jdy d + 0s j~u(; y)j2 (j'x (y)j + jr RR 2 = 3 D0 0<<1;jyj<1 j~u(; y)j3 d dy :
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Energy inequalities
3. The regularity criterion We may now introduce the celebrated regularity criterion of Ca arelli, Kohn, and Nirenberg [CAFKN 82]:
Theorem 30.1: (Ca arelli, Kohn, and Nirenberg's regularity criterion) There exist two positive constants CKN and C1 so that if ~u is a weak solution for the Navier{Stokes equations on (0; T ) IR3 , if x0 2 IR3 and 0 < t0 , p if for some r in (0; t0 ) we have (30:3)
Z Z
jx x0j
j~uj3 + jpj3=2 dx ds < 1 r2
where 1 < CKN and if ~u is a suitable solution on the cylinder (t0 B (x0 ; r), then
r2 ; t0 )
sup j~uj < C1 11=3 r 1 : jx x0 j
(30:4)
Proof: We rst notice that ~v(t; x) = r~u(r2 t; rx) is a weak solution for the Navier{Stokes equations on (0; T=r2 )IR3 (with pressure q(t; x) = r2 p(r2 t; rx)), 2 1; t0=r2 ) B (x0 =r; 1), and that we have the suitable onRthe R cylinder (t0 =r inequality jx x0 =rj<1;t0=r2 1<s
sup sup sup jxj< 12 k2 43 221k <s<1
23k
Z
j~u(s; y)j2 dy C12 21=3 jB1 j
jx yj<
1 2
k
with B1 = B (0; 1).
Step 1: The induction R We introduce the function px;k () = jB(x;21 k p2)j jx yj<2 k p2 p(; y) dy. We also de ne for k 2, x 2 B (0; 1=2) and s 2 (3=4 4 k ; 1) the sets Bk (x) =
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The Ca arelli, Kohn, and Nirenberg criterion
325
B (x; 2 k ) and Qk (x; s) = (s 4 k ; s) B (x; 2 k ). We are going to prove by induction on k that (if 1 is small enough) there exists two constants D1 and D2 such that for all x 2 B (0; 1=2), s 2 (3=4 4 k ; 1) and all k 2: ( R Ik = 23k Bk (x) j~u(s; y)j2 dy D1 21=3 (Ak ) RR ~ ~u(; y)j2 dy d D1 21=3 Jk = 23k Qk (x;s) jr and for all k 3
RR Kk = 25k Qk (x;s) j~u(; y)j3 dy d D2 21=3 (Bk ) RR Lk = 25k Qk (x;s) jp(; y) px;k ()j3=2 dy d D2 21=3 2k=2 In the proof of those inequalities, we nd constraints on D1 , D2 and 1 . We rst assume that 1 < 1. When the constraint concerns only D2 , we assume that D2 is big enough to ful ll the constraint. When the constraint concerns only D1 and D2 , we assume that D1 is big enough to ful ll the constraint (the size of D1 will depend on the choice of D2 ). When the constraint concerns the three parameters, we assume that 1 is small enough to ful ll the constraint (the size of 1 will depend on the choice of D1 and D2 ). The constants C , C 0 , : : : , which will appear in the computations, will not depend on D1 , D2 , 1 (nor on ~u, x, s, k). First, we take k = 2 and use the fundamental Rinequality, the fact that ~ ('x x;s;2 )j by D2 [Constraint 1 ] and R0<<1;jyj<1 j~u(; y)j3 + we control jr jp(; y)j3=2 d dy by 1 to get that I2 + J2 D0 21=3 + 3e1=4 (8)3=2 D2 1 . This gives (A2 ) if 1 < 1 and D1 D0 + 3e1=4(8)3=2 D2 [Constraint 2 ]. (
Step 2: Localisation of the pressure Before proceeding further, we need to have an expression for the pressure that uses only the knowledge of ~u on (0; 1) B (0; 1), since we control ~u only on this domain. The Navier{Stokes equations gives us local information on the Laplacian of p: p =
d X d X j =1 l=1
@j @l (uj ul ):
We x a function 2 D(IR3 ) with Supp B (0; 1) and = 1 on B (0; 3=4). We then write for y 2 B (0; 3=4): 1 p(t; y) = ( (y)p(t; y)): 1 The kernel of is the function K (x) = 43jxj . We write ( (z )p(t; z )) = ~ (z ):r ~ p(t; z ) and we get p = p1 + p2 with ( )(z )p(t; zP ) + (P z )p(t; z )) + 2r d d p1 (t; y) = j =1 l=1 @j @l K (uj ul ) and
p2 =
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P P Pd Pd 2 dj=1 dl=1 @j K (uj ul @l ) j =1 l=1 K (uj ul @j @l ) P K (p ) + 2 dj=1 @j K (p@j )
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Energy inequalities
If y 2 B (0; 5=8), we nd that, for all 2 IN3 and 2 IN3 , 6= (0; 0; 0), @ K (y z ) is bounded on the set fz 2 B (0; 1) = @ (z ) 6= 0g, uniformly with respect to y and z , and thus, for all 2 IN3 , sup0
Step 3: (Ak ) true for 2 k n implies (Bn+1 ) RR The control of Kn+1 = 25(n+1) Qn+1 (x;s) j~u(; y)j3 dy d is plain. We choose ! 2 D with ! = 1 on B (0; 1) and with Supp ! B (0; 2) and we write that k~u(; :)kL3 (Bn+1 (x)) C k!(2n+1 (x y))~u(; :)kH 1=2 C 0 k!(2n+1(x y))~u(; :)k12=2 k!(2n+1(x y))~u(; :)k1H=12 C 00 k~u(; :)k1L=22(Bn(x)) (2n k~u(; :)kL2 (Bn (x)) + kr~ ~u(; :)kL2 (Bn (x)) )1=2 : 2n=3 4=3 2n 4 We then use Young's inequality (ab 2 4a + 32 4 b ) and get
kr~ ~uk2L=3(Bn (x)) ): p n Due to the induction hypothesis (An ), we have 2 k~ukL (Bn (x)) D1 11=3 2 n , R ~ ~uk2L (B )(x) d D1 21=3 2 3n . 22n k~uk2L (Bn (x)) D121=3 2 n, and ss 4 n kr n Thus, we nd the estimate Kn+1 C21=3 (D13=2 11=3 + D13 41=3 ), and we have Kn+1 D2 21=3 if 1 is small enoughRR[Constraint 3 : C (D13=2 11=3 +D13 41=3 ) D2 ]. We now study Ln+1 = 25(n+1) Qn (x;s) jp(; y) px;n+1 ()j3=2 dy d. We
k~ukL (Bn (x)) C 3
+1
2 2 k~ukL2(Bn (x)) +22n k~uk2L2 (Bn (x)) +2 n
2
2
n
2 3
2
2
2
+1
write p = p1 + p2 (see Step 3). We decompose the average R pressure px;n+1 as px;n+1 = p1;x;n+1 + p2;x;n+1, with pj;x;n+1() = jBn+11 (x)j Bn+1(x) pj (; y) dy for p j 2 f1; 2g, and we similarly decompose Ln+1 as Ln+1 2(L1;n+1 + L2;n+1 ) RR with Lj;n+1 = 25(n+1) Qn+1 (x;s) jpj (; y) pj;x;n+1()j3=2 dy d. L2;n+1 is easily controlled. For x 2 B (0; 1=2) and n 2, Bn+1 (x) B (0; 5=8); moreover, we have on Bn+1 (x)
jp2 p2;x;n+1j C 2 (n+1) kr~ p2 kL1(B(0;5=8)) ; hence, we have Z
Qn+1
jp2 p2;x;n+1j dy d C 2 (x;s) 3 2
n
9 2
1
Z
0
kr~ p2 k3L=12 (B(0;5=8)) d C 0 2
n
9 2
1
Thus, we get L2;n+1 C 2(n+1)=2 1 . Pd Pd We now study L1;n+1. Recall that p1 (t; y) = j =1 l=1 @j @l K (uj ul ). We split P uj ul into u u 1 + u u (1 1 ) and get p1 = q1;n + r1;n with j l Bn j l B n d Pd @ @ K (u u 1 ). The average pressure p q1;n = j l j l B 1;x;n+1 may n j =1 l=1 be decomposed as p1;x;n+1 = q1;x;n+1 + r1;x;n+1 where q1;x;n+1 and r1;x;n+1
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The Ca arelli, Kohn, and Nirenberg criterion
327
R
are de ned as q1;x;n+1 () = jBn+11 (x)j Bn+1 (x) q1;n (; y) dy and r1;x;n+1() = p 1 R jBn+1 (x)j Bn+1 (x) r1;n (;RRy) dy. We write also L1;n+1 2(L1;q;n+1 + L1;r;n+1 ) with L1;q;n+1 = 25(n+1) Qn+1 (x;s) jq1;n (; y) q1;x;n+1 ()j3=2 dy d and L1;r;n+1 RR = 25(n+1) Qn+1 (x;s) jr1;n (; y) r1;x;n+1 ()j3=2 dy d. We have jq1;x;n+1 ()j 1 R 3=2 2=3 jBn+1 (x)j Bn+1 (x) jq1;n (; y)j dy ; hence, Z
Bn+1
jq1;n (; y) q1;x;n+1()j3=2 dy C kq1;n (; :)k33==22 C k j~uj2 1Bn k33==22 (x)
since the convolution with @j @l K is the Calderon{Zygmund operator Rj Rl and thus operates boundedly on L3=2 . This gives L1;q;n+1 CKn . But we know that Kn C1 D13=2 (1 + D13=2 1 ): for n 3, this is the same proof as for Kn+1 ; for n = 2, we write K2 26 1 CD1 1 [Constraint 4 : CD1 > 26 ]. To estimate L1;r;n+1, we write on Bn+1 that
jr1;n (; y) r1;x;n+1 ()j C 2 (n+1) kr~ r1;n (; :)kL1 (Bn ) : We then write for y 2 Bn+1 +1
j@m r1;n (; y)j
d X d Z X j =1 l=1 z2= Bn
j@j @l @m K (y z ) (z )uj (z )ul (z )j dz:
For jx yj 2 (n+1) and jx z j 2 n , we have j@j @l @m K (y z )j C jy z j 4 ~ r1;n (; :)kL1 (Bn+1 ) C Rjx zj2 n j~ujx(;zzj)4j2 j (z )j dz . 24 C jx z j 4 . Thus, kr We may choose to be nonnegative. As in the proof of Lemma 30.2, we use the local energy inequality for the nonnegative test function (; y) = (y)!s; (), and then we let go to 0; we get for 1=2 < s < 1: R RR j~u(s; y)j2 (y) dy + 1s jr~ ~u(; y)j2 (y) dy d RR
2
j~u(; y)j2 (y) dy d+ s 0 2 0 j~u(; y)j (y) dy d RRs ~ g (y) dy d + 0 (j~u(; y)j2 + 2p(; y))f~u(; y):r 2 =3 C1 1 + C2 1 : R 2=3 Since 1 < 1, we may write j~u(s; y)j2 (y) dy R C1 . By the induction hypothesis, we know that if 2 k n we have jx zj2[2 k ;2 k ] j~u(; z )j2 dz D1 21=3 2 3k for 2 (s 4 k ; s); thus, we have 3=2 R R j~u(;z)j d L1;r;n+1 C 2(n+1)=2 ss 4 n j x zj2 n jx zj j (z )j dz Rs 3=2 Pn 1 R j ~ u ( ;z ) j 0 ( n +1) = 2 C2 d k=0 2 k jx zj2 k jx zj j (z )jdz s 4 n R P 3 = 2 s n 1 C 00 1 2(n+1)=2 s 4 Rn 1 + k=2 D1 2k d C 000 D13=2 1 22(n+1) ss 4 n d = C 000 D13=2 1 : 2kkR1R
1 2
1
2
4
( +1)
2
( +1)
4
1
( +1)
( +1)
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Energy inequalities
We nd that Ln+1 C1 (D13=2 + D13 1 +2(n+1)=2) D2 21=3 2(n+1)=2 for 1 small enough [Constraint 5 : C (D13=2 11=3 + D13 41=3 + 11=3 ) D2 ] Remark: We see that the discrepancy between the local scaling of j~uj2 and the local scaling of jp px;k j is due to the non-local character of the pressure: the knowledge of p is not given by the local knowledge of ~u, and the ill-controlled part in p is the term p2 .
Step 4: (Bk ) true for 3 k n implies (An ) We use the fundamental inequality given in Lemma 2 and nd that In + Jn Un + Vn + Wn with RR Un = D0 0<<1;jyj<1 j~u(; y)j3 d dy 2=3 RR s Vn = e1=4 (8)3=2 0RR j~u(; y)j2f~u(; y):r~ g('x (y) x;s;n(; y)) dy d > : s 1 = 4 3 = 2 ~ g('x (y) x;s;n (; y)) dy d Wn = 2e (8) 0 p(; y)f~u(; y):r 8 > <
We prove that max(Un ; Vn ; Wn ) D1 2=3 =3. Indeed, we have Un D0 2=3 D1 2=3 =3 if D1 is big enough [Constraint 6 : 3D0 D1 ]. In order to estimate ~ ('x (y) x;s;n (; y)). We have Vn and Wn , we need to control the size of r j@j 'x (y)j j x;s;n (; y)j C 1jx yj1=4. On the other hand,
j'x (y)j j@j
x;s;n (; y )j C
4
n+s
1 : + jx yj2 2
This gives control on Vn :
Vn
RR
C (0;1)B(0;1) j~u(; y)j3 4 n +s 1+jx yj dy d RR C 0 (0;1)B(0;1) Q j~u(; y)j3 dy d RR RR P + nk=31 24k Qk Qk j~u(; y)j3 dy d + 24n Qn j~u(; y)j3 dy d RR C 00 (0;1)B(0;1) j~u(; y)j3 dy d + Pnk=3 2 k Kk 2
2
3
+1
Thus, Vn C (1 + D2 21=3 ) D1 21=3 =3 if D1 is big enough [Constraint 7 : 3C (1 + D2 ) D1 ]. The control on Wn is obtained in a similar way: we let be a smooth function on IR IR3 equal to 1 on (0; 1=4) B (0; 1=2) and equal to 0 outside from (0; 1) B (0; 1) and wePde ne x;s;k (; y) = (4k (s ); 2k (x y)). We write 1 = 1 x;s;3 (; y) + nk=31 x;s;k (; y) x;s;k+1 (; y) + x;s;n (; y) and
~ g ( x;s;k (; y) x;s;k+1 (; y))'x (y) x;s;n (; y) dy d 0 p(; y)f~u(; y):r RR ~g = Qk (p(; y) px;k ())f~u(; y):r ( x;s;k (; yRR ) x;s;k+1 (; y))'x (y) x;s;n (; y) dy d C 24k Qk jp(; y) px;k ()j j~u(; y)j dy d:
RR s
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329
This gives
Wn
RR
C (0;1)B(0;1) jp(; y)jj~u(; y)j RR Pn + k=3 C 24k Qk jp(; y) px;k ()j j~u(; y)j dy d P C kpkL3=2((0;1)B(0;1)) k~ukL3((0;1)B(0;1)) + nk=3 C 2 k Kk1=3 L2k=3 P C1 + nk=3 CD2 21=3 2 2k=3 :
Thus, Wn C (1 + D2 21=3 ) D1 21=3 =3 if D1 is big enough [Constraint 8 : 3C (1 + D2 ) D1 ]. Thus, Theorem 30.1 is proved.
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Chapter 31 On the dimension of the set of singular points The main result of Ca arelli, Kohn, and Nirenberg [CAFKN 82] is their estimate on the Hausdor dimension of the set of singular points of a suitable solution (generalizing results of Sche er [SCH 77]).
1. Singular times We begin by recalling the de nition of the Hausdor measure on IRd:
De nition 31.1: (Hausdor measure.) i) For a sequence balls B = (B (xi ; ri ))i2IN of IRd and for > 0, P of open we de ne (B) = i2IN ri . The Hausdor measure H on IRd is de ned for a Borel subset B IRd by
H (B ) = Ælim minf (B) = B = (B (xi ; ri ))i2IN ; B [i2IN B (xi ; ri ); sup ri < Æg !0 i2IN
ii) The Hausdor dimension dH (B ) of a Borel subset of IRd is de ned as
dH (B ) = inf f > 0 = H (B ) = 0g = supf > 0 = H (B ) = 1g: A classical result (which goes back to the description of the structure of turbulent solutions by Leray [LER 34]) states that the set of singular times for a restricted Leray solution is small:
Proposition 31.1: (Singular times) Let ~u be a weak Leray solution of the Navier{Stokes equations on (0; 1) IR3 which satis es the strong energy inequality. Then there is compact set t [0; 1) so that: i) ~u is smooth outside from t IR3 ii) H1=2 (t ) = 0. Proof: We proved in Chapter 15 that, for an initial data in (Lp (IR3 ))3 with p > 3, we are able to nd a solution of the Navier{Stokes equations in (C ([0; T ]; Lp))3 331 © 2002 by CRC Press LLC
332
Energy inequalities p
2
with T C (p)k~u0kp p 3 . If p = 3 and if k~u0 k3 is small enough (k~u0 k3 < 0 ), we may nd a global solution in (C ([0; T ]; L3))3 . If ~u isR a weak Leray solution that full lls the strong energy equality, we have that 01 k~u(t; :)k43 dt < 1; thus, there is a subset of positive measure of (0; 1) on which k~u(t; :)k3 is smaller than 0 . If t0 is a Lebesgue point of the map t 7! k~u(t; :)k22 so that k~u(t0 ; :)k3 is smaller than 0 , the theorems of Kato (existence of a global solution) and Serrin (uniqueness in the class of Leray weak solutions) show that ~u on [t0 ; 1) is given by the Kato solution associated with the initial value ~u(t0 ; :). Since this mild solution is smooth, we nd that t [0; t0 ]; thus, t is compact. Now, let us consider 2 t and let s < be a Lebesgue point of the map t 7! k~u(t; :)k22 . Let = k~u(s; :)k6 . If < 1, we know that there exists a mild solution ~v of the Navier{Stokes equations in (C ([s; s + T ]; L6))3 with T C 4 , with initial value ~u(s). Since ~u is a Leray solution on (s; s + T ), we nd that ~u = ~v on (s; s + T ) (due to the Serrin uniqueness theorem). Since ~v is smooth on (s; s + T ) IR3 , we nd that s + T < . Thus, k~u(s; :)k6 C ( s) 1=4 . If I = ( r; + r) is an open interval centered on 2 t andRif > r, we have for almost every s 2 [ r; ] k~u(s; :)k6 C ( s) 1=4 ; hence, I k~uk26 ds Cr1=2 . Now, let Æ > 0 and let us write t \ [Æ; 1) [ 2t ; Æ ( Æ=5; + Æ=5). The Vitali covering lemma (Proposition 6.1) gives us that we may nd N 2 IN and 1 ; : : : ; N 2 t \ [Æ; 1) so that t \ [Æ; 1) [N i=1 ( i Æ; i + Æ ) while min1i<jN j i j j 2Æ=5. We thus nd that for B =P(( i R Æ; i + Æ))0iN i 2 with 0 = 0, we have 1=2 (B) = (N +1)Æ1=2 Æ1=2 + C N i=1 i 2Æ=5 k~uk6 ds R 1 = 2 2 2 6 3 Æ + C d(s;t )2Æ=5 k~uk6 ds. Since we know that ~u 2 (L L ) , we nd that R H1=2 (t ) C t k~uk26 ds. In particular, H1=2 (t ) < 1; hence, the Lebesgue R measure of t is equal to 0; this gives t k~uk26 ds = 0 and nally H1=2 (t ) = 0.
2. Hausdor dimension of the set of singularities for a suitable solution To more accurately describe the singularities in IR IR3 , Ca arelli, Kohn, and Nirenberg used another Hausdor dimension, adapted to the scaling properties of the Navier{Stokes equations:
De nition 31.2: (Parabolic Hausdor measure) i) For a sequence of open parabolic cylinders Q = (Q((ti ; xi ); ri ))i2IN of IR IRd (where Q((ti ; xi ); ri ) =Pf(t; x) = jt ti j ri2 and jx xi j ri g) and for > 0, we de ne ~ (Q) = i2IN ri . The parabolic Hausdor measure P on IR IRd is de ned for a Borel subset B IR IRd by
P (B ) = Ælim minf~ (Q) =B [i2IN Q((ti ; xi ); ri ); sup ri < Æg !0 i2IN
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333
ii) The parabolic Hausdor dimension dH (B ) of a Borel subset of IR IRd is de ned as
dP (B ) = inf f = P (B ) = 0g = supf = P (B ) = 1g: We may now state the result of Ca arelli, Kohn, and Nirenberg:
Theorem 31.1: (Dimension of the singular set) Let ~u be a weak solution for the Navier{Stokes equations on (0; T ) IR3 , which is a suitable solution on the cylinder Q0 = (a; b) B (x0 ; r0 ). Let be the smallest closed set in Q so that ~u is locally bounded on Q0 . Then P 1 () = 0. The proof is based on the following proposition (which we shall prove in the next section):
Proposition 31.2: (The second regularity criterion of Ca arelli, Kohn, and Nirenberg) There exists a positive constant 0 so that if ~u is a weak solution for the Navier{Stokes equations on (0; T ) IR3 , if x0 2 IR3 and 0 < t0 , if ~u is a suitable solution on a neighbourhood of (t0 ; x0 ) and if
then we have
(31:2)
Z Z
lim sup r 1 r!0
(31:1)
lim sup r 2 r!0
jx x0 j
r <s
7 2 8
Z Z
jx x0j
r <s
7 2 8
jr~ ~uj2 dx ds < 0
j~uj3 + jpj3=2 dx ds < CKN
where CKN is the constant in the regularity criterion of Ca arelli, Kohn, and Nirenberg (Theorem 30.1). In particular, ~u is bounded on a cylinder Q((t0 + r2 =8; x0 ); r=2) = (t0 r2 =8; t0 + r2 =8) B (x0 ; r=2).
Before proving Proposition 31.2 in the next section, we prove that Theorem 31.1 is a consequence of Proposition 31.2.
Proof of Theorem 1: Let Æ > 0. Let (t; x)R2R . According to Proposition 31.2, we know that for ~ ~uj2 dy ds 0 r. Thus, we r small enough, we have jy xj
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Energy inequalities
we may apply the Vitali covering lemma and nd a countable subcollection Q[Æ] = (Q((ti ; xi ); ri ))i2IN of QÆ so that [i2IN Q((ti ; xi ); ri ) and i 6= j ) Qi \ Qj = ;. We then have:
~1 (Q[Æ] ) = 5
X
i2IN
ri
5 0
Z Z
(t;x)2Q ;dP ((t;x);)Æ 0
jr~ ~uj2 dx dt:
RR ~ ~uj2 dx dt. In particuler P 1 () < 1; hence, the This gives P 1 () 50 jr Lebesgue measure of is equal to 0 and nally P 1 () = 0.
3. The second regularity criterion of Ca arelli, Kohn, and Nirenberg The proof of Proposition 31.2 runs along the same lines as for the the regularity criterion of Ca arelli, Kohn, and Nirenberg (Theorem 30.1). We introduce the same normalized quantities on a cylinder Q((t; x); r) = (t r2 ; t) B (x; r): R 8 IQ = sups r2 <<s r 3 B(x;r) j~u(s; y)j2 dy > > RR > < ~ ~u(; y)j2 dy d JQ = r 3 QRRjr > KQ = r 5 Q j~u(; y)j3 dy d > > RR : LQ = r 5 Q jp(; y)j3=2 dy d The hypothesis is that lim sup r 1 r!0
Z Z
jx x0j
r <s
7 2 8
jr~ ~uj2 dx ds < 0
which we may rewrite as lim sup r2 JQ((t0 + r2 ;x0 );r) < 0 ; 8 r!0 and we want to prove that lim sup r3 (KQ((t0 + r82 ;x0);r) + LQ((t0 + r82 ;x0 );r)) < CKN : r!0 We write more concisely: Br = B (x0 ; r), Qr = Q((t0 + r8 ; x0 ); r), I (r) = r2 IQr , J (r) = r2 JQr , K (r) = r3 KQr and L(r) = r3 LQr . Thus, we attempt to prove that for a weak solution suitable in the neighborhood of (t0 ; x0 ), we have 2
lim sup J (r) < 0 ) lim sup K (r) + L(r) < CKN : r!0 r!0 As for Theorem 30.1, we shall try to estimate K (r) and L(r) from I () and J () with 2r and to estimate I (r) from K () and L() with 2r, but we have no need to estimate J (r), since its size is controlled by inequality (31.1).
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335
Step 1: Sobolev inequalities in the ball Lemma 31.1: R Let M f = jB1 j f (y) dy. Then: i) For 1 p 1 and f 2 Lp (B ), kf M f kLp(B ) 2kf kLp(B ) . ~ f kLp(B ) . ii) For 1 p 1 and f 2 Lp (B ), kf M f kLp(B ) Cp kr 3 p iii) For 1 < p < 3, f 2 W 1;p (B ) and r = 3 p , kf M f kLr (B7=8 ) ~ f kLp(B ) . Cp kr Proof: For i) and ii), we write
kf M f kpLp(B ) jB1 j
Z Z
jf (y) f (z )jp dy dz:
B B 2p (jf (y)jp + Rjf (z )jp ); t ~ p
We have jf (y) f (z )jp arrive at ii), we write jf (y) f (z )j
thus i) is obvious. In order to
0 jrf (ty + (1 t)z )jp jy z jp dt; thus, RR R kf M f kpLp(B ) j2Bpj B B 01=2 jr~ f (ty + (1 t)z )jp dy dz dt RR R j2Bpj B B 01=2 (1 t) 3 jr~ f (w)jp dy dw dt p ~ f kpLp(B ) : = 4 kr To prove iii), we select ! 2 D with ! = 1 on B (0; 7=8) and with Supp !
B (0; 1) and we de ne ! (y) = !(y=); then, conforming to the Sobolev inequalities, kf M f kLr (B7=8 ) k! (f M f )kW 1;p C (kr~ f kLp(B ) + 1 kf M f kLp(B ) ) C 0 kr~ f kLp(B ) : Step 2: Estimating K (r) RR The control of K (r) = r 2 Qr j~u(; y)j3 dy d is not diÆcult. We choose ! 2 D with ! = 1 on B (0; 5=4) and with Supp ! B (0; 7=4) and we write that k~u(; :)kL3 (Br ) C k!( x r y )~u(; :)kH_ 1=2 C 0 k!( x r y )~u(; :)k12=2 k!( x r y )~u(; :)k1H_=12 C 00 k~u(; :)k1L=22(B2r ) (r 1 k~u(; :)kL2 (B2r ) + kr~ ~u(; :)kL2 (B2r ) )1=2 C 000 1=4 I ()1=4 kr~ ~u(; :)k1L=22(B ) + r 1=2 k~u(; :)kL2 (B2r ) : ~ ~u in the last term of the estimate: We want to introduce the estimate on r R R R (; y)j2 dy = jB1 j y2B2r z2B j~u(; y)j2 j~u(; z )j2 + j~u(; z )j2 dy dz B2r j~uRR RR jB1 j ( B B j~u(; y) + ~u(; z )j2 Rdy dz ) 21 ( B B j~u(; y) ~u(; z )j2 dy dz ) 21 +( r )3 B j~u(; z )j2 dz
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Energy inequalities
~ ~u(; y + t(z y))j2 jy z j2 dt, hence We have j~u(; y) ~u(; z )j2 01 jr RR R1 2 2 2 RR 2 ~ BB j~u(; y ) ~u(; z )j dy dz 8 B B 0 jr ~u(; y + t(z y ))j dy dz dt 1 R RR R 82 BB 02 jr~ ~u(; w)j2 (1 t) 3 dw dz dt = 122jB j B jr~ ~u(; w)j2 dw: We thus obtain the estimate R
k~u(; :)kL (Br ) C
r
3
1=4 1=4 ~ r I () kr ~u(; :)k1L=22(B ) + I ()1=2 r
and nally Z t0 + r2 8
(31:3) K (r) = r 2
r
t0
7 2 8
k~u(; :)k3L (Br ) d C ( r )3 I () J () 3 4
3
Step 3: Estimating L(r) We repeat
d X d X
p =
j =1 l=1
3 4
r 3 +( )3 I () 2
@j @l (uj ul ):
Since ~u is divergence free, we may write as well d X d X
p =
j =1 l=1
@j @l uj (ul
M ul ) :
We then introduce (z ) = !(2(z x0 )=), where ! 2 D with ! = 1 on B (0; 5=4) and with Supp ! B (0; 7=4) and we write ( (z )p(; z )) = (z )p(; z )) + ~ :(p(; z )r ~ (z )) p(; z ) (z ). We again call K the kernel of 1 (K (x) = 2r 3 ), obtaining p = p1 + p2 with 4jxj
p1 (; y) =
d X d X j =1 l=1
and
p2 (; y) = 2
d X j =1
K @j @l uj (ul
M ul )
@j K (p@j ) K (p ):
For y 2 Br and r =2 and for z in the support of @ R(with 6= 0), we have jz yj 2 [=8; 11=8]. This gives jp2 (; y)j C 3 B jp(; z )j dz R C 0 3 B jp(; z )j3=2 dz 2=3 and thus
r 2
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Z Z
Qr
jp2 (; y)j3=2 dy d C r L():
Singular points
337
In order to estimate p1 , we write p1 = p3 + p4 with
p3 (; y) = and
p4 (; y) =
We have
d X d X j =1 l=1
@j @l K uj (ul
M ul )
Pd
Pd j =1 l=1 K (@j @l )uj (ul Pd Pd + j=1 l=1 @j K (@l )uj (ul P P + dj=1 dl=1 @l K (@j )uj (ul
M ul ) M ul ) M ul ) :
kp4 (; :)kL1 (Br ) C 3 k~ukL (B ) k~u M~ukL (B ) C 0 I () kr~ ~ukL (B ) ; ~ ~ukL (B ) C1=2 I ()1=2 kr ~
hence, kp4(; :)kL = (Br ) C r= I ()1=2 kr ~ukL (B ) . On the other hand, @j@l = Rj Rk , and we have kp3 (; :)kL (Br ) C k j~uj j~u M~uj kL (B ) C k~ukL (B ) k~u M~ukL (B ) 2
2
3 2
3 2
2
1 2
2
2
3 2
2
3 2
3 2
7 8
2
7 8
6
7 8
~ ~ukL2 (B ) . Thus, we get hence, kp3(; :)kL3=2 (Br ) C1=2 I ()1=2 kr r 2
Z Z
Qr
jp1 (; y)j3=2 dy d C ( r )3=2 I ()3=4 J ()3=4 :
We obtain the estimate: (31:4)
r r L(r) C ( L() + ( )3=2 I ()3=4 J ()3=4 )
Step 4: Estimating I (r) To estimate I (r), we use the local energy inequality. We recall that ~u is suitable on a cylinder [t0 7R2=8; t0 + R2 =8] B (x0; R). We x R independent from and r. We apply the local energy inequality to a test function !s; de ned as follows. We de ne (; y) = (( t0 )=2 ) ((y x0 )=) with 2 D(IR) with = 1 on [ 7=30; 1=30] and Supp [ 7=8; 1=8] and 2 D(IR3 ) with = 1 on B (0; 5=8) and Supp B (0; 7=8). Thus, = 1 on a neighborhood of Qr when r =2. R Now, we x 2 D(IR), Supp [0; 1], 0 and dt = 1. For s 2 [t0 7rR2 =8; t0 + r2 =8], r =2 ( < R) we introduce for > 0 the function !s; () = 1 R82 ( R82 ( t0 + 3R2 =4)) 1 ( s ) d . If is small enough 2 ( < 13 and bounded by 1. The local energy inequality 32 ), !s; is nonnegative gives then for s 2 [t0 7r2 =8; t0 + r2 =8]: RR jr~ ~u(; y)j2 !s; () (; y) dy d RR j~u(; y)j2 f@ + y g(!s; () (; y)) dy d RR ~ g(!s;() (; y)) dy d + (j~u(; y)j2 + 2p(; y))f~u(; y):r
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Energy inequalities R
We let go to 0; then, for all Lebesgue point s of s 7! B j~u(s; y)j2 dy, we get that R RR j~u(s; y)j2 (s; y) dy + ts 72 =8 jr~ ~u(; y)j2 (; y) dy d 0
2 RR C 4 tt00 752 ==88 j~u(; y)j2 (; y) dy d+ RRs 2 t0 72 =8 j~u(; y )j f@ + y g( (; y )) dy d RRs ~ g( (; y)) dy d + t0 72 =8 (j~u(; y)j2 + 2p(; y))f~u(; y):r Since ~u is weakly continuous from [t0 7r2 =8; t0 + r2 =8] into (L2 ((B )3 , we nd that this equality is valid for all s 2 [t0 7r2 =8; t0 + r2 =8]. Moreover, we have
Z
j~u(; y)j2 f~u(; y):r~ g( (; y)) dy =
Z
(j~u(; y)j2
jM ~uj2 ))f~u(; y):r~ g dy
since ~u is divergence free. Thus, we nd rI (r) C (A() + B () + C () with
RR A() = C 2 [t0 72 =8;t0 +2 =8]B7=8 j~u(; y)j2 dy d RR B () = 1 [t0 72 =8;t0 +2 =8]B7=8 j j~u(; y)j2 jM~uj2 j j~u(; y)j dy d RR > : C () = 1 [t0 72 =8;t0 +2 =8]B7=8 jp(; y)j j~u(; y)j dy d: We have A() CK ()2=3 . We estimate B () by writing j~uj2 jM~uj2 = (~u + M~u):(~u + M~u); hence, R 2 jM~uj2 j j~u(; y)j dy B7=8 j j~u(; y )j k~u + M~ukL2 (B ) k~u M~ukL5(B7=8 ) k~ukL3(B ) C k~ukL2 (B ) kr~ ~ukL2(B ) k~ukL3(B ) and thus B () CI ()1=2 J ()1=2 K ()1=3 . Finally, C () K ()1=3 L()2=3 . We obtain the estimate (31:5) I (r) C (K ()2=3 + I ()1=2 J ()1=2 K ()1=3 + K ()1=3 L()2=3 ) r 8 > <
Step 5: The recursion formula Putting together estimates (31.3), (31.4), and (31.5), we obtain for r 2 =2 =4 (and writing I () I ( ), J () J ( ), K () 2 K ( ) and 2 L() 2 L( )) 8 > > > > > > <
I (r)
C r (K ()2=3 + I ()1=2 J ()1=2 K ()1=3 + K ()1=3 L()2=3 ) = I ( )1=2 J ( )1=2 K ( )1=3 C 0 ( r I ( )1=2 J ( )1=2 + r I ( ) + r = 2
3
2
> > > > K (r) > > :
L(r)
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11 6 5 6
4=3 5=3 + r 2=3 I ( )1=4 J ( )1=4 L( )2=3 + r 1=3 I ( )1=2 L( )2=3 ) C ( r )3 I ( ) 34 J ( ) 34 + ( r )3 I ( ) 32 C ( r L( ) + ( r )3=2 I ( )3=4 J ( )3=4 ):
Singular points
339
Let M (r) = I (r) + K (r)2=3 + L(r)4=3 . We assume that is small enough to grant that J ( ) < 0. We nd that M (r) (r; ; )M ( )+ (r; ; )M ( )1=2 +
(r; ; )M ( )3=4 with 8 > > < (r; ; )
3 11=6 1=2 + 4=3 + r2 + r4=3 + 2 ) = C1 ( r 2 + r 5=6 0 r 1=3 2 4=3 r2 0 2 1=2 2 1=2 ) + ( r; ; ) = C ( 2 0 r r2 0 > > 5=3 1=4 : (r; ; ) = C3 r2=3 0 :
We want to prove that, for r small enough, K (r) + L(r) < CKN . We may =3 assume that CKN < 1. It is then enough to show that M (r) < 2 1=3 4CKN 1 = 4 3 = 4 (this will give K (r) < 1, hence K (r) + L(r) 2 M (r) < CKN ). We thus =3 assume that M ( ) 2 1=3 4CKN and we write
2=3 + (r; ; )21=12 1=3 )M ( ) M (r) ( (r; ; ) + (r; ; )21=6 CKN CKN We then choose to x the ratio =r = 1=2 and = = with small enough to grant that 2 4=3 C1 (22 + 21=3 + + 4=3 ) < 1=4 4 2 and then we require 0 < 1 to be small enough to grant that
C1 (
2 1 1 2=3 + C 2 21=12 1=3 1=4 < 1=4: + 2 ) + C2 ( + 2 )21=6 CKN 3 5=3 CKN 0 11 = 6 4 4 2
Thus, starting from an initial r0 so that ~u is suitable on Qr0 and so that J (r) < =3 or 0 for r r0 and de ning rn = (=2)n r0 , we nd that M (rn ) < 2 1=34CKN =3 1 1 = 3 M (rn+1 ) < 2 M (rn ). Thus, we may nd some n 2 IN with M (rn ) < 2 4CKN and Proposition 31.2 is proved.
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Chapter 32 Local existence (in time) of suitable local square-integrable weak solutions In this chapter, we attempt to prove how the local energy inequality for suitable solutions may be turned into a tool for proving the existence of solutions for the Navier{Stokes initial value problem for a locally square-integrable initial data, just as the Leray energy inequality was a tool for proving existence of weak solutions for a square-integrable initial data. We begin by de ning the local Leray solutions :
De nition 32.1: (local Leray solutions) ~ :~u0 = 0. A local Leray solution ~u on Let ~u0 2 (L2uloc (IR3 ))3 so that r (0; T ) IR3 for the Navier{Stokes initial value problem associated with ~u0 is a weak solution for the Navier{Stokes equations Ron (0; T ) IR3 so that: ) ~u 2 \t
) for all compact subset K of IR3 , limt!0+ K j~u ~u0 j2 dx = 0 Æ) ~u is suitable in the sense of Ca arelli, Kohn, and Nirenberg. We are going to prove in this chapter an existence theorem for local Leray solutions:
Theorem 32.1: (locally square-integrable initial value) ~ :~u0 = 0, there exists a positive real a) For all ~u0 2 (L2uloc (IR3 ))3 so that r number T and a local Leray solution ~u on (0; T ) IR3 for the Navier{Stokes initial value problem associated with ~u0 . b) If ~u0 belongs more precisely to E23 , where E2 is the closure of D(IR3 ) in 2 Luloc , then ~u 2 \t
8 u < @t ~ :
~ :((~u ! ) ~u ) IPr r~ :~u = 0 ~u(0; :) = ~u0
= ~u
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Energy inequalities
which we have introduced in Chapter 13. (We give here a proof slightly di erent from the proof we set forth in [LEM 98b]).
1. Size estimates for ~u In this section, we prove that, at least for small times, we may nd an estimate for the L2uloc norm of the solution ~u of (32:1), which depends only on the L2uloc norm of the initial value (and is independent from ):
Proposition 32.1: ~ :~u0 = 0. De ne 0 = k~u0kL2 Let ~u0 2 (L2uloc (IR3 ))3 be such that r uloc and 1 = min(1; 0 ). Then, there exists a positive constant C0 (which does not depend on ~u nor on ) so that the equations (32:1) have a solution ~u on 2 (0; T0) IR3 with T0 = min(1; 20 C1 04 ) and so that for all 0 < t < T0 we have (32:2)
p
k~u(t; :)kLuloc C0 k~u0 kLuloc 2
2
1
20 C04 1=4 t 21
Proof: We saw in Chapter 13 that equation (32:1) has a solution ~u de ned on (0; T) IR3 for some maximal T > 0 so that: i) ~u 2 \T
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Now, we introduce the functions (t) = sup'2B k~u(t; :)'(x)k22 , (t) = R ~ ~u(s; :))'(x)k22 ds. ~ ~u(s; :))'(x)k22 ds and ; (t) = sup'2B Rt k(r sup'2B 0t k(r We estimate and with the help of a series of technical lemmas.
Lemma 32.1: For 0 < < T < T, we have sup
1
and
2 (L1 ([; T ] IR3 ))3
and
In the next lemma, we estimate the size of the pressure p associated with
Lemma 32.2: For all ' 2 B (' = '0 (x x0 ) for some x0 2 IR3 ), there exists a function p;'(t) so that for all interval I = (T0 ; T1) with 0 < T0 < T1 < T: 2=3 3=2 I IR3 jp (t; x) p;' (t)j '(x) dx dt k~uk2L3(I;L2uloc) + k ~ukL6(I;L2 ) k ~ukL2(I;H 1 )
RR
C
where C does not depend neither on ' nor on T0 ; T1 nor on .
Proof: In order toPprove this P estimate, we recall that p is computed, at least formally, as p = 1j3 1k3 Rj Rk (u;j !)u;k , where Rj = p@j is the j th Riesz transform and u;j is theRj th coordinate of the vector ~u. Writing more compactly p = G :(~u ! ) ~u = G(x; y):(~u !)(t; y) ~u (y) dy, we thus compute p p;'(t) through the formula: p (t; x) p;'(t) = p1 (t; x) + p2 (t; x) with
p1 (t; x) = G 02 (y x0 )(~u !)(t; y) ~u(t; y) (x) p2 (t; x) = (G(x; y) G(x0 ; y)):(1 02 (y x0 ))(~u !)(t; y) ~u(t; y) dy R
It is easy to check that, on the support of '(x) = '0 (x x0 ), p2 (t; x) is bounded by C k~u(t; :)k2L2uloc . On the other hand, kp1 (x; t)kL3=2 ([T0 ;T1 ]IR3 ) C k 2 ((~u !) ~u)kL3=2 (G being a matrix of Calderon{Zygmund operators). But H 1 (IR3 ) L6 and thus we have L6 (L2 ) \ L2(H 1 ) L3 (L3 ). This nally gives the following control on p1 :
kp1 (x; t)kL = ([T ;T ]IR ) C k ~ukL ([T ;T ];L ) k ~ukL ([T ;T ];H ) : 3 2
0
1
3
6
0
1
2
We now prove the key local energy estimate for ~u.
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2
0
1
1
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Energy inequalities
Lemma 32.3: equality:
For all t
2 (; T )
2B
and all '
, we have the following
R
(32:3)
k~u(t; :)'(x)k22 + 2 t RR k(r~ ~u (s; :))'(x)k22 ds = k~u(; :)'(x)k22 + t j~uRRj2 ('2 (x)) dx ds RR t ~ '2 (x) dx ds +2 t (p p;')(~u :r ~ )'2 (x) dx ds + j~u j2 (~u !):r
Proof: Since ~u is smooth on (0; T) IR3 , we have ~ ~uj2 = j~u j2 @t j~uj2 + 2jr
r~ :(j~uj2 (u~ !) + 2p ~u)
and equality (32.3) follows by integration, since the divergence free condition ~ :(p ~u ) = ~u:r ~ p = ~u:r ~ (p p; '). on ~u gives that r
Lemma 32.4: inequality:
For all t Z Z t
2 (; T )
and all '
j~uj2 ('2 (x)) dx ds C1
2 B, Z t
we have the following
(s) ds
Proof: This is obvious. Lemma 32.5: inequality: ZZ t
For all t
2 (; T )
and all '
Z t
j~u j2 (~u ! ):r~ '2 (x) dx ds C2 (
2 B,
we have the following
(s)3 ds)1=4 ; (t)+
Z t
(s) ds 3=4
Proof: Let 0 be equal to 1 on the support of '0 and let us write, for '(x) = '0 (x x0 ) and (x) = 0 (x x0 ), ZZ t
j~u j2 (~u !):r~ '2 (x) dx ds C
Z Z t
j ~uj2 j ~u !j dx ds
We use the inclusions L2 (L6 ) \ L6 (L2 ) L3(L3 ) and H 1 (IR3 ) L6 to get ZZ t
j ~vj3 dx ds C
Z t
Z t
k ~vk62 ds 1=4 k ~vk2H ds 3=4 0 0
1
and we R must check 2the following four2 inequalities: ~ 0 (x x0 )j ) j~u(t; x)j2 dx C (t) (j (x x0 )j + jr RRt 0 2 ~ 2 j 0 (x x0 )j jr ~u (s; x)j dx ds C ; (t)
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345
R
~ 0 (x x0 )j2 ) j~u !(t; x)j2 dx C (t) (j 0 (x x0 )j2 + jr 2 2 ~ j 0 (x x0 )j jr (~u ! )(s; x)j dx ds C ; (t). The rst two inequalities are obvious, and the third inequality is easy, since L2uloc is a shift-invariant space of distributions, hence is stable through convolution with functions in L1 . RFor the last inequality, we notice that ;R(t) is equivalent ~ ~uj2 ds; now, recall that ! 0 and ! dx = 1, this to the L1uloc norm of t jr Rt ~ (~u ! )j2 ds Rt jr ~ ~uj2 ds ! , and we may conclude gives that jr 1 since Luloc is stable through convolution with functions in L1 . RRt
Lemma 32.6: inequality: ZZ t
For all t
2 (; T )
and all ' Z t
~ )'2 (x) dx ds C3 ( (p p;')(~u :r
2 B,
we have the following
(s)3 ds)1=4 ( ; (t)+
Z t
(s) ds)3=4
Proof: We use the results of Lemma 32.2 to get that Z Z t
C
ZZ t
(p
~ )'20 (x x0 ) dx ds p;')(~u :r
jp (t; x) p;' (s)j3=2 !0 (x x0 ) dx dt 2=3
ZZ t
j~uj3 !0 (x x0 ) dx dt 1=3
C 0 k~uk2L3 ([;t];L2uloc) + k 0 (x x0 )~u k2L3([;t]IR3 ) k 0(x x0 )~ukL3 ([;t]IR3 )
and we conclude since we know (as already used in the proof of Lemma 32.5) that we have
k
Z t
0(x x0 )~u kL3([;t]IR3 ) C 00 (
(s)3 ds)1=12 ( ; (t) +
Z t
(s) ds)1=4
and Z t
k~ukL ([;t];Luloc)k~uk1L=2([;t];Luloc)k~uk1L=2([;t];Luloc ) C 000( ds) 3
2
2
2
2
6
1 4
Z t
( 3 ds) 12 1
End of the proof: Now, clearly, these lemmas give us the following inequalities for tR 2 (; T) (just by looking at the suprema at time t of k~u(t; :)'(x)k22 ~ ~u (s; :))'(x)k22 ds): and of 2 t k(r (t) ()+C1
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Z t
Z t
Z t
(s) ds+(C2 +2C3 )( (s)3 ds) 4 ( ; (t)+ (s) ds) 4
1
3
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Energy inequalities Z t
Z t
Z t
2 ; (t) ()+C1 (s) ds+(C2 +2C3 )( (s)3 ds) ( ; (t)+ (s) ds)
1 4
Rt
Rt
3 4
We then use the Young inequality ( (s)3 ds) ( ; (t) + (s) ds) 34 Rt 3Æ4=3 1 Rt 3 4=3 4Æ4 (s) ds + 4 ( ; (t) + (s) ds) with 3=4 (C2 + 2C3 ) Æ < 1 to eliminate ; (t) and we obtain
(t) C4 ( () +
Z t
1 4
(s) ds +
Z t
(s)3 ds)
We then let go to 0. First, we notice that k~v !(t; :)k1 C k~v(t; :)kB13=2;1 , hence that
k
~ :(~v !) ~v dskL2 C pt sup k~v( ; :)k IPe(t s) r uloc
Z t
0
sup k~v( ; :)kL2uloc
B12 ;1 0<
0<
3
Thus, the Picard iteration scheme provides a solution not only in the space (L1 ((0; T0 ()); B13=2;1 )3 ) for some positive T0 (), but more precisely in the space (L1 ((0; T1 ()); B13=2;1 )3 ) \ (L1 ((0; T1 ()); L2uloc )3 ) for some positive T1 (). We then write:
p
p
(t) C k~u(t; :)kL2uloc C ket~u0 kL2uloc + C (~u ) t C 0 + C (~u ) t Thus, we get that (t) is bounded in the neighborhood of t = 0 (with a bound depending on ) and that
(t) C5 ( 0 +
Z t
0
(s) ds +
Z t
0
(s)3 ds):
We then write (s) 0 + (s20) and get for t < 1: 3
(t) C0 ( 0 +
(s)3 ds) 0 21
Z t
Thus, the function (t) = 0 + 0t (s21) ds is a solution of the di erential inequality 0 C03 3 = 21 . Thus, we obtain 1= 20 1= (t)2 C03 t= 21 , so that (t) C0 (1= 20 C03 t= 21 ) 1=2 and this completes the proof of Proposition 32.1. R
3
2. Local existence of solutions We may now begin the proof of Theorem 32.1. In this section, we prove the existence of a suitable weak solution in (0; T ) IR3 for some positive T .
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Local existence
347
~ :~u0 = 0. We proved the We consider an initial value ~u0 2 (L2uloc )3 with r following important estimates on the solutions ~u of the molli ed equations: sup k~u(t; :)'(x)k22 C0 (1= 20
'2B
sup
'2B
Z t
0
C03 t= 21 ) 1=2
k(r~ ~u(s; :))'(x)k22 ds C6 (1= 20 C03 t= 21 ) 1=2
which are valid on (0; T0). Then, we may use the limiting process described in Theorem 13.2 and get that a subsequence ~un is convergent to a solution ~u of the Navier{Stokes equations associated to ~u0. More precisely, when n converges to 0, we have for all 2 D((0; T0 ) IR3 ) strong convergence of ~u in Lp ((0; T0); (L2 )3 ) for all p < 1 and weak convergence in L2 ((0; T0); (H 1 )3 )). Moreover, if we use the formula:
p=
X
j<0
pj (x) pj (0) +
X
j 0
pj (x)
we see that for all t < T the pression p is bounded in L1((0; t); B13;1 ), which is a dual space of a separable Banach space; hence, we may have weak convergence of pn . The weak limits (~u; p) are then a solution of the Navier{ Stokes equations in (0; T0 ) IR3 . We demonstrate that this solution is suitable. Since the space L2uloc of uniformly locally square-integrable functions is a dual space, the control on the L2uloc norm of ~uR gives ~u 2 \t
~ ~u j2 = j~uj2 @t j~uj2 + 2jr
r~ :(j~u j2 (u~ !) + 2p ~u) We then use the following convergences in D0 : j~un j2 ! j~uj2 , j~un j2 (u~n !n ) ! j~uj2~u and r~ :(pn ~un ) ! r~ :(p ~u). Hence, 2jr~ ~un j2 converges in D0 and we ~ ~un j2 ! 2jr ~ ~uj2 + where is a nonnegative locally nite have that 2jr
measure. Now, we must prove that we have strong convergence of ~u(t) to ~u0 in (L2loc (IR3 ))3 when t goes to 0. We already know that we have convergence in (S 0 (IR3 )3 and thus, since the norm of ~u(t; :) in L2uloc is controlled and since L2uloc is a shift-invariant Banach space of distributions, we get that we have weak L2 convergence on any compact subset of IR3 . To prove that we have strong local L2 convergence, we Rjust have to show that R for any nonnegative ' 2 D(IR3 ) we have lim supt!0 j~u(t; x)j2 '(x) dx j~u0 (x)j2 '(x) dx. We
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Energy inequalities
know that we have strong convergence in (L2loc ((0; T0 ) IR3 ))3 of ~un toward ~u; hence, (at least for a subsequence of the sequence n ) we have, for almost R every t 2 (0; T0), limn!p j ~ u ( t; x ) ~u(t; x)j2 '(x) dx = 0. Moreover, since 0 n k~u R et~u0 kL2uloc C t and ~u is smooth on (0; T0 ) IR3 , we nd that tR 7! j~u (t; x)j2 '(x) dxR is continuous on [0; RT0R) and smooth on (0; T0); thus, j~u (t; x)j2 '(x)RdxR = j~u(0; x)j2 '(x) dx + 0t @t j~u (s; x)j2 '(x) dx ds. We write I ('; t) = 0t @t j~u(s; x)j2 '(x) dx ds, and we have
I ('; t)
R Rt
2 0 j~uj '(x) dx ds
RR ~ '(x) dx ds +R R 0t j~uj2 (u~ !):r t ~ '(x) dx ds +2 0 (p p;'(s))~u :r
Now, we write (t) = sup>0 (t) and (t) = (t) = sup>0 (t); we know that those two functions are bounded on every [0; T ] with t < T0. The computations R in the proof of Proposition 32.1 give us that I ('; t) C (')( 0t (s) ds + R Rt ( 0 (s)3 ds)1=4 ( (t) + 0t (s) ds)3=4 ) = A('; t). This gives that we have for almost every t 2 (0; T0) Z
j~u(t; x)j2 '(x) dx
Z
j~u(0; x)j2 '(x) dx + A('; t): R
Since A('; :) is a continuous function on [0; T0) and since j~u(:; x)j2 '(x) dx is a lower semicontinuous function on [0; T0) (by *-weak continuity of t 7! ~u(t; :) 2 (L2uloc )3 ), we nd that this equality holds for every t. Since limt!0 A('; t) = 0, this gives the required estimate.
3. Decay estimates for suitable solutions In this section, we consider a local Leray solution associated with ~u0 2 (E2 )3 and we prove that we may nd an estimate for the decay at in nity of the solution that depends only on the decay rate of the initial value and on its L2uloc norm:
~ :~u0 = 0 and let ~u be Proposition 32.2: Let ~u0 2 (E2 (IR3 ))3 be such that r 3 be a local Leray solution on (0; T ) IR for the Navier{Stokes initial value problem associated with ~u0 . Let 2 D(IR3 ) be equal to 1 in a neighborhood of 0 and de ne R (x) = 1 (x=R). Then, for all T < T , there exists a positive constant CT so that for all 0 < t < T and all R > 1 we have (32:4)
k~u(t; :) R kE CT (k~u0 R kE
r
2
2
+
1 + ln R ) R
R ~ ~u(s; :))'(x)k22 ds and The constant CT depends only on T , sup'2B 0T k(r 2 supt
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Local existence
349
Proof: We mimic the proof of Proposition 32.1. Thus, we de ne (t) = Rt 2 ~ sup'2B k~u(t; :)'(x)k2 and (t) = sup'2B 0 k(r ~u(s; :))'(x)k22 ds. We de ne moreover the truncated estimates R (t) = sup'2B k~u(t; :)'(x) R (x)k22 and R ~ ~u(s; :))'(x) R (x)k22 ds. Lemmas 32.2 to 32.6 then are R (t) = sup'2B 0t k(r changed in those lemmas that follow: Lemma 32.7: We have the following estimate valid for all R > 1 and all ' 2 B ('(x) = '(x x0 ) for some x0 2 IR3 ): (p p' ) 2R can be decomposed into (p(t; x) p' (t)) 2R (x) = p';R(t; x) + q';R(t; x) so that ZZ
jp';R (t; x)j3=2 '(x) dx dt 2=3+
I IR3
ZZ
jr~ q';R (t; x)j3=2 '(x) dx dt 2=3
I IR3
C k R~uk2L (I;E ) + k R~ukL (I;L ) k R~ukL (I;H ) + lnRR k~uk2L (I;E ) + R1 k ~ukL (I;L ) k ~ukL ([T ;T ];H ) 3
6
2
6
2
2
2
2
0
1
1
3
2
1
where we write I = [T0 ; T1 ] and where C is a constant which depends neither on x0 , nor on R, nor on T0 ; T1.
Proof: As with the proof of Lemma 32.2, we split p p' into p1 + p2 with
p1 (t; x) = G 02 (y x0 )(~u(t; y) ~u(t; y) (x) p2 (t; x) = (G(x; y) G(x0 ; y)):(1 02 (y x0 )) ~u(t; y) ~u(t; y) dy R
Now, we just write p1 (x; t) 2R (x) = q1 + q2 where
q1 = G 02 (y x0 ) 2R (y)~u(t; y) ~u(t; y) (x) q2 = [G ; 2R ] 02 (y x0 )~u(t; y) ~u(t; y) (x) and similarly p2 = q3 + q4 where R q3 (t; x)R= (G(x; y) G(x0 ; y))(1 02 (y x0 )) 2R (y)~u(t; y) ~u(t; y) dy q4 (t; x)= (G(x; y) G(x0 ; y))(1 02(y x0))( 2R (x) 2R (y)) ~u(t; y) ~u(t; y) dy On the support of '0 (x x0 ), q3 (t; x) is bounded by C k~u(t; :) R k2E2 and q4 by C lnRR k~u(t; :)k2E2 . On the other hand, we control q1 by kq1 (x; t)kL3=2 ([T0 ;T1 ]IR3 C k R ~ukL6 ([T0 ;T1 ];L2 ) k R ~ukL2 ([T0 ;T1 ];H 1 ) . The term q2 is the most diÆcult ~ q2 as r ~ q2 = q5 + q6 where one. We compute r ~ 2R )(x)G 02 (y x0 )~u(t; y) ~u(t; y)(x) q5 (x; t) = (r ~ G ; 2R ] 02 (y x0 )~u(t; y) ~u(t; y)(x) q6 (x; t) = [r q5 is controlled by kq5 kL3=2([T0 ;T1 ]IR3 C R1 k ~ukL6 ([T0 ;T1 ];L2) k ~ukL2([T0 ;T1 ];H 1 ) . ~ G ; 2R ] is a Calderon commutator beFor q6 , we notice that the commutator [r tween an operator of order 1 and a Lipschitz function; hence, it is a CalderonZygmund operator (see Chapter 9) whose operator norm is controlled by the L1
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Energy inequalities
~ 2R ; we thus obtain a control on q6 , namely, that norm of r kq6 (x; t)kL3=2 ([T0 ;T1 ]IR3 ) C R1 k ~ukL6 ([T0 ;T1 ];L2 ) k ~ukL2 ([T0 ;T1 ];H 1 ) . This proves the lemma. Lemma 32.8: For all t > 0 and all ' 2 B , we have the following inequality:
k~u(t; :) R (x)'(x)k22 + 2
Z t
0
k(r~ ~u(s; :)) R (x)'(x)k22 ds ZZ t
ZZ t
k~u0 R 'k22 + j~uj2 ( 2R '2 ) dx ds + (j~uj2 +2(p p ))(~u:r~ )( 2R '2 ) dx ds 0 0 Proof: We start from the local energy inequality 2
ZZ
jr~ ~uj2 (t; x) dx dt
ZZ
j~uj2 (@t + ) dx dt+
ZZ
~ ) dx dt (j~uj2 +2p)(~u:r
applied to (t; x) = 2R (x)'2 (x)g ; (t) with g ;(t) = G(t=) G((t )=), where G is a xed smooth function equal to 0 on ( 1; 1] andRto 1 on [2; +1). We may R ~ '(x) dx = r ~ :('(x) ~u(t; x)) dx = 0. replace p by p p' since we have ~u(t; x):r We x 2 (0; T ) and let go to 0. We obtain that R lim sup!0 0T Rk~u(s; :) R (x)'(x)k22 1 G0 ( s ) G0( s ) ds ~ ~u(s; :)) R (x)'(x)k22 ds +2 0 k(r RR RR 0 j~uj2 ( 2R '2 ) dx ds + 0 (j~uj2 + 2(p p' ))(~u:r~ )( 2R '2 ) dx ds
If is a Lebesgue point of t ! k~u(t; x) R (x)'(x)k22 , we obtain that
R lim!0 0T k~u(s; :) R (x)'(x)k22 1 G0 ((s )=) G0 (s=) ds = k~u( ; x) R (x)'(x)k22 k~u0 R (x)'(x)k22
Thus, when ' and R are xed, the inequality stated in Lemma 32.8 is true for almost every t in (0; T ). But t ! ~u(t; :) is weakly continuous and thus if n is a sequence of times converging to t, we have k~u(t; x) R (x)'(x)k22 lim inf k~u( n ; x) R (x)'(x)k22 . Finally, we may conclude that the inequality is valid for all t.
Lemma 32.9: For all t > 0, all ' 2 B and all R > 1, we have the following inequality: Z Z t
0
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Z t
Z t
0
0
j~uj2 ( 2R (x)'2 (x)) dx ds C1 ( R (s) ds + R1
(s) ds)
Local existence
351
Proof: This is obvious. We just have to consider whether we di erentiate R (getting the O(1=R) term) or not (getting the term involving R ). Lemma 32.10: For all t > 0 and all ' 2 B, we have the following inequality: ZZ t
0
j~uj2 (~u:r~ )( 2R '2 ) dx ds C2 M (t) MR (t)2 + R1 M 2 (t)
where M is the function M (t) = ( 0t (s)3 ds)1=12 (t) + 0t (s) ds 1=4 and R R MR is the function MR (t) = ( 0t R (s)3 ds)1=12 R (t) + 0t R (s) ds 1=4 . R
R
Proof: We again consider whether or not we di erentiate R (getting the O (1=R) term). When R is not di erentiated, we have to estimate the integral RRt 2 ~ )'2 (x) dx ds, and we proceed as in Lemma 32.5 to control 0 j~u3 R (x)j (~u:r the L norm of (x) ~u and of (x) R (x)~u. Lemma 32.11: For all t > 0, all ' 2 B and all R > 1, we have the following inequality: ZZ t
~ )( 2R (x)'2 (x))dx ds C3 M (t)MR (t)2 + 1 + ln R M (t)2 (p p' )(~u:r R 0
Proof: We consider one more time whether or not we di erentiate R R(getting R an O(1=R) term). When R is not di erentiated, we have to estimate 0t (p ~ )'2 (x) dx ds. We then decompose (p p' ) 2R (x) into px0 ;R (t; x)+ p' ) 2R (x)(~u:r qx0 ;R (t; x) according to Lemma 32.7 and we write Z Z t
0 =
Z Z t
0
~ )'2 dx ds = (p p') 2R (x)(~u:r
~ )'2 (x) dx ds px0 ;R (s; x)(~u:r
Z Z t
0
~ )qx0 ;R (s; x) dx ds '2 (x) (~u:r
Putting together estimates of Lemma 32.7 and Lemma 32.6 gives the required estimate.
End of the proof: Now, clearly, these lemmas give us the following inequalities (just by looking at the extrema of the functions k~u(t; :) R (x)'(x)k22 and of Rt ~ ~u(s; :)) R (x)'(x)k22 ds and by controlling the size of M (t) on (0; T )): 2 0 k(r R R (tR) R (t) and 2 R (t) R R (t) where R (t) = R (0) + C1 0t R (s) ds + R AT ( 0t R (s)3 ds)1=6 ( R (tR)+ 0t R (s) ds)1=2 + BT 1+ln R R . We use the Young inequality to control ( 0t R (s)3 ds)1=6 ( R (t) + 0t R (s) ds)1=2 by
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352
Energy inequalities
Rt 1 Rt 3 1=3 T 2 2T ( 0 R (s) ds) + 2 ( R (t) + 0 R (s) ds) with AT T =2 < 1 to remove R (t) and we obtain 2
2
R (t) DT R (0) +
Z t
0
R (s) ds + (
Z t
R
1 + ln R R (s)3 ds)1=3 + R 0 R
Using the Holder inequality, 0t R (s) ds T 2=3( 0t R (s)3 )1=3 , we nally obtain: Z t 1 + ln R 3 ) 3R (t) ET 3R (0) + R (s)3 ds + ( R 0 Thus, the function (t) = 3R (0) + R 3 inequality 0 ET ( + ( 1+ln R ) ).
Rt
3 0 R (s) ds is a solution of the di erential
This gives Proposition 32.2.
Final proof: We may now end the proof of Theorem 32.1 by proving that, when ~u0 2 (E2 )3 , ~u 2 \0
sup
Z
R!1 0
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j~u(t; x)j2 dx = 0:
Chapter 33 Global
existence
of
suitable
local
square-
integrable weak solutions
In this chapter, we complete the result of Chapter 32 (where we got locally in time the existence of local Leray solutions for uniformly locally squareintegrable initial data) by proving the existence of global weak solutions for locally square-integrable initial values vanishing at in nity: Theorem 33.1: (locally square-integrable initial value) ~ :~u0 = 0, there exists a local Leray solution For all ~u0 2 (E2 (IR3 ))3 so that r 3 ~u on (0; 1) IR for the Navier{Stokes initial value problem associated with ~u0 .
Existence of global weak solutions, generalizing the result of Leray for [CAL 90] and later by Lemarie-Rieusset [LEM 98a]. Theorem 1 was proved in [LEM 98b].
~u0
2 (L2 )3 was rst established for ~u0 2 (Lp )3 (2 p < 1) by C. Calderon
1. Regularity of uniformly locally
L2
suitable solutions
In the preceding chapter, we proved a local existence result of weak solutions for a locally square-integrable initial value: ~ :~u0 = 0, there exists a Local existence: For all ~u0 2 (E2 (IR3 ))3 so that r positive T and a local Leray solution ~u on (0; T ) IR3 for the Navier{Stokes initial value problem associated with ~u0. Moreover, ~u 2 \t
2
uloc
2
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354
Energy inequalities (B) for all [Æ; t] (0; T ), ~u 2 L4 ((Æ; t); (E3 )3 ). Proof: Point (A) is easy. We use the energy equality on suitable solutions to get that for all ' 2 B and for all Lebesgue points t, of k~u'k2 with t < we have the inequality k~u( ; :)'(x)k22 + 2 k~u(t; :)'(x)k22 +
Z Z
Z
t
k(r~ ~u(s; :))'(x)k22 ds Z Z
(j~uj2 +2(p p'))(~u:r~ )' ds dx Then, we use the weak continuity of t ! ~u(t; :) to conclude that this equality is valid for almost all t and for all > t. Thus, for all ' 2 B and for all Lebesgue points t of k~u'k2 we have lim >t; !t k(~u( ) ~u(t))'k2 = 0. Moreover, Proposition 32.2 implies a good uniform control of the decay of k~u( )'0 (x k)k2 when k goes to 1, whereas we have a good control of k(~u( ) ~u(t))'0 (3x k)k2 on the points t which are Lebesgue points for all k~u'0 (x k)k2; k 2 ZZ ; hence, for almost all t. Since Pk '0 (x k) = 1, this gives (A). In order to prove point (B), we use again Proposition 32.2 and we see from the estimates on ~u and on p that for all t < T we have lim sup
R!1 jx0 j>R
t
Z Z
j~uj2 ('2 (x)) dx ds +
j~uj jx x0j<min(pt;1);sup(0;t 1)<s
t
3 + jp(s; x) px (s)j3=2 dx ds = 0 0
and thus (due to the Ca arelli, Kohn, and Nirenberg regularity criterion) sup j~uj < C1 min(11; pt) : p jx j>R(t);jx x j<min(1; t)=2;sup(3t=4;t 1=4)<s
0
2. A generalized Von Wahl uniqueness theorem
In this section, we generalize the Von Wahl theorem, replacing the Leray
L2 solutions by local Leray solutions (as de ned in the preceding chapter) and the L3 condition by a uniformly locally L3 condition: Theorem 33.2: (locally L3 initial value) If ~u 2 C ([0; T ); (E3)3 ) is a solution for the Navier{Stokes equations, then the velocity ~u and Rthe pressure p are smooth on (0; T ) IR3 and for all t < T R ~ ~uj2 dx ds < 1. If ~v is a local Leray we have supx0 2IR3 0<s
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Global existence Proof: Step 1:
p
355
t~u is bounded
There is a direct proof by May [MAY 02] that a solution in C ([0; T ]; (E3)3 ) is bounded: sup0
uloc
3
0
0
0
0
0
0
0
3
0
0
0
0
0
0
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0
2
3
0
0
356
Energy inequalities
Step 2: Suitability of the E3 solutions Now, we prove that a solution ~u 2 C ([0; T ); (E3)3)
rst prove that the problem
is suitable. We shall
8 z < @t ~
= ~z r~ :(~u ~z) r~ q (33:1) r~ :~z = 0 : ~z(0; :) = ~u0 (which has (~u; p) as solution) has a solution (~z; q) so that, for all t < T , (33:2)
sup
Z
j~z(s; x)j2 dx + sup
x0 2IR ;s
ZZ
x0 2IR
0<s
3
jr~ ~zj2 dx ds< 1
x0 j1
Indeed, let us approximate ~u0 by2 ~u3 0 = ~v + w~ with divergence-free vector elds ~v and w~ so that ~v 2 (L ) and kw~ kE < . Equation (33:1) with ~v as initial value instead of ~u0 may be solved by the Leray method, which provides a solution ~z. We know that this solution is in \t
(t) =
and
(t) =
sup
sup
Z
x0 2IR3 jx x0 j1
j~z (t; x)j2 dx
Z Z
x0 2IR
0<s
3
x0 j1
jr~ ~z j2 dx ds:
We show that for all t < T we have sup(sups
Z
jq (s; x) Cs;x ; j2 dx C (f jx x j<2 jx 0
0
x0 j<3
j~z(s; x)j6 dxg1=3 + (s))k~u(s)k2E
3
where C does not depend on x0 norR on s nor on and where the constant Cs;x ; is formally given by Cs;x ; = jy x j>3 R(x0 ; y)~u ~z (s; y) dy. Thus, if we integrate the energy equality against 1(0;t)(s)'(x x0 ) where ' 2 D(IR3) is equal to 1 on the unit ball B(0; 1) and to 0 outside from the ball B(0; 2), we get 0
0
Z
j~z(s; x)j2 '(x x0 ) dx + 2
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0
Z Z t
0
jr~ ~z(s; x)j2 '(x x0 ) dx
Global existence
357
Z
j~z (0; x)j2 '(x x0 ) dx + C1 Z t
+C2 0<s
0
(s) ds)1=2 (
Z t
C3 (0) + C4 (1 + sup k~u(s)k2E )
0
Z t
0
(s) ds+
(s) ds + (t))1=2
Z t
(s) ds + (t) 0<s
we have
(t) 2 (C3 (0) + C4 (1 +
sup k~u(s)k2E ) 0<s
Z t
0
(s) ds)
and thus (t) 2 C3 (0)e2 C (1+sup k~u(s)k )t. This gives the desired inequalities on and . Now, we again use the Leray weak convergence lemma (Theorem 13.3) to get that for a sequence n going to 0 ~z converges strongly in (L2loc((0; T )IR3 )3 to a solution ~z of problem (33.1) with energy estimate (33.2). For such a solution, we may compute @tj~zj2 as 2 h@t~zj~zi and get that ~ ~zj2 = j~zj2 r ~ :(j~zj2~u) 2r ~ :(q ~z): @t j~zj2 + 2jr In order to prove the suitability of ~u, we only need to show that ~z1= ~u. Thus, we will show that Problem (33.1) has a unique solution in \t
0<s
2
E3
n
0 ~ For T2 < T , we split ~u in X + Y~ where sup0<s
kw~ (t)kML
1
2;
p C sup kw~ (s)kML 1 ( sup kX~ k3 + t t0 sup kY~ k1 ) 0<s
which is enough (if C < 1) to grant that w~ = 0. The estimate on Y~ is obvious: 1 since pointwise multiplication by a L function operates on ML2;1 and since
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358 Energy inequalities the norm of ML2;1 is shift-invariant, we have ke(t Rs) IPr~ :(Y~ w~ )kML 1 t 1 ~ ~ :(X~ w~ ) ds, C pt s kY (s)k1 kw~ (s)kML 1 . For the estimate on 0 e(t s)IPr we write that X~R t w~ 2 L1((0; T2); (ML6=5;1)9), so that it is enough to check that f~ ! 0 e(t s)IPr~ :f~ ds is bounded from L1((0; T2); (ML6=5;1)9 ) to L1((0; T2); (RML2;1)3). We are going to prove in an equivalent way that ~ :f~ ds is bounded from L1 ((0; T2 ); (ML6=5;1)9 ) to f~ ! A(f~) = 0T es IPr 2 ; 1 3 (ML ) . We write 2;
2;
2
A(f~) = A(f~1[ 2;2]3 (y x0 )) + A(f~(1
1[
2;2] (y x0 ))) = f~1 + f~2 3
Now, we estimate f~1 by checking that A is a bounded linear operator from L1 ((0; T2 ); (L6=5;1 )9 ) to (L2;1 )3 and we estimate f~2 by checking that f ! A(f~(1 1[ 2;2] (y x0 ))) operates boundedly from L1((0; T2 ); (M 1;1 )9 ) to (L1(x0 + [0; 1]3)3 ). More precisely, we have estimates that, for 1 < q < 6, kesIPr~ :~gkL Cq k~gkq s (3=q+1=2)=2 so that R01 kesIPr~ :~gkLP ds C k~g(x)kL (because each g 2 L2;1 can be decomposed into g = n2IN gn with Pn2IN kgnk11=2kgnk11=2 C kgkL ) and by duality we see that A operates boundedly from L1((0 ; T2)p; (L6=5;1 )9 ) to (L2;1 )3 . On the other hand we have R that jesIPr~ :~gj C IR jx syj j~g(y)j dy and this gives the control on f~2. 3
6;1
6;1
2;1
2;1
3
4
Step 3: Local Leray solutions We consider (~u; p) and (~v; q) two local Leray solutions on (0; T ) IR33 for the Navier{Stokes initial value problem associated with some ~u0 2 (E3 ) and we assume that ~u 2 C ([0; T ); (E3)3 ). 3First, we notice that @th~uj~vi = h@t~uj~vi + h~uj@t~vi since for all 2 D((0; T ) IR we have ~v 2 L2 ((0; T ); (H 1 )3 ) 2 @t~v 2 L ((0; T ); (H 1)3 ) + L1 ((0; T ); (L3=2 )3 )
and
~u 2 L2 ((0; T ); (H 1 )3 ) \ C ([0; T ]; (L3)3 ) @t ~u 2 L2 ((0; T ); (H 1)3 ): This gives for some nonnegative local measure : ~ ~vj2 = j~v j2 r ~ :(j~vj2~v) 2r ~ :(q ~v ) @t j~vj2 + 2jr 2 2 2 2 ~ ~uj = j~uj r ~ :(j~uj ~u) 2r ~ :(p ~u) @t j~uj + 2jr
and
~ ~ujr ~ ~vi = h~uj~vi r ~ :(h~uj~vi~u) + h((~u ~v):r ~ )(~v ~u)j~ui+ @t h~uj~vi + 2hr 1 2 ~ ~ ~ + 2 r:(j~uj (~u ~v)) r:(p ~v) r:(q ~u)
Hence, we get
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~ (~v ~u)j2 = @t j~v ~uj2 + 2jr
Global existence 359 = j~v ~uj2 r~ :(j~v ~uj2~u) r~ :(h~v ~uj~v + ~ui(~v ~u)) 2r~ :((q p) (~v ~u)) with q p = R(~v (~v ~u) + (~v ~u) ~u). Now, we write ~u ~v = w~ , and we do the same estimations as in the preceding chapter for (t; w~ ) and (t; w~ ), and we get Z t
Z t
Z t
(t; w~ ) (0; w~ )+C1 (s; w~ ) ds+C2( (s; w~ )3 ds)1=6 ( (t; w~ )+ (s; w~ ) ds)1=2 0 0 0 Z t
Z t
Z t
2 (t; w~ ) (0; w~ )+C1 (s; w~ ) ds+C2( (s; w~ )3 ds)1=6 ( (t; w~ )+ (s; w~ ) ds)1=2 0 0 0 where C2 depends on ~u and ~v and satis es Z T
C2 C3 (1+ T )3=2f(
0
Z T
(s; ~u)3 ds)1=3 + (T; ~u)+(
0
(s; ~v )3 ds)1=3 + (T; ~v)g
Then, we can get rid of (t; w~ ) and, since (0; w~ ) = 0, get (t; w~ ) C4 (T )(
Z t
0
(s; w~ )3 ds)1=3
which gives (t; w~ ) = 0 and, thus, ~u = ~v. Thus, Theorem 33.2 is proved. Besides, Theorem 33.2 provides an useful insight into the structure of global solutions in L3: Theorem 33.3: ( Global solutions in L3 .) If ~up2 C ([0; 1); (L3 )3 ) is a solutionpfor the Navier{Stokes equations, then sup0<s s k~u(s; :)k1 < 1 and limt!1 t k~u(t; :)k1 = 0. Proof: We now consider the case of a global L3 solution ~u 2 C ([0; 1); (L3)3 ). We split ~u0 in ~v0 + w~ 0 where r~ :~v0 = r~ :w~ 0 = 0, ~v0 2 (L2)3 and kw~ 0k3 < where > 03 is3 small enough. Then, we write W~ the global Kato solution in C ([0; 1); (L ) ) of the Navier{Stokes equations with w~ 0 as initial value; ~v = ~ is solution of the equation ~u W 8 ~ :(~v ~v) r ~ :(W ~ ~v) r ~ :(~v W ~) r ~p v = ~v r < @t~ r~ :~v = 0 : ~v(0; :) = ~v0 Since ~v0 2 (L2)3 and supt>0 kW~ k3 < 2, we know that this problem has a Leray solution (~z; q) so that: sup k~zk2 < 1 and t>0
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1
Z
0
kr~ ~zk22 ds < 1
360 Energy inequalities and (for some nonnegative local measure ) ~ ~zj2 = @t j~zj2 + 2jr 2 2 2 ~ ~ ~ ~ j~zj r:(j~zj ~z) r:(j~zj W ) 2r:((~z:W~ ) ~z) + 2((~z:r~ ) ~z):W~ 2r~ :(q ~z) Moreover, we know3that ~u and W~ are suitable and, therefore, that ~v is locally 2 1 L (H ) and is C (L ). We may then apply the Von Wahl uniqueness theorem [WAH 85]. Indeed, we have for ~y = ~z ~v: ~ ~yj2 = j~yj2 r ~ :(j~zj2~z) r ~ :(j~v j2~v) + 2((~z:r ~ ) ~z):~v+ @t j~yj2 + 2jr +2((~v:r~ ) ~v):~z r~ :(j~yj2W~ ) 2r~ :((~y:W~ ) ~y)+2((~y:r~ ) ~y):W~ 2r~ :((q p) ~y) and we write ((~z:r~ ) ~z):~v + ((~v:r~ ) ~v):~z = ((~y:r~ ) ~z):~v + r~ ((~v:~z) ~v); now, all the quantities can be integrated (against 1[0;t] !(x=R) and letting R go to 1) to give k~y(t)k2 +2
ZZ t
ZZ t
jr~ ~yj2 ds dx 2
ZZ t
((~y:r~ ) ~z):~v ds dx 0 0 For nite T , we split ~v on [0; T ] in X~ + Y~ where supt kX~ k3 < and Y~ 2 (L1)3 . Then, we get a Gronwall inequality on k~yk22, which allows one to conclude that R1 ~ k43 dt < 1 while supt kW ~ k3 < 2; thus ~y = 0. This implies that 0 k~u W k~uk3 take small values and the Kato theorem for small datas in L3 gives us the behavior of ~u as t goes to 1. 3.
0
Global
existence of
((~y:r~ ) ~y):W~ ds dx+2
uniformly
locally
L2
suitable
solutions
We may now nish the proof of Theorem 33.1 about the existence of global Leray solutions. Step 1: Local resolution in E2
We consider the initial value ~u0 in (E2 )3 and we solve the Navier{Stokes equations as explained in Chapter 32. We then have a local Leray solution ~u(0) 3 on (0; T0) IR . Step 2: Local resolution in E3
Assume we have a local Leray solution ~u(N ) on some (SN ; TN ) IR3 . Then, according to Proposition 1, we may choose SN +1 2 (sup(SN ; TN 1); TN ) so that ~u(N )(SN +1 ) 2 (E3 )3 and lim k~u(N )(s) ~u(N ) (SN +1 )kE = 0 s>S ;s!S N +1
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N +1
2
Global existence 361 Then, due to the Kato theory of mild solutions (Chapter 17), we may nd a time UN +1 > SN +1 and a solution ~v(N +1) 2 C ([SN +1; UN +1]; (E3 )3) with ~v(N +1) (SN +1 ) = ~u(N ) (SN +1 ). Step 3: Resolution in E3 + L2 Using the approximation lemma (Chapter 12), we know that we may split ( N ) ~ :X~ N +1 = r ~ :Y~N +1 = 0, Y~N +1 2 (L2 )3 ~u (SN +1 ) into X~ N +1 + Y~N +1 with r ~ and kXN +1kE < , where we choose small enough to ensure that we may nd a Kato solution X~ (N +1) 2 C ([SN +1; SN +1 + 2]; (E3)3 ) with X~ (N +1)(SN +1) = X~ N +1 . Then, using the Leray theory developed in Chapters 14 and 21, we know that we may nd a Leray solution Y~ (N +1) of 8 ~ = Y~ r ~ :(Y~ Y~ ) r ~ :(X (N~ +1) Y~ ) r ~ :(Y~ X (N~ +1)) r ~p < @t Y r~ :Y~ = 0 : Y~ (SN +1 ; :) = Y~N +1 3
Then, ~u(N +1) = X~ (N +1)+Y~ (N +1) is a local Leray solution on (SN +1; TN +1)IR3 (with TN +1 = SN +1 + 2) of the Navier{Stokes initial value problem associated with ~u(N )(SN +1). Step 4: Identi cations of solutions
If we use the generalized Von Wahl uniqueness theorem (theorem 33.2), we nd that on [SN +1; UN +1], we have ~u(N ) = ~v(N +1) = ~u(N +1). Step 5: Construction of ~u By induction on N , we have a sequence of local Leray solutions ~u(N ) on (SN ; UN +1) IR(0)3 so that: i) S0(N=) 0 and(N ~+1) u (0) = ~u0 ii) ~u = ~u on (SN +1; UN +1) IR3 iii) SN + 1 < SN +1 < UN +1 < SN + 2 so that (0; 1) = [N3 2IN(SN ; UN +1) This de nes a unique distribution ~u on (0; 1) IR . It is obvious that we have the desired estimates on ~u, as it can be seen by using a nonnegative partition of unity adapted to [0; U1) [[N 1 (SN ; UN +1), proving Theorem 33.1.
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Chapter 34 Leray's conjecture on self-similar singularities
In 1934, Leray [LER 34] proposed to study self-similar solutions to the Navier{Stokes equations in order to try to exhibit solutions that blow up in a nite time. Leray's self-similar solutions on (0; T ) IRd are given by ~u(t; x) = (t)U~ ((t)x) with (t) = p2a(1T t) for some positive a. Recently, Necas, Ruzicka, and Sverak [NECRS 96] proved that the only solution U~ 2 (L3(IR3 ))3 was U~ = 0. Their result was extended by Tsai [TSA 98] to the case U~ 2 (C0(IR3))3 or to the case of Rsolutions ~u with local energy estimates near the blow-up point: supT
0
1. Hopf 's strong maximum principle
In this section, we prove a lemma of Tsai, which is a variation on the maximum principle of Hopf: Proposition 34.1: (Tsai's lemma) Let , b1 ,: : : , bd be real-valued functions so that 2 C 2(IRd) and, for j = 1 : : : ; d, bj 2 CIR (IRd ) and limx!1 jb jx(xj )j = 0. Let a > 0, ~b = (b1 ; : : : ; bd) ~ be the identical vector eld on IRd : X~ (x) = (x1 ; : : : ; xd ). If and let X (~b + aX~ ):r~ 0 and lim j(x)j = 0; j
x!1
363 © 2002 by CRC Press LLC
jxj2
364
Energy inequalities
then is constant.
Let us de ne L as the di erential operator Lf = f (~b + aX~ ):r~ f . The hypothesis is then L 0. Let us assume that is not constant. We select two points X0 and X1 so that (X0 ) < (X1). We select two real numbers R0 and R1 so that 0 < R0 < jX0 X1 j < R1 and so that, for jx X0 j R1 , j~b(x) + aX~ 0 j ajx X0 j=2 and, for jx X0j R0 , (x) ((X0 ) + (X1))=2. Moreover, we require that aR12 > 2d. We then consider the function V (x) = (x) + (e jx X j jx X0j2 ), where the positive numbers , and are chosen in the following way: i) rst, we choose big enough to ensure that the function (x) = e jx X j satis es L > 0 on jx X0j R0 . This is always possible, since we have L = e jx X j ((4 +2a)jx X0 j2 2d +2(~b + aX~ 0 ):(X~ X~ 0 )); since j~b(x)j may be bounded by C (R0 )jx 2X0 j for jx X1 0j R0, it is enough to take > 0 with 4 + 2a > C (R0 ) + 2dR0 + ajX0jR0 . ii) next, we choose small enough to ensure that the function (x) = jx X0 j2 satis es supR jx X jR jL (x)j < minR jx X jR L (x). iii) nally, we choose small enough to ensure that sup j (x) (x)j < ((X1 ) (X0 ))=4: Proof:
0
2
0
0
0
2
2
0
0
1
0
1
jx X0 jR1
The function V is C 2; we have V (x) jxj2 when x ! 1, thus V has a maximum at some point X2; at1 thisR point, we must have r~ V (X2 ) = 0 and V (X2) 0 (since V (X2) (S ) S V (X2 + ) d() = V (X2) + 2 2d V (X2 ) + o( )). Thus, we have LV (X2 ) 0. 3 For jx X0j R0, we have V (x) (X0 ) + 4 ((X1 ) (X0)) < V (X1 ). Thus, we cannot have jX2 X0j R0 . For R0 jx X0j R1 , we have jL (x)j < L (x) and L(x) 0, hence LV (x) > 0. Thus, we cannot have R0 jX2 X0j R1 . For jx X0j R1, we have L (x) > 0, L(x) 0, and L (x) = 2d 2(~b + aX~ 0):(X~ X~ 0) 2ajx X0 j2 2d ajx X0 j2 < 0, hence LV (x) > 0. Thus, we cannot have jX2 X0j R1 . Therefore, must be constant. 2. The C0 self-similar Leray solutions are equal to 0 We may now study self-similar solutions to the Navier{Stokes equations. Recall that a weak solution of the Navier{Stokes equations is a locally squareintegrable vector eld ~u(t; x) in (L2loc((0; T ) IRd)d, which satis es, for some distribution p 2 D0 ((0; T ) IRd), the equations ~ :(~u u~) r ~p @t ~u = ~u r (34:1) r~ :~u = 0: d
2
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1
d
1
Self-similar singularities 365 A self-similar solution is a solution that can be written as 1 (34:2) ~u(t; x) = (t)U~ ((t)x) with (t) = p 2a(T t) for some positive a and some locally square-integrable U~ 2 (L2loc(IRd))d . We then nd the equation: r~ p(t; x) = (t)3 Q~ ((t)x) with Q~ = U~ r~ :U~ U~ ~r ~ )U~ . Thus, we shall have p(t; x) = (t)2 P ((t)x) + q(t) for some aU~ a(X: distributions P 2 D0(IRd) and q 2 D0 ((0; T )). We may assume with no loss of generality that q(t) = 0. Thus, self-similar solutions (~u; p) are given by the equations 8 ~u(t; x) = (t)U~ ((t)x) > > > > p (t; x) = (t)2 P ((t)x) < (34:3) > ~r ~ )U~ + r ~ :U~ U~ + r ~P > U~ = aU~ + a(X: > > : ~r:U~ = 0 A direct consequence of Tsai's lemma is then the following theorem: Theorem 34.1: Let U~ 2 (C 2 (IRd ))d and P 2 C 2 (IRd ) be solutions of Leray's equations: ~r ~ )U~ + r ~ :U~ U~ + r ~P U~ = aU~ + a(X: ~r:U~ = 0 ~ with a > 0. If moreover limx!1 jUj(xxj )j = 0 and limx!1 jPjx(jx2)j = 0 then U~ is constant. ~ U~ and L the di erential operator Proof: We de ne = 12 jU~ j2 + P + aX: Lf = f (U~ + aX~ ):r~ f . We apply Proposition 34.1 by proving that L 0. ~r ~ )U~ is divergence free since r ~ : (X: ~r ~ )U~ = Let us rst notice that (X: ~r ~ )(r ~ :U~ ). Thus, we have r~ :U~ + (X:
P =
d X d X j =1 k=1
@j @k (Uj Uk ) =
d X d X j =1 k=1
@j Uk @k Uj
~ U~ ) = X: ~ U~ + (by taking the divergence of Leray's equations). We have (X: ~r:U~ = X: ~ U~ and ( 12 jU~ j)2 = U~ :U~ + jr ~ U~ j2 . On the other hand, we have (U~ + aX~ ):r~ = (U~ + aX~ ): r~ P + aU~ + ((U~ + aX~ ):r~ )U~ = (U~ + aX~ ):U~ : We nd that d X d d X d X 1X L = jr~ U~ j2 @j Uk @k Uj = j@j Uk @k Uj j2 0: 2 j =1 k=1 j =1 k=1
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366
Energy inequalities We may then apply Proposition 34.1. We have that is a constant function, hence L = 0. This gives @j Uk = @k Uj for j; k 2 f1; : : : ; dg; since U~ is divergence free, we nd that U~ = r~ F for some harmonic function F . Thus, for 1 j d, Uj is a harmonic function on IRd that is o(jxj) at in nity; thus Uj is a harmonic tempered distribution, hence a harmonic polynomial, and since it has a very slow growth at in nity, it is constant. Corollary 34.1: (Tsai's theorem) If ~u = (t)U~ ((t)x) is a self-similar Leray solution of the Navier{Stokes equations with U~ 2 (C0 )d , then ~u = 0.
Since ~u vanishes at in nity, we may apply Theorem 11.1 to get that ~u is solution of the Navier{Stokes equations: Proof:
(34:4)
~ :(~u u~) @t ~u = ~u IPr r~ :~u = 0
It means that we may choose p = (Id IP)(r~ :~ud ~u). Moreover, we know that a solution of (34.4) with ~u 2 (L1((0; T ) IR ))d is smooth: for all T0 < T and all k 2 IN we have ~u 2 (L1 ((T0; T1); B1k;1 (IRd)))d. Thus, U~ is smooth. We have to check that P is smooth and is o(jxj2 ) at in nity. It is enough to check that,Pfor a xed t0, r~ p(t0 ; x) is smooth and is o(jxj) at in nity. Let Id = S0 + j2ZZ j be the Littlewood{Paley decomposition; (Id IP)(Id S0 ) maps B1k;1 to B1k;1 , so that (Id S0 )r~ p belongs to all the Besov spaces k;1 , hence is smooth and bounded. Since S0 r ~ P is smooth (because it has B1 ~ a compactly supported Fourier transform), rP is smooth. Moreover, we have k(Id IP)r~ S0 (~u ~u)k1 C k~uk21 , and thus, r~ P is bounded. We may apply Theorem 34.1 and get that U~ is constant. Since it goes to 0 at in nity, we have U~ = 0. Corollary 34.2: If ~u = (t)U~ ((t)x) is a self-similar Leray solution of the Navier{Stokes equations on (0; T ) IRd with U~ 2 (Ld )d , then ~u = 0. Proof: The result of Necas, Ruzicka, and Sverak is now easy to prove. Since ~u vanishes at in nity, we may apply Theorem 11.1 to get that ~u is solution of
the Navier{Stokes equations (34:4)
~ :(~u u~) @t ~u = ~u IPr ~r:~u = 0
Moreover, we have uniqueness of the solutions of (34.4) in (C ([0; T ]; Ld))d (Chapter 27); Kato's algorithm provides a solution that belongs to (C ([0; T ]; Ld))d
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Self-similar singularities 367 \(C ((0; T ]; C0))d ; thus, we get that U~ 2 (C0 )d . Tsai's theorem then gives U~ = 0. We conclude this section with other results of Tsai: Corollary 34.3: Let q 2 (d; 1). If ~u = (t)U~ ((t)x) is a self-similar Leray solution of the Navier{Stokes equations on (0; T ) IRd with U~ 2 (Lq )d , then ~u = 0. The same conclusion holds for the weaker hypothesis U~ 2 (Eq )d , where Eq is theR closure of the test functions in Lquloc , i.e. if U~ is locally Lq and limx!1 jy xj<1 jU~ (y)jq dy = 0.
We use the same proof as for Corollary 34.2, since we have uniqueness of the solutions of (34.4) in (C ([0; T ]; Eq ))d . Proof:
3. The case of local control
In this section, we deal with d = 3. We study self-similar Leray solutions under some energy estimates. If we assume that we have a global control as for the square-integrable Leray solutions, i.e. ~u 2 (L1((0; T ); L2(IR3)))3 \ (L2((0; T ); H_ 1(IR3)))3 , then we have that ~u 2 (L4((0; T ); L3(L3)))3 , thus U~ 2 (L3)3 , and according to Corollary 34.2 we may conclude that ~u = 0. Tsai [TSA 98] proved that the same results hold under the weaker assumption that we have control of the energy in the neighborhood of the blowing up point (T ; 0): Theorem 34.2: Let ~u be a self-similar solution of the Navier{Stokes equations on (0; T ) 3 IR : 8 ~u(t; x) = (t)U~ ((t)x) > > > > p (t; x) = (t)2 P ((t)x) < > > > > :
~r ~ )U~ + r ~ :U~ U~ + r ~P U~ = aU~ + a(X: ~r:U~ = 0
where (t) = p2a(1T t) for some positive a. If we assume that for some T0 < T and some r0 > 0, we have the energy estimates
sup
Z
T0
then ~u = 0.
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j~u(t; x)j2 dx < 1 and
Z T Z
T0
jxj
jr~ ~u(t; x)j2 dx dt < 1
368
Energy inequalities ~ U~ and L as the di erential operator Proof: If we de ne = 12 jU~ j2 + P + aX: ~ ~ ~ Lf = f (U + aX ):rf , we nd again that 3 X 3 X L = 21 j@j Uk @k Uj j2 0: j =1 k=1 Thus, our strategy is clear: we check that U~ and P~ are smooth and that they can grow only at a slow rate at in nity, then we apply Theorem 34.1. Step 1: Regularity of U~ and of P
We3 start from the following classical estimate: if f 2 D0 (IR3) and if f 2 3 +2 (IR ) for some 2 IR, then f 2 Hloc (IR ). Indeed, if is an open bounded s ( ) for some s 2 IR; now, just write for ! 2 D( ) subset of IRd, we have f 2 Hloc 1 ~ !:r ~ f , which gives !f = (Id ) f (Id )! !f r s ( ) and f 2 H ( ) ) f 2 H min(s+1;+2) ( ) f 2 Hloc loc loc and nally the3 required regularity on f . Now, the hypothesis on ~u gives that 1 (IR ))3 : in order to prove that r ~ U~ is square-integrable on B (0; R), U~ 2 (Hloc 2 r 1 just consider ~u(t; :) with T t 2a R . Now we write P = P3j=1 P3k=1 @j @k (Uj Uk ) ~r ~ )U~ + r ~ :U~ U~ + r ~P U~ = aU~ + a(X: Hloc
0
s (IR3 ) with s 1, we have fg 2 H s (IR3 ); and we check that for f and g in Hloc loc s 3 3 s 3 ~ thus, U 2 (Hloc(IR )) with s 1 gives that P 2 Hloc (IR ) and then U~ 2 s+ (Hloc (IR3))3 . Thus, U~ and P are C 1. 1 2
1 2
1 2
Step 2: Global estimates on U~
We use the energy inequalities. Let A = supT
0
0
Z
B (0;r0 )
thus,
R
j~u(t; x)j2 dx = (t) 1
~ (x)j2 dx AR r0
B (0;R) jU
B (t)
=
Z
B (0;(t)r0 )
jU~ (x)j2 dx;
for R > (T0 )r0 . Similarly, we have
R T
R ~ U~ (x)j2 dx ds (s) B(0;(s)r0 ) jr R (T t)(t) B(0;(t)r0) jr~ U~ (x)j2 dx
t
thus, RB(0;R) jr~ U~ (x)j2 dx p2aBr(t)R for R > (T0)r0 and T gives that kU~ kL (B(0;R) is o( R) as R goes to +1. 0
6;2
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2 t = 2Rar02 .
This
Self-similar singularities
369
Step 3: Global estimates for P
We rst check that we may de ne a pressure by the formula 3 X 3 1X k=1 l=1 @k @l(Uk Ul);
P0 =
then we shall3 compare P0 to P . Let us see how P0 behaves on B(0; R): we x ! 2 D(IR ) equaly to 1 on B(0; 1) and supported by B(0; 2). We de ne, for R 1, !R (y) = !( 4R ). Then, for x 2 B (0; R), we write P0 (x) = P1 (x) + P2 (x) with P1 = P3k=1 P3l=1 Rk Rl (!RUk Ul). The distribution kernel of 1 @k @l is bounded by C jx yj 3 outside P from the diagonal x = y, hence jP2(x)j R R ~ 1 2 0 ~ C jyj4R jx yj jU (y)j dx C n2IN 4 Rjyj4 R j4U (yR)j dx C 00 R 2 . Thus, kP2kL (B(0;R)) CR 1. On the other hand, the Riesz transforms operate on L3;1, hence kP1kL C k!RjU~ j2kL C 0kU~ k2L (B(0;8R)) C 00 (R)R with limR!1 (R) = 0. This gives that P0 is well de ned and that kP0 kL (B(0;R)) = o(R). Moreover, @j P0 = Q1 + Q2 with Q1 = P3k=1 P3l=1 Rk Rl(!R @j (Uk Ul )). On one hand, we have kQ1kL C k!RjU~ j jr~ U~ j kL C 0 kU~ kL (B(0;8R)) kr~ U~ k2L (B(0;8R)) C 00 (R)R with limR R!1 (R) = 0. On the other hand, for x 2 B(0; R), we have jQ2(x)j C jyj4R jx 1yj jU~ (y)j jr~ U~ (y)j dx R P ~ ~ ~ C 0 n2IN 4 Rjyj4 R jU (y)4j jrR U (y)j dx C 00 R 2 . Thus, kQ2kL (B(0;R)) C . This gives that kr~ P0 kL (B(0;R)) = o(R). We now compare P and P0 . We have P = P0; moreover, P and P0 are tempered distributions (because U~ has a small rate of growth P at in nity). Thus, P P0 is a harmonic polynomial. Let @ ( P P ) = j 0 j jN c ;j x . R For R j j N , let ' 2 D so that x ' dx = 1 and, for j j N and 6= , x ' dx = 0 and let 2 D so that = 1 on a neighborhood of the support of ' . Then, we have 2
3
n
n+1
3n
3
3;1
3;1
3;1
6;2
3;1
3=2;1
6;2
3=2;1
2
3
3n
n+1
n
3=2;1
3
3=2;1
Z
' (
x )(@ P (x) @j P0 (x)) dx = c ;j R3+j j: R j
On the other hand, we have Z
j ' ( Rx )@j P0 (x) dxj k' ( Rx )kL 1 k ( Rx )r~ P0 kL 3;
3=2;1
= o(R2)
and R
j ' ( Rx )@j P (x) dxj ((a + d)k' ( Rx )k2 + k R1 (' )( Rx )k2 )k ( Rx )U~ k2 + R1 kr~ ' k1(k ( Rx ) jxj jU~ jk1 + k ( Rx )jU~ j2k1) 2
=
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O(R2 )
370 Energy inequalities and thus we get c ;j = 0. Thus, P P0 is constant, and we may assume with no loss of generality that P = P0 . Step 4: Local estimates for U~ We now estimate the size of U~ . We cannot use the Leray equations for U~ , since the estimate of the size of X~ U~ cannot be o(jxj). We thus go back to
the Navier{Stokes equations. Let T = T =2. We x ! 2 D(IR3), which is equal to 1 on the ball B(0; 1) and which isy supported in the ball B(0; 2), and we de ne, for jxj 1, !x as !x(y) = !( 4jxj ). We have for jxj 1 sup0tT k~u(t; y)kL (B(0;8jxj)) (x)jxj1=2 sup0tT kp(t; y)kL (B(0;8jxj)) (x)jxj with limx!1 (x) = 0. We then write the equations @t (!x~u) = (!x~u) 2 P3j=1 @j ((@j !x)~u) + (!x)~u r~ :(!x (~u ~u)) + (~u:r~ !x)~u r~ (!xP ) + P r~ !x This gives for t 2 (0; T ] 6;2
3;1
!x (y)~u(t; y) = et (!x~u(0; :)) +
Z t
0
e(t )( ~x ( ; :)
3 X j =1
@j ~ j;x ( ; :)) d
with ~x = (!x)~u + (~u:r~ !x)~u + P r~ !x and ~ j;x = 2(@j !x)~u + !xuj ~u + !xP ej . Let us write Bx = B(0; 8jxj). We have ( k ~xkL C k~ukL (B ) jxj 2 jxj3=2 + (k~uk2L (B ) + kP kL (B ) )jxj 1 jxj k~ j;x kL C k~ukL (B ) jxj 1 jxj1=2 + k~uk2L (B ) + kP kL (B ) Thus, sup0tT k ~x(t; :)kL =R o1(jxj) and sup0tT k~ j;x(t; :)kL = o(jxj). We now useR theRestimates 0 ketp f k1 Rdt C kf kL (IR ) (Proposition 5.3)1 and 0T 3k;1 0t e(t s)f (s; :) dsk1 dt C 0T kP f (s; :)kL ds: indeed, ifPf 2 L ((0; T ); L ), it can be written as f ( s; y ) = n2IN n (s)fn (y ) with Rt k k k f k C k f ( s; : ) k ds ; we have L n2IN n 1 n L 0 6;2
3=2;1
3;1
6;2
x
6;2
6;2
x
3;1
x
x
3;1
x
x
3;1
3=2;1
3;1
3
3;1
3;1
Z T
Z t
XZ T
Z T s
j n (s)j ke fn k1 d ds: 0 0 0 n2IN Since p 1 maps L3=2;1 to L3;1, we nd that the local tendency w~ x(t; y) = !x(y)~u(t; y) et(!x~u(0; :)) satis es RT R C 0T k ~x(t; :)kL + P3j=1Pk~ j;x(t; :)kL dt 0 kw~ x (t; y)k1 dt CT sup0
k
e(t s)f (s; :) dsk1 dt
3;1
3;1
3=2;1
3=2;1
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3;1
Self-similar singularities where limx!1 T (x) = 0. Similarly, Z T
0
371
ket(!x~u(0; :))k1 dt CT 1=4k!x~u(0; :)k2 = o(jxj):
RT Thus, we nd that p 0 k!x~u(t; :)k1 dt = o(jxj). Since we have, on [0; T ], (0) (t) (T ) = 2(0), we have 1 j!x( (T ) x)~u(t; (T ) x)j; jU~ ((T )x)j = (1t) j~u(t; ((Tt)) x)j (0) (t) (t) hence, jU~ ((T )x)j T 1(0) R0T k!x~u(t; :)k1 dt = o(jxj). Since (T ) is a constant, we nd that limx!1 jU~j(xxj )j = 0.
~ U~ Step 5: Local estimates for r
We may proceed in the same way in order to estimate the size of the derivatives of U~ . We keep the notations T = T =2 and !x(y) = !( 4jyxj ). Di erentiating the formula for !x~u gives for t 2 (0; T ]
Z t
e(t )@l~Æx ( ; :) d 0 with ~Æx = ~xP P3j=1 @j ~ j;x = (!x)~u + (~u:r~ !x)~u (r~ !x:~u)~u !x(~u:r~ )~u ~ P 2 3j=1 @j !x @j ~u. Let us write Bx = B (0; 8jxj). We have !xr @l !x(y)~u(t; y)
k~ÆxkL
3=2;1
=
@l et
(!x~u(0; :)) +
C k~ukL (B ) jxj 2 jxj3=2 + k~uk2L (B ) jxj 1 jxj +k~ukL (B )kr~ ~ukL (B ) + kr~ P kL (B ) +kr~ ~ukL (B )jxj 1 jxj1=2 : 6;2
6;2
6;2
x
2
x
2
x
3=2;1
x
x
x
Thus, sup0tT k~Æx(t; :)kL = o(jxj). R 1 p We again use Proposition 5.3 to get R0 ket f kL dt C kf kL (IR ) R T R t (t s) and 0 k 0 e f (s; :) dskL dt C 0T kf (s; :)kL ds. We nd that the local tendency w~ x(t; y) = !x(y)~u(t; y) et(!x~u(0; :)) satis es RT R C 0T k~Æx(t; :)kL dt 0 k@p w~ x (t; y)kL dt CT sup0
3;1
3;1
3=2;1
3;1
3=2;1
3=2;1
Z T
0
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k@p et(!x~u(0; :))kL dt CT 1=4 k!x~u(0; :)k2 = o(jxj): 3;1
3=2;1
3
372 Energy inequalities Thus, we nd that R0T k@p !x~u(t; :)kL dt = o(jxj). On [0; T ] B(0; 2jxj), we have 1 j@p !x~u(t; :)( (T ) y)j; j@p U~ ((T )y)j = (1t)2 j@p~u(t; ((Tt)) y)j (0) 2 (t) 3;1
hence, k@pU~ kL (B(0;2(T )jxj)) T 1(0) R0T (t)k@p (!x~u(t; :))kL ~ ~ = 0. This gives that limR!1 kr U k R 3;1
3;1
2
dt
= o(jxj).
L3;1 (B (0;R))
Step 6: Local estimates for P We have !xP = 1 ((!xP )) = 1 ( P !x + !xP + 2r~ :(P r~ !x)), and thus k!xP k1 C k(!xP )kL3 2 1 PC 0 (kPPkL3 1(B )jxj 1 + kP kL3 2 1(B ) + kr~ P kL3 2 1(B ) jxj 1 ); since P = 3j=1 3k=1 @j Uk @k Uj , we have = ;
= ;
;
= ;
x
x
x
kP kL
3=2;1
~ U~ k2L (B ) = o(jxj2 ): (B ) C kr 3;1
x
x
Thus, we nd that limx!1 Pjx(jx) = 0. 2
Step 7: End of the proof
We may now papply Theorem 34.1 and we get that U~ ~ kU kL2 (B(0;R) is O( R) as R goes to +1, this gives U~ = 0.
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is constant. Since
Chapter 35 Singular initial values 1. Allowed initial values
The Navier{Stokes equations have a long history. Classical solutions were studied by Oseen at the beginning of the 20th century [OSE 27]. Then Leray introduced the notion of weak solutions [LER 34] and proved by using weak compactness the existence of global solutions associated with an initial value in (L2(IR3 ))3 (see Chapter 14). Uniqueness in the class of Leray solutions remains an open question. Thirty years later, Kato and Fujita introduced mild solutions associated with more regular initial values [FUJK 64] (i.e., initial values in (H s (IR3))3 with s 1=2) (see Chapter 15). Then several authors tried to eliminate regularity in the search for mild solutions: Fabes, Jones, and Riviere [FABJR 72] and Giga [GIG 86] worked in LptLqx spaces, then Weissler [WEI 81] and Kato [KAT 84] worked in Lebesgue spaces (see Chapter 15), the next step was Morrey{Campanato spaces (Giga and Miyakawa [GIGM 89], Taylor [TAY 92], Kato [KAT 92], Federbush [FED 93], Cannone [CAN 95]) (see Chapters 17 and 18), then Besov spaces with negative regularity (Cannone [CAN 95], Kozono and Yamazaki [KOZY 97]) (see Chapter 20). All these spaces in which mild solutions are constructed are contained in the Besov space B11;1 (or, when searching for global solutions, in the smaller space B_ 11;1 ). This is related to the scaling of the Navier{Stokes equations: if ~u(t; x) is a solution2 on (0d ; T ) IRd with initial value ~u0, then ~u(2 t; x) is a solution on (0; T= ) IR with initial value ~u0(:). Thus, a theorem of existence up to a time T modulo a size condition k~u0kE C (T ) would be consistent with an estimate for dilations of the type sup0<1 k~u0(x)kE k~u0kE .pIf, moreover, e is a bounded operator from E to L1 , we nd that supt2(0;1) tketf k1 kekL(E;L1) kf kE , hence E B11;1 . There is a strong suspicion that the problem of nding mild solutions in the space (B11;1)d is ill-posed (see the recent remarks of Montgomery-Smith [MOS 01] on a simpli ed model). Auscher and Tchamitchian [AUST 99] proved that the model of adapted spaces developped by Cannone [CAN 95], Meyer and Muschietti [MEY 99] and Furioli, Lemarie-Rieusset, and Terraneo [FURLT 00], which was essentially based on a Littlewood{Paley decomposition of the1;1initial value, could not be extended to the limit case of the Besov space B1 , nor to the smaller space 375 © 2002 by CRC Press LLC
376
Conclusion bmo 1 . Auscher and Tchamitchian's result did not close the question of searching weaker conditions to grant the existence of mild solutions, since at the very same time Koch and Tataru [KOCT 01] were able to1 dprove the existence of mild solutions associated with initial values in (bmo ) . While Auscher and Tchamitchian focused on a priori estimates obtained by looking rst for estimates in the space variable, then integrating in time, Koch and Tataru obtained a more precise estimate by integrating rst in time, then estimating the size of the result in the space variable. Such an approach is customary for other nonlinear equations (as the Korteweg de Vries equation, for instance). In a similar way, one may rst integrate in time then estimate the size of the result in the frequency variable, as Chemin and co-workers do in some limit cases (see for instance Chemin and Lerner [CHEL 95] or Chemin [CHE 99]). Thus, bmo 1 has replaced B11;1 as the maximal space for mild solutions. The Besov space B11;1 is still used in some results as a tool to prove uniqueness or stability in smaller subspaces, as in the uniqueness theorem of Chemin [CHE 99] (see Chapter 28) or in the stability theorem of Cannone and Planchon [CANP 00] (see Chapter 29). p Similarly, the estimate limt!0 tk~uk1 = 0, which is satis ed by mild solutions associated with smooth initial values, may often be used to prove uniqueness of solutions in subcritical classes (for example, if X = (Lp((0; T ); E1))d, with p > 2 and E1 the closure of the testd functions in the Morrey space of 1 uniformly locally nite p measures Mulocp(IR ), and if Y1 = f~u 2 (C ((0; T ) d d IR )) = sup0
In most cases, when studying the existence of mild solutions associated with a Banach space E d of initial values, we used regularizing properties of the heat kernel and obtained estimates not in L1((0; T ); E d) but in the smaller space L1 ((0; T ); E d) \ Y1 . We discussed the properties of the solutions in the whole class L1((0; T ); E d) at only two points: when discussing the generalization of Kato's theorem to shift-invariant Banach spaces of local measures (Theorem 17.1) and when discussing uniqueness of solutions in C ([0; T ); E d) (Chapter 27). When considering the uniqueness problem in C ([0; T ); E d), we had to assume that E was continuously embedded into L2loc (in order to give sense to the nonlinear term r~ :~u ~u in the di erential equation); then, the scaling condition sup0<1 k~u0(x)kE k~u0kE and the assumption that E is a shift-invariant Banach space of test functions give2;dE m2;d, the closure of the test functions in the Morrey{Campanato space M . When E is supercritical (E M 2;q with q > d or sup0<1 k ~u0(x)kE k~u0kE with 2 [0; 1)), the uniqueness is proved by 1a=2direct computation based on the estimate ke(t s)IPr~ :~u ~vkM d= (2 q ) Cq jt sj (1 + jt sj )k~ukM k~vkM . When E is critical, we nd an 2;q
2;q
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2;q
Singular initial values 377 estimate ke(t s)IPr~ :~u ~vkE = O(jt sj 1)k~ukE k~vkE , which we cannot integrate. This diÆculty may be circumvened in three ways: i) by using the maximal LpLq regularity of the heat kernel: the integral we deal with is considered as a singular integral and studied through the theory of Calderon{Zygmund operators (Chapter 7). The control on the size of the integral is provided by the oscillations of the kernel, not only by its decay. This approach of Monniaux [MON 99], extended by May2;d[MAY 02], allowed one to prove uniqueness in spaces close to the limit case m { in fact, in the closure of the test functions in the multiplier space X 1, hence, in mp;d for every p > 2 (see Chapters 27 and 21). ii) by using Besov spaces Bps;q with q = 1: the integral we deal with is considered as a sum of bandpass lters focusing on separate bands; since we do not sum the contributions of di erent frequencies (q = 1), we suppress the probR t ds lem of the divergence of the integral 0 t s . This was the approach of Furioli, Lemarie-Rieusset, and Terraneo [FURLT 00]. iii) by considering the integral as a representation in a sum of Banach spaces A0 + A1 . Then, the use of the J -method of real interpolation of Lions and Peetre (Chapter 2) will allow an estimation in [A0 ; A1];q with q = 1. The choice of q = 1 suppresses the summation over the whole line when estimating the A0 norm or the A1 norm. This approach may be used with Besov spaces (see Lemma 27.3) or with Lorentz spaces (see Lemma 27.6). The use of Lorentz spaces in this setting was proposed by Meyer [MEY 99]. The role of real interpolation spaces with q = 1 has been underlined as well in the setting of Kato's theorem for critical spaces of local measures (Theorem 17.1). Moreover, the role of Besov spaces or Lorents spaces may be viewed as borderline cases of the maximal LpRLt q regularity theorem. Indeed, this maximal regularity theorem states that k 0 e(t s)f dskL L C (p; q)kf kL L for 1 < p < 1 and 1 < q < 1. For theR t limit cases p = 1 or p = 1, we may use Besov Rspaces and Lorentz spaces: k 0 e(t s)f dskL1B_ 1 C (q)kf kL1B_ 1 and k 0t e(t s)f dskL RB_ Cp(q)kf kL B_ , or, for d=(d 1) < q < 1 and 1 =q = 1=r 1=d, k 0t e(t s) f dskL1L 1 C (q)kf kL1L 1 and R p k 0t e(t s) f dskL L C (q)kf kL L . Notice that the last estimate was useful in proving Tsai's theorem [TSA 98] in Chapter 34. p t
q x
t
1 t
s;1 q
1 t
t
q;1
1 t
s; q
q x
t
s; q
s;1 q
t
1
p t
q;
t
r;
r;1
3. Mixed initial values
Leray's weak solutions are provided by energy inequalities and weak compactness. Kato's mild solutions are provided by the Picard iteration scheme. There is a possibility of combining the two methods to prove the existence of weak solutions in new classes. For instance, Calderon [CAL 90] and LemarieRieusset [LEM 98a] discussed the case of an initial value in (Lp)d with 2 < p < d. Their idea was to split the initial value ~u0 into the sum of two divergence-free vector elds ~v0 and w~ 0 with ~v0 in (Ld)d with a small norm and w~ 0 2 (L2)d.
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378 Conclusion One may then compute by Kato's algorithm a solution ~v 2 (Cb([0; 1); Ld))d to the Navier{Stokes equations with initial value ~v0 : 8 ~ :~v ~v < @t~ v = ~v IPr ~ :~v = 0 r : limt!0 ~v = ~v0 and, thereafter, prove by Leray's method the existence of a solution w~ 2 \T >0 (L1 ((0; T ); L2) \ L2 ((0; T ); H 1))d to the perturbated equations 8 ~ :(w~ w~ + ~v w~ + w~ ~v) ~ = w~ IPr < @t w r~ :w~ = 0 : limt!0 w~ = w~ 0 This result is easily extended to a large class of singular initial values: Theorem 35.1: (Singular initial values) Let X~r be the the closure of the test functions in the multiplier space Xr
and let E2 be the the closure of the test functions in the Morrey space L2uloc . ~ :~v0 = r ~ :w~ 0 = 0, w~ 0 2 (L2 )d and ~v0 2 A) Let ~u0 = ~v0 + w~ 0 with r 1+ r; 2 = (1 r ) d (BX~ ) and 0 < r < 1. Then, there exists a weak solution ~u 2 2 \T >0 L ((0; T ); (E2 )d ) for the Navier{Stokes equations on (0; 1) IRd so that limt!0+ ~u = ~u0. ~ :~v0 = B) If d = 3, the same conclusion holds when ~u0 = ~v0 + w~ 0 with r 1+ r; 2 = (1 r ) r~ :w~ 0 = 0, w~ 0 2 (E2 )3 and ~v0 2 (BX~ )3 and 0 < r < 1. Moreover, ~u may be chosen to be suitable, in the sense of Ca arelli, Kohn, and Nirenberg. r
r
We rst prove (A). The theory of mild solutions in Besov spaces of negative regularity (Chapter 20) allows us to compute by Kato's algorithm a solution ~v to the Navier{Stokes equations with initial value ~v0; this solution will be de ned on a strip (0; Td ) IRd and satisfy the following regularities: i) ~v is smooth on (0; T ) IR ii) for all t 2 (0p; T ), ~v(t; :) belongs to (C0(IRd))d iii) sup0
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Singular initial values 379 This has been proved in Chapter 21 (Theorem 21.3). Thus, we proved the existence of a solution at least for small times. To get a global solution is an easy task: when we have a solution ~uN on (0; TN )IRd, we choose SN < TN and we write ~u(SN ) = ~v(SN ) + w~ (SN ) where ~v(SN ) and w~ (SN ) are divergence free, ~v(SN ) 2 (C0 )d and w~ (SN ) 2 (L2 )d . Then, the approximation lemma (Chapter 12) allows us to split ~v(SN ) into ~ N + ~N where ~ N and ~N are divergence free, ~ N 2 (C0 )d with an arbitrarily small norm and ~N 2 (L2)d. We may compute by Kato's algorithm a solution ~vN +1 to the Navier{Stokes equations on (SN ; TN +2) with initial value ~ N . We then use Leray's method to prove the existence of a solution w~ N +1 2 (L1 ((SN ; TN +2); L2) \ L2((SN ; TN +2); H 1))d to the perturbated equations (with perturbation IPr~ :(w~ N +1 ~vN +1 + ~vN +1
w~ N +1 )). If we de ne ~uN +1 by ~uN on (0; SN ] and by ~vN + w~ N on (SN ; TN + 2), we still have a solution of the Navier{Stokes equations: by weak continuity of t 7! ~uN (t; :), we may di erentiate by pieces the distribution ~uN +1. We now prove (B). This is the very same proof as for (A), following the lines of Chapter 32 and 33. We may use a smoother patching in the de nition of the global solution, using the suitability of the solution and Serrin's uniqueness theorem to enforce ~uN and ~vN +1 + w~ N +1 to coincide not only at t = SN but on a strip [SN ; SN + ) (see Chapter 33).
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