Advances in nonlinear partial differential equations and related areas : a volume in honour of Professor Xiaqi Ding

0. We show the effect of the strict local maxima and minima of Q(x) on the existence of multi-peaked positive solutions as h becomes small. Using variational methods, we construct positive solutions condensing at the extremal points of Q(x). An explicit expression for the dominant parts of the solutions is also obtained.

1

Introduction and Statement of the Main Result

The aim of this paper is to establish the existence of single and multi-peaked positive solutions to the following problem -h2Au + u = Q(x)up-1, lim u{x) = 0,

xeKN,

(1.1) (1.2)

|x|—>oo

where A is the Laplace operator, Q(x) is a positive function of locally Holder continuous on RN(N > 1), 1 < p < $±§ if TV > 3; 1 < p < +oo if TV = 1, 2, and h > 0 is a constant. The existence of positive solutions of (1.1)-(1.2) and its various generaliza­ tions have been studied by many authors in the past twenty years. In addition to the papers mentioned below, the readers may consult the survey articles [9,11] and the references cited therein. Most of them provide the existence of solutions for arbitrary h > 0. When Q(x) has one or more nondegenerate critical points, Cao, Noussair, and Yan [5] established single-peaked and/or multi-peaked positive solutions concentrating near a neighborhood of those points. In [4], Cao and Noussair obtained the existence of k positive solutions with energy close to that of leastenergy solutions, when Q(x) has k global maxima and ft > 0 is sufficiently small.

16

It is then natural to ask whether we can still establish the existence of single-peaked and/or multi-peaked positive solutions, when Q(x) has one or more local maxima or minima which may be degenerate critical points. Similar problem was considered by Gui in [8]. Following the methods originally intro­ duced by Sere [17] and Coti-Zelati and Rabinowitz [7], Gui used the method of minimaxing the penalized functionals for a special family of "mountain passes" in [8] and obtained the existence of multi-bump solutions for the problems sim­ ilar to (1.1)-(1.2). The existence of periodic solutions was also considered by Alama and Li [1] using similar arguments of [17,7]. The purpose of this paper is to answer the question raised above. To state our result more precisely, some preparations are in order. First we recall some known facts about positive solutions to the problem -Au + u = up~\ xeRN, lim u(x) = 0, u(0) = maxxeKNu(x).

(1.3) (1.4)

|x|-»oo

It is proved that the problem (1.3)-(1.4) has a unique positive solution [/(ground state) satisfying that U is spherically symmetric and its first deriva­ tives decay exponentially. Next we introduce some notations. For U , I J G i7 1 (R i V ), let (u, v)h = / (h2 Vu\7v + uv),

(1.5)

INlX = /(ft 2 |Vu| 2 + |ti|2).

(1.6)

Denote r m m = {a € R N

Fmax = {a e K

| a is a strict local minima

ofQ(x)},

| a is a strict local maxima

ofQ(x)}.

Define, for a,y G R ^ ,

Eh,a,y = {ue &{*") I (u,u(x~l~y))h {U

'

dXi

= o,

)^ = 0,i = l,.--,N}.

The main result in this paper is the following.

(1.7)

17

Theorem 1.1. Suppose there exist k distinct points a 1 , • • • , afc such that {a , • • • ,ak} C Tmin or {a 1 , • • • ,a fc } C Tmax. Then, for h small enough, (1.1)(1.2) has a positive solution Uh of the form 1

uh

= J2
a

[ yt*)+u,h{x),

^

where ylh € R ^ , ^ > 0 , t = l , " - , t , ( j / l e rii=i^M',j/i» ajl->(Q(ai))"^2,

(1.8)

'

|i£|—>0,

5WC

^ ^ a * ' ^ o r ^ "^ °'

\\u;h\\h = o(h%).

Our strategy in proving Theorem 1.1 is to use a type of Lyapunov-Schmidt reduction to reduce the problem we are dealing with to a finite dimensional one and then solve the finite dimensional problem by considering an auxilinary minimization or maximization problem. Our method is a combination of the variant of those used in [2,15] and energy comparison used in [12,13]. Throughtout this paper all integrals are over R ^ . We shall use the same letter C to denote various generic positive constants and use 0{t) to mean \0(t)\ < C\t\. o{\) will denote various quantities that tend to zero as h —> 0. 2

Proof of the Main Result

We shall only give the proof of Theorem 1.1 for the case k = 2 since the case k = 1 is similar but simpler and the case k > 3 can be proved similarly.By change of variables x —> x/h we see that Uh is a solution of (1.1) (1.2) if and only if Vh = Uh(x/h) is a solution of -Au + u = Qh\u\p~2u, lim u(x) = 0,

xeRN,

(2.1) (2.2)

|sc|—>oo

where Qh(%) — Q(hx). Corresponding to (•, -)h,Eh,a,y defined in Section 1, we define < u, v >= / (VuVv + uv), and let ||.|| be the usual norm introduced by (•,•). Fh,aty = {ue HX{RN)

| (u,Uhiaiy)

= 0,

( t i , ^ ^ > = 0 , » = l,-,JV}>

(2.3)

18

where Uh^y = U(x - S±*). Theorem 1.1 is equivalent to the following result. Theorem 2.1. Under the same assumptions of Theorem has a positive solution Vhof the form

Vh

1.'1,(2.1)(2.2)

(x)=£aiUh9aiiyih+uh

(2.4)

i=l

with alh > 0, ylh G R

, i = 1, • • •, k, Uh G f]i=1 Fhaiyi

satisfying as h —> 0

ai->(Q(a'))-^,

(2.5)

\V\\ "► 0,

(2.6)

IKII -»• 0.

(2.7)

Let a 1 , a 2 be two points in Tmin or r m a x , cr0 = (§fet)

P

and £ > 0 be a

number to be determined. Denote by Br the open ball in R ^ of radius r > 0 centered at the origin. Our arguments are based on establishing the existence of critical points of the functional Kh : H1(RN) \—> R defined by

KM

_ J(iv~l' + lf) (JQM'V

via seeking critical points of Jh{yl,y2,cr,uj)

= Kh(Uhjaijyi

+ (cr0 + o-)Uh,a2,y2 + ^)

(2.9)

in the manifold M* = {

(z/1,?/2,^,^) | (y\y2,a) 2

G A^,

2=1

where TV* = {(z/1,2/2, cr)!^^ e Bs,i = l , 2 ; a G R , | a | < 6} The one to one correspondence of critical points of Jh in M$ and Kh in jff 1 (R JV ) when h and £ are small enough has been established in [5].We note that (y1 ,y2 ,a,u) G Ms is a critical point of Jh if and only if there exist

19

(ai, a 2 ) G R 2 , 0% = (/?j, • • •, /3^) G R N , 2 = 1,2 satisfying the following ' ^

= 0,

(*)c

TV

(*) { 2=1

2 = 1 Z = l

y i

Similar to [5],our approach in solving (*) is a combination of the implicit function theorem method used in [2],[16] and the energy comparison method used in [12,13].We first solve (*)u; and (*)a for each fixed pair (y 1 ,?/ 2 ) G B2§ x B25 by arguments used in [16] and then solve (*)yi via considering the following minimization problem (if a 1 , a 2 G Tmin) (**) \ni{Jh{y\y\a{y\y2),u{y\y2))

I (vl,V2)

eBsx

Bs }

or via considering the following maximization problem (if a1, a2 G Tmax) (* * *) s u p i J ^ y 1 ^ 2 , ^ 1 ^ 2 ) , ^ 1 ^ 2 ) ) | (y\y2)

eB6x

B6}

where a(yx,y2),uj(yl,y2) are determined by (*)
(2.11)

for u = (t,uj),v = (s,w) in Fh^yijy2 and endow Fhjyi^y2 with the norm intro­ duced by [•,•]. Let Qaiiyi = Q{a{ + y{) - Q{al),A = / ( | W | 2 + U2) = / E7*. The first preliminary result is the following which enable us to reduce our problem to a finite dimensional one. Proposition 2.2. There exist h0 > 0, S0 > 0 such that for h G (0, ho], S G (0,<J0] there is a uniqueCl map {y1 ,y2) G B2SxB2S '—> (°h{yl,y2),uh{yl,y2)) G i^ ) 3 / i } 3 / 2 si/c/i £/m£

dJ

for all w G Fh,ai,yi

h(y\y2,Vh(y\y2),Vh(y\y2)),,

^Q

,212,

C\Fh,a2,y2,

dJh(y\y2,crh{y\y2),ujh(y\y2)) <9cr

= 0

,2 1 3 x

20

Furthermore we have 2

2

2

\\My\y )\\ + \Mv\v )\ = o ( £ \Qai,yi\) + o(i).

(2.14)

2=1

Proof. Denote (

t>h{y1,y2)

- Uh,ai,yi

+voUhi(l2:y2

and expand J/^y 1 ,?/ 2 ,^,u) with respect t o u = (cr,u;) at ^ = (0,0):

Jh{yl,y2,cr,u) = Jh(y1,y2,0,0) + fh,y\yz(u>) + Q/i,2/i,y2(u) + Rhjyiiy2(u)

(2.15)

where fh,yl,y2 (u) is the linear ternijQ/^i )2/ 2 (ix) is the quadratic form and Rh^1,y2 (u) collects the remaining higher order terms.

(IQHK)+'

J

(fQ»KV

-Uf

a) j Qh^Uh,**^},

(2.16)

here and in the sequel

Qh,y\Au)

= Qly^y2(uj)

(2-18)

+ Qltylty2((T) + Qliyliy2(*,Uj),

with 2 2

U Qh(f>h)"

J

Hp+2)

(

n

(219)

m^ I^

(/Gfc#)

"

-

J

21

~Mh,Uh,a2,y2) / Qh4>ph 1uh>a2^

j ' Qhr2uia2>y2

-(jp-i)uhf

+ lp+2)lh{4>h)(jQh
Qfc,»i,»»(*»<") =

r/^

g<7

°

1+a{-

4

},

(2.20)

< ^ , ^ , a » , » 0 / Qft0fc_1W

-2(p-l)||^||2|Q^-2^)a2,y2a;

J Qkfi-1*},

+2(p+2)lh(h) J Q^U^y*

(2.21) J2fciWi,wa(u) = 0(||«||S- +1 ) >

(2.22)

< y », y 2(«) = 0(||«||S-),

(2-23)

1

<„!,„>(«) = OdMIS-- )

(2.24)

with p* = min{p — 1,2}. To estimate fh^1^2 we need to establish estimates of each term in (2.19).

+0{j{Ul^yiUh^,y,

j Q»KXy^ = j ^

h x + ai +

+ Uh^,viU^-ltya)\w\). 1

»') - Q(ai *))U{x)>- u>(x +

= 0(|g„S s < |)IM|+o(l)|M|,

(2.25)

?-^L) (2.26)

which together with (2.25) implies 2

r

/ Q^r'iU

= 0 ( £ \Qa',y* DIMI + 0(1)\\W\\.

(2.27)

2=1

**

JQHK

=

jQhUlaKyl+ap0JQhUla^2+o(l)

= [Qia1) + ap0Q(a2) + Qal + ap0Qa^y2 + o(l)}A. (2.28)

22

(h,Uh,a*,yz) = J Kj,yiUh,a*,yZ

+ °0 J K,a2,y*

= a0A + o(l),

(2.29)

a

i U)_ J|v^l + # M fc) * " SQ^h

+0(Y/\Qaityi\2)

+ o(l)}.

(2.30)

2=1

/ QhVuh^,y.

= Q{a2)al~lA

+ o J - ' Q ^ , ^ A + o(l).

(2.31)

(2.27)-(2.31) yield 2

l/ fc ,»i,^(«)l < C ( X ; i Q a ' ^ l + o(l))IHI.-

(2-32)

2=1

By the same argument as in [5] we can obtain a positive constant p inde­ pendent of yl,y2 such that Qlyl,y2(u)>p\\LU\\2

(2.33)

for all u e Fh^ai,yi H^M 2 ,*/ 2 Similarly, by direct computations we have

{ {P 2) Qly*A°) = (n( (7l + a ) ^-^ - ~ {Q{a S ))p(l l

0

2

+OC£\Qa<,v'\)+o{l)},

(2-34)

2=1

2

\Ql„wM\
+ o(l))\W\.

(2.35)

2=1

Since fhy\y2 is a bounded linear operator in the space F/^1^2,there exists a unique element A,3,1,3,2 £ Fh^y\^y2 such that A,?,1 ,y2 (w) = [A>yl ,3,2 , U]

(2.36)

23

for all u e Fh^yi^2 and by (2.32) 2

l l A ^ I I * = ° ( E l ^ , ^ l ) + o(l).

(2.37)

i-l

Also there exists a unique element qh,yi,y* £ Fh^ai^yi P|^)i,a2,y2 such that 0 3 h,yi^(l ) w) = (g M i >y 2,w)

(2.38)

for all CJ € i ^ , ^ , ^ f]^,a 2 ,y 2 and by (2.35) 2

ll<7M*,y»II = 0 ( 5 Z \Q^,r I) + o(l).

(2.39)

By (2.33) there exists a unique symmetric continuous and coercive operator \,y\y2

:Fh,a\y^

f]Fh,a2,y2

'

> -F/i.a1,!/1 f)Fh,a2,y2

Such t h a t

A

Ql

yi,y2(u)

= (Alyl^u,u)

for all u e Fh^aijyi f|^,a 2 ,t/ 2 Define Ah^yiy2 : Fh^y\^y2 \—> Fh^yi^y2

(2.40)

as follows

2

^,yi,l/ W= W l y i ^ ^ i j / i ^ w + ^ y y )

(2.41)

for w = (
(

5/i,y 1 , 2 / 2 ( C r )'

Then Qh,yi,y*(u) = [Ahjyiiy2U,u]. Let 1?^ 1

2

: Fh^yijy2 i—>> F^^i^ B

Mi,y* w =

(2.42)

be defined as follows faOL^'^Ly1,*2")

(2.43)

for w = (cr,a;), B

ly\y*U

= (Ah,y\y* ~ Bh,y\y*)U

= (M^y1,*/2)-

Then l l ^ . y 1 ^ 2 ! ! * = lltf/1,1/1 ,y2H

(2-44)

24

which can be as small as one wants provided S and h are small enough. So, exists and can be represented by A

W,v>

= (S/Lvi.v') - 1 £ ( ( * L ^ r X y ^ ) \

Ah*12

(2-45)

2=0

which yields that H ^ 1 ! 2 || < C for some positive constant C independent of h^y1 ,y2. We can now complete the proof by arguing as in [5]. Proposition 2.3. Suppose a1, a2 G Tmin{resp.al ,o? G Tmax).Let a = Ohiy1 ,y2),w = u)h{yl,y2) be as in Proposition 2.2.Then,for h small,the follow­ ing problem max{Jh{y\y2,ah{y\y2),uh{y\y2))

\ (y\y2)

G Bs x Bs}

(2.46)

min{Jh(y\y2,ah(y\y2),ujh(y\y2))

\ (y\y2)

G B5 x £*}

(2.47)

h > 0 is sufficiently

small.

(resp.

j /ms a solution (y^y^) Furthermore as h —> 0

€ Bs. x Bs. provided

toil, li£|-+o. Proof. Consider the case a1,a2 G r m i n .Setting i ^ = we have My1,y2,Th(y1,y2),cjh(y1,yi))

= Jh(y1,y2,0,0)

+fh,yi,v2(uh)

{ah{yl,y2),uh(yl,y2)) +

= (d+aoW-t Q-i (a>){i -

0(\\uh\\l) [Q

«;-: + 2 ff y

(H-ag)Q(a 1 )

2

+O(£|g oii!/i | 2 ) + o(l)}-

(2.4!

i=l

From (2.49) we can see that problem (2.47) has a solution which we denote by (yLyl)Jt follows from Jh{ylyl°h(ylyh),uh(yly2h))

> A(o, 0 , ^ ( 0 , 0 ) ^ ( 0 , 0 ) ) - ( ( l + a 0 ) A ) 1 - f g - i ( a 1 ) { l + o(l)} (2.49)

and the fact that a 1 , a 2 are strict local minima that \y\\ -> 0, \y%\ -» 0 as

25

Having established all the preliminaries we are now in a position to prove Theorem 2.1. and u Denote yl^yl^hiyhvl) h{ylh^y2h) obtained in Proposition 2.3 by l 2 y ,y ,a and u simply.It follows from (2.12) that there exist (a\, a 2 ) GR 2 ,/5 2 = (0{, • • • ,/?V) € R N ( i = l , 2 ) such that

OUJ

ayl

i=l 1=1

i=l

that is, (*)u is satisfied. Since (a,uj) is a solution of (2.47)(resp.(2.48)) we have for i = 1,2,/ = 1,-..,JV DyiJh(y1,y2,a,u)=0, (2.51) where r / 1 2

n

dJh(y\y2,cr,u})

s

h da dw8 dvl

8J

J

(2.52) h

On the other hand, by ^d^da — 0, and dJh

2

du

~

y-^

du *l

j=l N

2

s^ST j=l

N

m,dlJh,ai,yi

du

s=l

&1

= -E«(^f.">

^,^), <"»

for i = 1, 2; / = 1, • • •, N, since ( — ^ ^ - , u ) = 0, which yields

(^%¥,.) = -(% dy\dy\

dy\

(2.54)

dy\

So ^ ( ^ V ^ ) dy\

=

^ j£

g ^ . ^ dy\dy\

.- = 1,2;/ = 1

JV,

(2.56)

26

that is, (*)yi in system of equations (*) are satisfied. Hence WH = Uhaiyih + {(Jhiy^yl) + ^ 0 ) ^ 2 ^ 2 +Uh(yh>yl)) i s a c r i t i c a l P ° i n t o f Kh for sufficiently small h.Therefore there is a Lagrange multiplier 6h such that

-Awh + wh = ehQh \wh\p~2 wh,

x e nN,

(2.57)

Multiplying the above equation by Wh and integrating by parts over R ^ we get ff = /|V«; fc |

"

2

+ Kl 2

SQh\wh\p

Hence Uh can be written as uh = a\Uh^aKyih -f a2hUh^^y2 + uh with a{ >0,yiheKN,Cjhe

Fhjaiiyih f]Fh,a^yl

a{ -> Q~i±*{a%

\y\\ -> 0,

satisfying for i = 1,2 \\uh\\ -> 0,

(2.58)

as h —>> 0. The positivity of ?i^ can be obtained the same as in [5]. Thus Theorem 2.1 is proved. Acknowlegment. visiting the Institute of a research fellow of the foundation for partially

Part of this work was carried out while the author was Mathematics of the University of Mainz, Germany as Alexander von Humboldt Foundation. He thanks the financial support of this work.

References 1. S. Alama and Y. Y.Li, On "multibump" bounded states for certain semilinear elliptic equations, Indiana Univ. Math. J. 41(1992), 983-1025. 2. A. Bahri and J. Coron, On a nonlinear elliptic equation involving critical Sobolev exponents, Comm. Pure Appl. Math. 41(1988), 225-294. 3. A. Bahri, Y. Li, and O.Rey, On a variational problem with lack of com­ pactness: the topological effect of the critical points at infinity, Cal. Var. 3(1995), 67-93. 4. D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in R N , Ann. Inst. H. P. Anal. Non lineaire, 13(1996), 557-588.

27

5. D. Cao, E. S. Noussair and S. Yan, Solutions with multiple "peaks" for nonlinear elliptic equations, to appear in Proc. Royal Soc. Edinburgh, Sect.A. 6. D. Cao, N. Dancer, E. S. Noussair and S.Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete and Continuous Dynamic Systems, 2(1996), 221-236. 7. V. Coti-Zelati and P. Rabinowitz, Homoclinic type solution for a semilinear elliptic PDE on R N , Comm. Pure Appl. Math. 45(1992),12171269. 8. C. Gui, Existence of multi-bump solutions for Schrodinger equations via variational method, Comm. PDE. 21(1996), 787-820. 9. P. L. Lions, On positive solutions of semilinear elliptic equations in un­ bounded domains, In: Nonlinear Diffusion Equations and Their Equilib­ rium States, Ni, W.-M., Peletier, L. A., Serrin, J. (eds.), New York,Berlin: Springer 1988. 10. M. K. Kwong, Uniqueness of positive solutions of Au — u -f up — 0 in Rn, Arch. Rat. Mech. Anal. 105(1989), 243-266. 11. W.-M. Ni, Some aspects of semilinear elliptic equations. Lecture notes, National Tsing Hua University, Taiwan, 1988. 12. W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44(1991), 819-851. 13. W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70(1993), 247-281. 14. W.-M. Ni and J. Wei, On the location and profile of spike-layer solu­ tions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math.48(1995), 731-768. 15. P. Rabinowitz, On a class of nonlinear Schrodinger equations, Z. Angew. Math. Phys. 43(1992), 270-291. 16. 0 . Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89(1990), 1-52. 17. E. Sere, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209(1992), 27-42.

28

LARGE-TIME B E H A V I O R OF E N T R O P Y SOLUTIONS IN V FOR MULTIDIMENSIONAL CONSERVATION LAWS GUI-QIANG CHEN Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730, USA E-mail: [email protected] HERMANO FRID Instituto de Matemdtica, Universidade Federal do Rio de Janeiro C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil E-mail: [email protected] Dedicated to Professor Xiaqi Ding on the Occasion of His 70th Birthday A b s t r a c t . We study the large-time behavior of entropy solutions in L°° of the Cauchy problem for inviscid scalar conservation laws in any space dimension, using the approaches developed in [2,3,4]. For the initial data that are periodic in each space variable, we show that the corresponding periodic solution u(t, x) decays to the average of the initial data over the period in Lj , oc (M n ), as t —> oo, provided that the linearly degenerate set of the nonlinear flux function has measure zero. For the initial data that are any L1 nL°° perturbation of Riemann data determining a piecewise Lipschitz Riemann solution R(£), we prove that the solution u(t, t£), con­ sidered as a function of the variables t and £ = x/t, converges to R(£) in Lfoc(Rn), as t —> oo. These results significantly extend the known asymptotic results, which mostly apply only for the one dimensional case, to the invicid multidimensional case.

1

Introduction

In this paper we are concerned with the large-time behavior of entropy solutions in L°° of the Cauchy problem for a scalar conservation law in several space TTO •pi O V) 1 QC •

f dtu + 6ivxf(u) \u\t=0=u0(x),

= 0,

x € Rn, * > 0,

(1) )

where u G l and f(u) = (fi(u), • • •, fn(u)) is a vector function in C3(M; Rn). We consider two distinct cases: the initial data that are periodic in each space variable and the ones that are L1 C\ L°° perturbation of Riemann data. For the first case, we show that the solution u(t, x) decays to the average of the

29 initial data over the period in Lf oc (M n ), as t -» oo, for 1 < p < oo, provided that the set of linear degeneracy of the nonlinear flux function has measure zero (see (8)). In the second case, we prove that the solution u(t,t£), considered as a function of the variables t and £ = x/t, converges to the corresponding Riemann solution R(£) in Lf o c (E n ), as t -» oo, for 1 < p < oo, provided that the Riemann solution is piecewise Lipschitz in £. A pair (n(u),q(u)), q — (g 1 ,.••> qn), is called an entropy-entropy flux pair for (1) if 7] and q are Lipschitz continuous and satisfy q'(u) = r,'(u)f'(u).

(2)

Definition. A bounded measurable function, u{t,x), is called an entropy solu­ tion of (1) in UT = W1 x (0,T) if, for any entropy-entropy flux pair (77, g) of (1) with convex 77(14), /

{r)(u)(j)t + q(u) • V x 0 } dxdt + /

n(uo(x))(j)(0, x) dx > 0,

(3)

for any nonnegative function (f>(t,x) 6 Co(M n + 1 ). The existence of global entropy solutions of (1) with u0 G Loc(Wl) was first proved by Kruzkov [20], by improving an earlier result of Volpert [33] for UQ G BV{Wl). The large-time behavior of entropy solutions for the one dimensional case has been extensively studied. For example, see [8,15,18,22,26] and the references cited therein and in [14,16,19,27, 29,35]. In this paper, we apply our new approaches, developed in Chen-Frid [2,3,4], to deal with the asymptotic problems for the inviscid equations (1) in any space dimension for periodic initial data and general L1 C\ L°° initial perturbations of Riemann data. In Section 2, we show how the large-time decay of L°° periodic entropy solutions of (1) is achieved in any space dimension. Here we give a direct simpler proof of the compactness result of Lions, Perthame, and Tadmor [25], which allows the application of Theorem 2.1, leading to the decay result of Theorem 2.3. The decay of entropy solutions for scalar conservation laws was also established by Lax [21] and Dafermos [7] for the one-dimensional case and by Engquist-E [11] for the two-dimensional case with periodic initial data of locally bounded variation. In Section 3, we study the large-time behavior of entropy solutions of (1), when no is a general L1 D L°° perturbation of Riemann initial data. The convergence of any entropy solution u(t, ££) to the corresponding Riemann so­ lution R(£) was established in the sense of time-average, as t —» 00, in [3] for general initial data. For initial data that are an L 1 D L°° initial perturbation of Riemann data producing a planar Riemann solution, the convergence of the

30

corresponding entropy solution to the Riemann solution was also proved in Lf 0C (R n ), 1 < p < oo, as t —> oo in [3]. Here we extend these results to general initial data that are L1 fl L°° initial perturbation of Riemann data determining piecewise Lipschitz Riemann solutions. We remark the generality of such Rie­ mann data which generate piecewise Lipschitz Riemann solutions. Efforts in studying the piecewise Lipschitz structure of multidimensional Riemann solu­ tions for (1) have been made in many recent publications (cf. [1,6, 24,34,36]). 2

Large-Time Decay of Periodic Solutions

In this section we study the large-time decay of periodic solutions of multidi­ mensional scalar conservation laws, without local BV restriction on the L°° periodic initial data UQ G L°°(Rn) with period P — IIf =1 [0,p;]: u0(x+piei)

- u0(x),

z = l,---,n.

(4)

We first recall the basic theorem of [2] (see [3] for a detailed proof). This theorem holds not only for scalar equations but also systems of conservation laws. Let u(t,x) be a periodic solution of (1) as a hyperbolic system so that u € E m and f(u) e (M m ) n . We define the scaling sequence uT, T > 0, associated with u(t, x) by uT(t,x)

=u(Tt,Tx).

(5)

Theorem 2.1. Let the system be endowed with a strictly convex entropy. As­ sume that u(t,x) £ L°°(M" +1 ) is its periodic solution, with u0 G L°°(Rn) satisfying (4), and that the associated scaling sequence uT(t,x) is compact in Llc(Mn++1). Then esslim / \u(t,x) — u\pdx — 0,

for any

1 < p < oo,

(6)

t-»oo Jp

where u is given by 1

r

\p\ JP fp

uo(x)dx.

(7)

We apply Theorem 2.1 to analyze the large-time behavior of periodic solutions. First, we investigate the L\oc compactness of the entropy solu­ tion operator of (1). In this connection, we prove the following theorem first established by Lions, Perthame, and Tadmor in [25]. Here we give a direct proof of this compactness result, motivated by their ideas. Denote a(v) = (ai(v),---,a"(v)) = (f{(v),---, / » ) .

31

Theorem 2.2. Assume that, for any (r, jfc) G E x W1, with r 2 + \k\2 = 1, meas { v G E | r + a(v) •fc= 0 } = 0.

(8)

Then the solution operator u(t,-) = StUo(-) : L°° —> L°°, determined by (1), is compact in L}oc{(0,T) x E n ) . Proof. Let u(t,x) be any entropy solution of (1) in the sense of (3). Set f{t,x,v)

= Xu(t,x)(y),

where X\(v)

= H(v)H(X -v)-

H(-v)H(v

- A),

and W

\1,

S>1,

is the Heaviside function. We first prove that such a function f(t,x,v)

satisfies

dtf(t, x, v) -f a(v) - Vzf (t, x, v) — dvm(t, x, v) in the sense of distributions in (0,T) x E n x E for certain m(t,x,v) that m(t,x,v)

(9) satisfying

is a nonnegative Radon measure in (0,T) x E n x E.

(10)

Specifically, by entropy inequality (3), we deduce dt(u(t,x)

- v)+ +divxH(u(t,x)

- v)(f(u(t,x))

- f(v)) - - m i ( t , r r , v ) , (11)

and dt(v -u(t,x))+

+divxH(v

-u(t,x))(f(v)

- f(u(t,x)))

= -m2(t,x,v),

(12)

in the sense of distributions in (0, T) x E n xE, where mi and rri2 are nonnegative Radon measures (the Schwartz lemma [30]). Here we use the notation (5)+ = sH(s), s G E. Now we proceed formally (and we will justify our approach later) in the following way. We take the derivative of (11)-(12) with respect to v to find dtH(u(t, x) — v) + a(v) - VxH(u(t, x) — v) = dvm\ (t, x, v),

(13)

dtH(v — u(t,x)) + a(v) • VxH(v - u(t,x)) = —dvm2(t,x,v).

(14)

and

32

We then multiply (13) by H(v) and (14) by H(-v), to obtain dtf + a(y) • Va-f = dvm(t,x,v)

- (mi(t,x,0)

and take their difference

-m2(t,x,0))

®5v=:o,

(15)

where Sv=0 is the usual Dirac measure concentrated at v = 0 and m(t, x, v) = H{v)m\ (t, x, v) + H(—v)rri2(t, x, v).

(16)

Now we have mi(*,x,0) = ft (*(*,*))+ + div x #(iz(t,x))(/(u(*,:r)) - /(0)), m 2 (t,a;,0) = ft(-ti(t,x))+ + divxH{-u(t,x))(f(0) f(u(t,x))), which implies mi(t,x,0)

— ra2(£,x,0) = dtu(t,x)

+ div x /(i/(^,x)) = 0.

Therefore, we have proved (9) at least formally. Now we verify that all our formal procedures can be rigorously justified with the usual arguments in the theory of distributions. The only part that requires more explanation is the use of the formula H(v)dvmi

(t, x, v) = dvH(v)mi

(t, x, v) — mi (t, x, 0) 5v=o

in the sense of distributions and the analogous one for rri2(t,x,v) and H(—v). We notice that the measures m* may be written into sliced form as m,i(t, x, v) = filv(t,x) dv1 i = 1,2, where, for each v G M, \xxv is the Radon measure in (0,T) x Mn, which is given by the same expression defining m*, when v is viewed as a parameter for i = 1,2. Therefore, we can easily justify the above formula using a mollifying sequence Hs(v) for H(y) and then taking limit in the sense of distributions when 6 —> 0. The details can easily be filled out. Given a sequence of initial data UQ(X),SUP£>0 \\UQ\\L<*> < oo, there exist uniform bounded solutions u£(t,x) of (1). It suffices for Theorem 2.2 to show that ue(t,x) is compact in Zj oc ((0,T) x Rn). Now, for each u£(t,x), we associate the function f£(t,X,v)

=Xu*(t,x)(V)>

which satisfies (9) in the sense of distributions in ( 0 , T ) x l n x E, for a nonnegative Radon measure m£(t,x,v) obtained from (11), (12), and (16) by sub­ stituting u(t,x) by u£(t,x). Note that / X\(v)i/>(v) dv = / JK

JO

ip(v)dv,

33

for any function ijj G L / 1 oc (E u ). In particular, u£{t,x)

= / f£(t,x,v)dv.

(17)

Therefore, we need to prove that / f£(t,x,v)dv

is compact in L}oc((0,T)

x En).

(18)

We notice that f£(t,x,v) = 0, if \v\ > Ro, for any R0 > 0 larger than the uniform bound of u£{t,x) in L°°((0,T) x E n ) . In particular, the integral in (18) may be taken over a fixed finite interval (—RQ,RQ). Now, from (11), (12), and (16), we see that the measures m£ have uniformly bounded total variation over any compact subset of (0, T) x E n x E. This follows from the uniform boundedness of mf and m | in Mioc((0, T)xRn xK), which, in turn, follows from the proof of the Schwartz lemma [30]. Hence, by the Sobolev embedding, we see that m£ form a compact set in Wz~c1,p((0,T) x E n x E) with 1 < p < g±f- (cf. [12]). Also, by (11), (12), and (16), we see that m£ are uniformly bounded in W~ 1 , o o ((0,T) x E n x E). Hence, using the interpolation arguments (see [28]), we obtain that m£ form a compact set in Wi; c l j 2 ((0,T)xlir x E ) . Next, we localize our problem in the following way. Given any compact set in (0,T) x E n , we multiply (9) by a C°° function that identically equals one over this compact set and has compact support contained in (0,T) x E n . For simplicity, we keep the notation f£ as the product of the original f£ by this test function. We then see that the new functions f£ satisfy an equation like (9) with m£ belonging to a compact set in W /r_1 ' 2 ((0,T) x E n x E) and have supports contained in a fixed compact set in (0, T) x E n x E. We may then write dvm£(t,x,v)

= (I - d2v)(I -

At,xy/2g£■

Then f£ satisfy dtf(t, x, v) + a(v) • V*f(*, x, v) = (I-

82V)(I - At,x)1/2&,

(19)

with g = g e belonging to a compact set in L 2 ((0,T) x E n x E). Let g£l be a subsequence of g£ converging in L 2 to certain g G £ 2 ((0, T) x E n x E). We may assume, passing to a subsequence if necessary, that f£l converges weak-star to certain function f G L°°((0,T) x E n x E) with compact support contained in (0,T) x E n x ( - i ? 0 , # o ) . Clearly, f and g satisfy (19) as well.

34

Now we prove that / f£l(t,x,v)dv

-> / f(t,x,v)dv,

JR.

in

L 2 ((0,T) x E n ) .

(20)

JH

Denote ~ the Fourier transform in (t,x).

It suffices to prove that

f F~i(T,k,v)ip{v)dv->

ff(T,k,v)il;(v)dv,

Jn

JR.

in L 2 (E x E n ) ,

(21)

for any \j) G CQ°(—RI,RI), with i?i > i?o- By Plancherel's identity, one has (20) while taking ip G CQ°(—RI, RI) that identically equals one over (—R0, Ro). Let f(t,x,v), g(t,x,v) G L 2 ( E x E n x E) satisfy (19). We take C € C£°(E) so that ( = 1 in (—1,1) and £ = 0 outside the interval (-2,2). Let S > 0 denote a number to be suitably chosen later. As in [10], we write /

fy(v)dv

= I1+I2,

(22)

JR

where

(23) For (r, k) G E x E n , with r 2 + |&|2 = 1, let //(£) be the distribution function li(t) = fiT>k(t) = m e a s { v G ( - i ? i , # i ) | |r + a(v) • A;| < * } , * > 0. Applying the Cauchy-Schwarz inequality to Ii, we obtain that, for r 2 4\k\ > 0, 2

|f|2dv)

\h\
m e a s { t ; € (-Ri,Ri)\

\T + a(v) ■ k\ < 5}1'2

On the other hand, using the Cauchy-Schwarz inequali (19) to I2, one finds

ty again and applying

1/2

\h\
7I KJ\v\
2

| g |dv\^ \%\ 2

)

2

2

2 /2 (l+T +| *\k\ (1 | 2 ) 1 )^ + T +

(25)

35

i + \k\yp + \k\*/§* M<*i I l r + a ( v ) • k\2

(/ {■ \J\v\
;

|fc|2 ; 4 \r + a(v) • A:|

I

\ k\ | r + a(v) ■ fc|6

1/2 Xl

{|r+a(v)-fc|>^}^

Now, for any TV > 0, rRi

J-Rl

d//(t) Ir + a i v ) - ^ , ^(T

1

^ ^ ^ ^ ^ ^ -

8 2

2

v1

2

2

(r + |jfc| )"/ / _ ■

JV

• '

A

"

( r 22 ++ | fc | f c|2)l/2 |2)l/2

(T

00

|i(t) t JV+1 dflC 4jJV+1

|Jfc|2 )tf/2 // + |Jfe|2)V2^JV + (rT22 ++ |Jfe|2)JV/2

JV /-00 - (r + |Jb|2)^/2 / , 2

^(t) *"+! * ^ C *

(T2 +

•

|fc|2)l/2

Therefore, from (25), we find ( 2 + |A,|2)l/2 T

1/2

|/2|
\g\2dv)

\/M<*i

(l +r 2 + | fc |2)l/2 ( J - l + | fc |J-2 + | fc |2J-3) _

(26)

/

We now replace f and g by f — f£l and g — g e/ in (24) and (26). We notice that f and f£l are uniformly bounded. Then, given 6 > 0, we can find 7 > 0 so that / I / ( f - f^)^(v) dvl 2 drd/c < J. 2 2 2 •/r + |A;| <7 7 Fixing 5 and 7 as above, we prove / \ (f-&)il;(v)dv\2 drdk^O, .Ar2 + |fc|2>72 «/

(27)

as et -> 0.

Now, for r 2 + \k\2 > 7 2 , we choose 5 = S'(r2 + I&I2)1/2, where 8' > 0 is a small number. We then get \h\
f \J\v\
\f-f'\2dv)

,

(28)

)

|/ 2 | < C ( ( ^ ) _ 1 + (^')" 2 + (5')"3)M^A;) ( / \g-&\2dv) \J\v\
,(29)

36

where h(r,k) is a function in L°°({T2 + \k\2 > 7 } ) ^ We conclude that \I2\ converges to zero in L 2 ({ r 2 + \k\2 > 7 }), since f - g £/ -> 0 in L2(R x R n ) . As for \hI, we have the following. Let tu: 5 n x [0, <%] -> K, with 0 < S'0 < 1, be given by tu(r,Jb,(J / )=Ai r,fc (* / )We claim that tu is uniformly continuous on Sn x [0,5'0]. Indeed, condition (8) implies that meas{v G (--Ri,i?i) | \T + k - a(y)\ = 6' } = 0, for any (r, A;) G 5 n and 0 < 5f < 1. This follows from the fact that {ve

(-RuRi)

I \r+k-a(v)\=6'}

= \J{ve

(-RuRi)

\r±5' + k>a(v) = 0],

(30) and, by condition (8), both sets in the right-hand side have measure zero. If (T£, ki, 5'e) is a sequence in Sn x [0,5'0] converging to (r^,, A^, 8^) G Sn x [0, <%L then it is easy to see that 1

{\Te+a(v)-ke\<6'£}

-> 1 { | r u , + a ( v ) - f c u , | < ^ } 5

a.C in ( - i ? i , i ? i ) .

By the Dominated Convergence Theorem, we then get the continuity, and consequently the uniform continuity, of w(r, k, 5') on Sn x [0,6'0]. In particular, we have sup nT>k(5')->0, when 6' -> 0. (r,k)eSn

Therefore, we conclude that | i i | is bounded in L2({r2 0(5'), where we also used the fact that

+ \k\2 > 7}) by an

1 / *

If

\f-P~i\2dv

\J\v\
is uniformly bounded in L2(R x E n ) . Since 6' > 0 can be taken arbitrarily small, we obtain that f(f-

F')tl>(v)dv -► 0,

in L2{{T2

+ |A:|2 > 7 } ) .

This fact together with (27) gives the desired convergence (21) as si ->• 0. Consequently (20) follows by Plancherel's identity. This concludes the proof of the theorem. □

37

Remark 2.1. Condition (8) is weaker than the following generalized genuine nonlinearity condition: meas{ v G K | k • f"(v) = 0 } = 0,

for all \k\ = 1.

(31)

This is a consequence of the fact that the derivative of r + k • f'{v), viewed as a function of v, i.e., k • f"(v), equals 0 a.e. on the set where r + k • f'(v) = 0, from a well-known result of real analysis (see, e.g. [13]). Remark 2.2. For n = 2, condition (31) is clearly satisfied by the vector func­ tions f(v) — (v\v\p,v\v\q), with p ^ q and p, q > 1. We then have the following immediate corollary of Theorem 2.2. Corollary 2 . 1 . Assume that UQ satisfies the periodic condition (4) and that condition (8) holds. Let u(t,x) be the entropy solution of (1) in E ^ + 1 . Set uT(t,x) — u(Tt,Tx). Then the self-similar scaling sequence uT is compact in Llc((0,oo)xW). Proof. Since u(t,x) e L°°(IR^ +1 ), then I|UT||LOO < C < oo,

where C is independent of T. Since u(t,x) is the periodic entropy solution, it satisfies dtri(u) + Vx • q(u) < 0 in the sense of distributions, for any convex entropy-entropy flux pair (rj^q). Thus uT also satisfy dtv{uT) + V x • q{uT) < 0, which implies that uT(t, x) is a sequence of entropy solutions of (1) in [0, oo) x Rn with oscillatory initial data u0(Tx). Theorem 2.2 implies the result we expected. □ Corollary 2.1 together with Theorem 2.1 yields the main result of this section. T h e o r e m 2.3. Let u(t,x) be the entropy solution of (1) in [0, oo) x Rn with uo satisfying (4)- Assume that condition (8) holds. Then esslim \\u(t, •) -

U\\LP(P)

where 1 < p < oo and u — rpr Jp UQ(X) dx.

= 0>

(32)

38

This can be seen as follows. From Theorem 2.1, it suffices to show that the corresponding self-similar scaling sequence uT(t,x) = u(Tt,Tx) is compact in L L ( M + + 1 ) - T h i s directly follows from Corollary 2.1. 3

Large-Time Behavior of General Entropy Solutions in L°°

We are now concerned with the large-time behavior of entropy solutions of the initial value problem for scalar conservation laws (1), with the initial data u0 G L°°(Rn) satisfying u0(x) = Ro(x/\x\) + Po(x), for a.e. a ; E l n , R0 G L ^ S " " 1 ) , P0 G L1 H L°°(M n ). Denote by R(x/t)

,^\ (M)

the unique entropy solution of (1) with initial data: u\t=o = R(x/\x\),

and call it the Riemann convergence u(t,tt;) —> average. Theorem 3.1. [4]. Let Riemann solution of (1)

(34)

solution of (1) and (34). In [4], we established the i?(£)> m ^ / / 1 o c (^ n )' as t -> oo, in the sense of timeu be the entropy solution of (1). Let R(x/t) be the and (34)- Assume uo G L°° satisfies (33). Then

lim ^ / \u{t,tO-R(0\dt T->oo 1 J0

= 0,

fora.e.^eW1.

(35)

We now prove that the convergence in the sense of (35) actually implies the convergence in the usual sense as t goes to infinity. This is given by the following result, which is motivated from Theorem 2.1 and [2,31]. Theorem 3.2. Let u(t,x) be the entropy solution of (1). Let R(£), £ = x/t, be the Riemann solution of (1) and (34)- Assume R(£) is piecewise Lipschitz continuous in the variable £, in the sense that R G jBV/0C(Mn); the closure of the jump set has measure zero, and outside the null set the first-order derivatives of R(£) are uniformly bounded. Then, for any 1 < p < oo, esslim / \u(t, t£) - R(0\pd£

= 0,

for any 17 <e

(36)

Proof. Let (ry(ix), q(u)) be a strictly convex entropy pair of (1). Denote (a(u, i>), P(u,v)) a family of entropy pairs, parametrized by v and formed by the quadratic parts of n and q at v: a(u,v) — 7](u) — r)(v) — Vr)(v)(u — v), 0{u,v) = q(u) - q{v) - Vn(v)(f(u) - f(v)).

39 Since u is an L°° entropy solution of (1), one has dtniu) + V* • q(u) < 0

(37)

in the sense of distributions. Let J be the closure of the jump set of R(£), which is a null set in the £-space by assumption, and B be any open ball in Rn — J . For (£, x) in the cone {(t, x) | x/t E B}, one has dtR + Vx-f{R) = 0,

(38)

dtr](R) + V* • q(R) = 0.

(39)

Then we obtain dta(u,R)

+ V x • /?(ii,i?) < -\/2r](R)(VxR,Qf(u,R))

(40)

in the sense of distributions, where Qf(u,v) = /(it) - /(i>) - Vf(v)(u — v) is the quadratic part of / at u. Now, since u is just an L°° function, we consider a mollifying kernel u £ Q>°(-1,1), CJ > 0, fRu(t)dt = 1, and set c^(*) = u(t/6)/6, 6 > 0. We will use the notation h6 = h*ws, for any function /i depending on t. Then, from (40), we get dta6(u,R)

+ Vx.(3d(u,R)

< - {\72r)(R)(VxR,Qf(u,R)))d

.

(41)

We now use the change of coordinates (t,x) *-> (£,£), € = xl^- Inequality (41) then becomes dta5(u,R)-^^as(u,R)+-tVc0s(u,R)

< - QvVW{12,Q/(u,.R))) .

(42) The derivatives with respect to £ in (42) should be taken in the sense of distributions (except those applied directly to R). We consider a nonnegative smooth function of £, 0 £ CQ°(B), such that (£) = 1, in {£ E i? | dist(£, c?I?) > e } , e > 0 sufficiently small. Applying (42) to the test function 0(£) yields

It*

-

t

'

for some constant C > 0, where we denote

Y$(t)= [ as(u(t),R)
(43j

40

Denote Y(t)=

f

a(u(t),R)d£.

JB IB

Theorem 3.1 especially implies

..... r

lim 1 / Y(t)dt -+°° 1 Jo T->oo T J0

(44)

= 0.

We will prove that esslim Y(t) = 0.

(45)

t—KX)

Indeed, we have (* - f ) ^ ( ' ) 2 = 2 Jjs

- | ) ( F / ) ' ( S ) r / ( S ) ds + ^

Y«(s)2 da,

and thus use (43) to get ^Y*(T)2

< Cj^

^Yfit)

dt + J^ Y*{t? dt.

(46)

Now, in the above inequality, we can make —> 1 in B, keeping \\(/>\\oo and Var{0} bounded, and then make 5 -> 0 to get Y(T)2
2

JT

-Y{t)dt+

t

f

Y(t)2dt,

(47)

JT

assuming that T is a Lebesgue point of Y(t). Inequality (47), valid for all Lebesgue points of Y(t), immediately leads to (45) by using (44) and the boundedness of Y(t). Next, we consider the case where B is any bounded open set contained in E n — J. For any £ € B, we take a closed ball B$ with center in £ and radius r*£ < dist(f, dB). Let B be the family constituted by all these balls. We recall that, by Besicovitch's Covering Theorem (e.g. [37]), B contains a finite number of subfamilies Bi, i — 1 , . . . , AT, such that each B% is a countable collection of disjoint balls {Bij}JLl, and AT

B=\J{U™1Bij

'-

BijeBi}.

41

Therefore, we have

^)<EE/

<*Wt),R)d£.

(48)

The uniform boundedness of u and R guarantees the convergence of the last sum on the right-hand side of (48), uniformly with respect to t. Hence, one has iV

es

oo

„

N

oo

f e Yl J2 / <*(*(*)>£) ^ = S S e s s A m / <*(*(*)>R) ^ = °> i=l

j=i

^^i

i=i

j=i

~*°° JBij

which, together with (48), gives (45) again. To extend (45) to the case where B is any bounded measurable set, possibly intersecting J, we observe that, in this case, B will be the union of a finite number of bounded measurable open, disjoint sets in E n - J, plus a null set. Then, the integral of \u(t, t£) -i?(f)l over B is equal to the sum of the integrals of this function over those sets, each of which, as proved, goes to zero when t —> +oo. Hence (36) is completely proved. □ Asymptotic Problems and Multidimensional Riemann Problems. The assumption that the Riemann solution i?(f) is piecewise Lipschitz in f in The­ orem 3.2 is quite general. This assumption is related to the structure problem of multidimensional Riemann solutions. The structure problem has been ex­ tensively studied in recent years for multidimensional scalar conservation laws. See [1,6,17,24,34] and the references cited therein for two dimensional Rie­ mann problems. In Zhang-Zheng [36], piecewise Lipschitz solutions were constructed under the assumption that f"(u) ^ 0, j — 1,2, and (fi/fy'Y ^ 0 for the Riemann problem that the space axes are all initial discontinuities and the initial value in the four quadrants are different constants. In Chen-Li-Tan [6], the Riemann problem, whose data are three constants in three fan domains forming different angles, was considered, and the dependence of the piecewise structure of the solutions upon the value of the constants as well as the angles was studied. All the solutions of these Riemann problems are piecewise Lipschitz and, therefore, these Riemann solutions are always asymptotically stable under L1 DL°° initial perturbation with the aid of Theorem 3.2. It would be interesting to study further the piecewise Lipschitz structure of solutions of multidimensional Riemann problems.

All the re

sults established in this paper hold for the viscous servation laws in several space variables by using the arguments developed in Chen-Frid [4]. Acknowledgments: Gui-Qiang Chen's research was supported in part by the National Science Foundation grants DMS-9623203 and DMS-9708261, and by an Alfred P. Sloan Foundation Fellowship. Hermano Frid's research was supported in part by CNPq-Brazil, proc. 352871/96-2. REFERENCES

1. Chang, T. and Hsiao, L., The Riemann Problem and Interaction of Waves in Gas Dynamics, Longman Scientific and Technical (Pitman Mono­ graphs No. 41), Essex, 1989. 2. Chen, G.-Q. and Frid, H., Asymptotic decay of solutions of conservation laws, C. R. Acad. Sci. Paris, Serie I, 323 (1996), 257-262. 3. Chen, G.-Q. and Frid, H., Decay of entropy solutions of nonlinear con­ servation laws, Arch. Rat. Mech. Anal. 1998 (to appear). 4. Chen, G.-Q. and Frid, H., Large-time behavior of entropy solutions of conservation laws, J. Diff. Eqs. 1998 (to appear). 5. Chen, G.-Q. and Frid, H., Divergence-measure fields and conservation laws, submitted (1997). 6. Chen, G.-Q., Li, D., and Tan, D.-C, Structure of the Riemann solutions for two-dimensional scalar conservation laws, J. Diff. Eqs. 127 (1996), 124-147. 7. Dafermos, C M . , Applications of the invariance principle for compact processes. II. Asymptotic behavior of solutions of a hyperbolic conserva­ tion law, J. Diff. Eqs. 11 (1972), 416-424. 8. Dafermos, C. M., Regularity and large time behavior of solutions of a conservation law without convexity, Proc. Royal Soc. Edinburgh 99A (1985), 201-239. 9. Ding, X. and Liu, T.-P., Nonlinear Evolutionary Partial Differential Equations, Proceedings of the International Conference held in Beijing, June 21-25, 1993, AMS/IP Studies in Advanced Mathematics 3, Amer. Math. Soc. Providence, RI; International Press, Cambridge, MA, 1997. 10. DiPerna, R. J. and Lions, P. L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729-757. 11. Engquist, B. and E, W., Large time behavior and homogenization of solutions of two-dimensional conservation laws, Comm. Pure Appl. Math. 46 (1993), 1-26.

43

12. Evans, L. C , Weak Convergence Methods for Nonlinear Partial Differ­ ential Equations, CBMS 72, AMS, Providence, 1990. 13. Evans, L. C. and Gariepy, R. F., Lecture Notes on Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, Florida, 1992. 14. Freisttiler, H. and Serre, D., L1 stability of shock waves in scalar viscous conservation laws, Comm. Pure Appl. Math. 51 (1998), 291-301. 15. Glimm, J. and Lax, P., Decay of solutions of systems of nonlinear hy­ perbolic conservation laws, Amer. Math. Soc. Memoir 101, A.M.S.: Providence, 1970. 16. Goodman, J., Stability of .viscous scalar shock fronts in several dimen­ sions, Trans. Amer. Math. Soc. 311 (1989), 683-695. 17. Guckenheimer, J., Shock and rarefactions in two space dimensions, Arch. Rat. Mech. Anal. 59 (1975), 281-291. 18. Il'in, A. M. and Oleinik, O. A., Asymptotic behavior of solutions of the Cauchy problem for some quasilinear equations for large values of time, Mat. Sbornik 51 (1960), 191-216. 19. Jones, C. K. P. T., Garner, R., and Kapitula, T., Stability of travelling waves for nonconvex scalar viscous conservation laws, Comm. Pure Appl. Math. 46 (1993), 505-526. 20. Kruzkov, S. N., First-order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243. 21. Lax, P., Hyperbolic systems of conservation laws, Comm. Pure Appl. Math. 10 (1957), 537-566. 22. Lax, P., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS. 11, SIAM, 1973. 23. Li, B.-H. and Wang, J.-H., The global qualitative study of solutions of a conservation law (I),(II), Sci. Sinica, 1979, Special Issue on Math. I (1979), 12-24, 25-38 (Chinese. English summary). 24. Lindquist, W. B., Scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown, SIAM J. Math. Anal. 17 (1986), 1178-1197. 25. Lions, P. L., Perthame, B., and Tadmor, E., A kinetic formulation of mul­ tidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), 169-192. 26. Liu, T.-P., Invariants and asymptotic behavior of solutions of a conser­ vation law, Proc. Amer. Math. Soc. 71 (1978), 227-231. 27. Matsumura, A. and Nishihara, K., Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity Comm. Pure Appl. Math. 165 (1994), 83-96. 28. Murat, F., L'injection du cone positif de H~l dans W~l,q est compacte

44

pour tout q<2,J. Math. Pures Appl. 60 (1981), 309-322. 29. Osher, S. and Ralston, J., L1 stability of travelling waves with applica­ tions to convective porous media flow, Comm. Pure Appl. Math. 35 (1982), 737-751. 30. Schwartz, L., Theorie des Distributions, Vol.1, Actualites Scientifiques et Industrielles 1091, Herman, Paris, 1950. 31. Serre, D. and Xiao, L., Asymptotic behavior of large weak entropy solu­ tions of the damped p-system, J. Partial Diff. Eqs. 10 (1997), 355-368. 32. Tartar, L., Compensated compactness and applications to partial differ­ ential equations, In: Research Notes in Mathematics, Nonlinear Analysis and Mechanics ed. R. J. Knops, 4(1979), Pitman Press, New York, 136211. 33. Volpert, A. I., The space BV and quasilinear equations, Mat. Sb. (N.S.) 73 (1967), 255-302, Math. USSR Sbornik 2 (1967), 225-267 (in English). 34. Wagner, D. H., The Riemann problem in two space dimension for a single conservation laws, SIAM J. Math. Anal. 14 (1983), 534-559. 35. Weinberger, H. F., Long-time behavior for a regularized scalar conserva­ tion law in the absence of genuine nonlinearity, Ann. Inst. H. Poincare Anal. Nonlineire 7 (1990) 407-425. 36. Zhang, T. and Zheng, Y.-X., Two-dimensional Riemann problem for a scalar conservation law, Trans. Amer. Math. Soc. 312 (1989), 589-619. 37. Ziemer, W. P., Weakly Differentiable Functions, Sobolev Spaces, and Functions of Bounded Variation, Springer-Verlag New York Inc., New York, 1989.

45 F O R M A T I O N OF SHOCK IN P O T E N T I A L FLOW SHU-XING CHEN and LI-MING DONG Institute of Mathematics, Fudan University Shanghai 200433, China Email: sxchen@fudan. ac. en

1

Introduction

In the study of a quasilinear hyperbolic system, one important phenomena is that the discontinuity of the solution can be produced in the domain where the system is considered, no matter how smooth the data are given to determine the solution. A natural problem is then how the discontinuity is produced and developed in the domain. In gas dynamics, the corresponding problem is how a shock could be produced in a continuous flow, and then the strength of the shock increases graduately. Many papers are devoted to predict the occurrence of shock (see [1,2] and the references cited there). But generally it is more interesting and more difficult to give a detailed description of the formation of shock and the behaviour of the solution near the point where the shock forms. In the case of scalar equation, the formation of the shock has been clearly described (see [3-5]). Recently, M. P. Lebaud in [6] also discussed the case of 2 x 2 system with one space variable under some assumptions on initial data. In this paper we will discuss the process of formation of shock for plane potential flow. Consider a constant two-dimensional supersonic flow coming from infinity along a straight wall. If the wall keeps smooth but becomes concave starting from some point, then the flow will be forced to turn its direction, and a shock will be formed inside the flow. The flow near the wall forms a simple wave and all parameters of the flow keep constant along characteristic lines with positive slope (the wall is assumed to locate below the flow and the flow comes from the left side). Because the wall is concavely bended, these characteristic lines are compressed and form an envelope in the flow. Generally, the envelope has two branches starting from a cusp. Then a shock will be produced starting from the cusp of the envelope. It has zero strength at the starting point and increases the strength in the process of its propagation. Our aim in this paper is to give a detailed construction of the shock and the flow field near the starting point of the shock. In this paper we describe the flow by using the potential equation, which

46

is reduced from the general system in gas dynamics under the assumption that the flow is isentropic and irrotational. Because the shock is very week near the point of its formation, then the potential flow is a good approximation of the real flow. The equation of stationary potential flow takes the form as ( see [7,8] ) -($XH) + — (* y ff) = 0,

(1.1)

where H = hi(C - | | V $ | 2 ) , C = \ql + h(p0) = \ql + ^ , 7 is the adiabatic exponent, a is sonic speed, and h\ is the inverse function of the enthalpy h depending on p. The relation of potential and the velocity is u — §x,v — $y. Taking (u,v) as unknown functions, equation (1.1) is equivalent to J (a 2 — u2)ux — uv(uy + vx) + (a 2 — v2)vy — 0, \ Uy - VX .

(1.2)

The characteristic directions of equation (1.2) is uv ± ay/u2 + v2 — a2 A± =

5

2

^'3^

Denote the Riemann invariants of (1.2) by r(u,v) and s(u,v), then they take constant on characteristics of first class and second class, respectively. Along the wall the normal component of the velocity of the flow vanishes. Assume that the flow locates above the wall and comes from the left side, then it forms a simple wave of second class near the wall and can be described by r(u,v) = const. If all characteristics of second class form an envelope with its cusp at (xo,2/o)? then the shock will be produced at (xo,yo). The conclusion of this paper is the following. Theorem 1.1 Assume that the characteristic lines of second class form a en­ velope with its cusp at (xo,2/o); then there exists a weak entropy solution of (1.2) in a neighbourhood Q, of (xo,yo) with a shock T : y — (f>(x) starting from ixo,yo) The solution is continuous on fl\T. Moreover, we have estimates: (j)(x) - y0 + a(x - x0) + 0((x - x0)2), r(x,2/) = O ( ( x - x 0 ) 3 / 2 ) , s(x, y) = s0(yo) + 0{(x - x0f + (y - 2/ 0 ) 2 ) 1/6 ),

(1.4)

where a is the slope of the characteristics of second class passing through the cusp (x0,y0).

47

The whole paper is devoted to prove the above main theorem. In Section 2 we discuss the properties of simple wave and shock for potential equation (1.2). In Section 3 we recall some facts on the formation of shock for scalar quasilinear equation. In Section 4 we set up the iterative scheme of approximate solution. Then we establish the uniform boundedness of the sequence of approximate solutions in Section 5, and establish the convergence of the sequence in Section 6, where the proof of Theorem 1.1 is also finally completed. 2

Basic Properties of the Shock for Potential Equation

Let us first recall some basic facts for potential equation (see [9]). Denote q — yju2 + v2 and 6 — arctan(i>/?/), the Riemann invariants for the potential equation (1.2) have expression s = 9-F(q), where F(q) = f ^

a

r = 6 + F(q),

(2.1)

dq. The Riemann invariants satisfy the system

\

dr

,

A

dr _

n

(2-2)

which implies r = const, on A_ characteristics, and s = const, on A+ char­ acteristics. Therefore, in the region of simple wave of second class r = const. (the constant can be simply taken as 0) and all A+ characteristics are straight line. Similarly, in the region of simple wave of first class s = const, all A_ characteristics are straight line. Above the curved wall the solution of (1.2) is a simple wave of second class. In the region of simple wave the unknown functions u(x,y),v(x,y), as well as both Riemann invariants only depend on one parameter, and then the whole image (u(x,y),v(x,y)) under the map T : (x,y) -» (u,v) is located on a curve on (u,v) plane. The curve is epicycloidal with its equation as (see [7,10]) u = a*(cos/i(a; - CJ*)COSCL; + pT1 sin/i(cj — a;*)sinu;), v = a*(cos/x(cj — CJ*) sincj — fi~1 sin/i(u; — a;*) COSCJ),

, . (2.3)

where ji = i2^)1^2', a* is the critical sound speed, a* and LJ* are determined by the coming flow. Assume that the equation of the bended wall is y — f(x), which is smooth function and satisfies f(x) = 0 if x < 0, and f'(x) > 0, f"{x) > 0 if x > 0. For

48

any x > 0, the direction of the velocity at (x, f(x)) is u : v = 1 : / ' ( # ) , which determines the relation between UJ and x as ,,,_,. (cos/x(o; - cj5(e)sina; - / i _ 1 sin/i(o; - a;*)cosu;) , N / W — 7 7 \ i—: 7 r~: 7 l^-4) _i (cos//(o; — u;*jcosu; + / i sin//(a; — a;*) sin a;) The geometric meaning of UJ is that it is the angle between the normal direction of characteristics and positive x-axis. Then the equation of characteristic line on (x, y) plane is

{

x = x + tcos(uj — | ) ( = i + tsiriuj)

. ( ,2.5)

y = / ( £ ) + *sin(o; - f ) (= / ( * ) - t cos a;) ^'5j Due to the bending of the wall all characteristic lines given by (2.5) are com­ pressed and forms an envelope. The fact let the structure of simple wave breaks down at some distance away from the wall. The envelope of the family of characteristics is determined by A =

Xx Xf,

yx yt

= 0

(2.6)

Substituting (2.5) into (2.6) we obtain A = cos UJ — tu' -f / ' sin UJ Therefore, on the envelope we have t = -^(/'sinu; + of the envelope with x as its parameter is

COSCJ),

{ x — x — ^r (/' sin UJ 4- cos UJ) sin u \ y — f(x) + ^-(/'sincj + COSCJ)COSO;

and the equation

, . ^ ' '

Denote the right side of the first equality of (2.7) by h(x), where u and u/ are functions of x determined by (2.4). If ti{x) = 0, h"(x) > 0

at

x = x0

(2.8)

then x takes minimum XQ at x0, and the envelope has its cusp at (x0,yo), where yo is obtained by substituting x = XQ into (2.7). In the sequel in order to simplify computation we will move the origin of coordinate system to (x 0 ,yo)- Moreover, we will also straighten the shock and place it onto the new x-axis. The corresponding coordinate transformation will be described later.

49 Next, let us describe some properties of the shock for potential equation. The Rankine-Hugoniot relation for potential equation is [u] + a[v] = 0,

a[pu] - [pv] = 0

(2.9)

where [•] means the jump of the corresponding quantity in the bracket, a is the slope of the shock. Therefore, any state (ix,v,p), which can be connected by a shock with the given state (UQ,VO,PO) must satisfies (pu - p0u0){u - u0) + (pv - p0v0)(v - v0) = 0

(2.10)

By using the variable 0 and q, (2.10) can be written as coS(0-80)=™l+™2° (P + Po)qoq

(2.11)

which is also called equation of shock polar. Lemma 2.1 The jump of the parameters of the flow on the both sides of shock satisfies [0\ = k[F(q)] (2.12) where k is a smooth function of 6-,8+,q-,q+, [q] tends to zero. Proof: From (2.11) we have

sin2[0] = l -

and k tends to 1 when the jump

M+P-£)a ((p++

p-)q+q-)2

(qi-ql)(pl(ql-ql)+ql(pl-pl)) (q+ + q^ipj

((/>+ + p-)q+q-)2 + qUpj - pl)/(q2+ - q2-)) [q]2

[ f l ( ]2>

[F(q)}2 I

{p++p^q\ql

In view of sin[0] = [0] — |r[0] 3 + • ■ ■ we only need to show the coefficient of [F(q)]2 tends to 1 as [q] —> 0. In fact, the limit of the coefficient is 4 < ? y + <7 2 f^) 4p2q4(F'(q))2

{P2 + Pqdj) p2q2F'(q)2 '

On the other hand, from Bernoulli's law we have qdq + jp7~2dp

= 0.

50

Then ^ = —pq/a2, which implies (P2+P
2 2

p q (q

p2-p2q2a~2 = 1. - a2)a~2q~2

2

This completes the Lemma. Lemma 2.2 If r- =0 on a given shock, then r+=f(s+,s-)[s}3

(2.13)

where / ( s + , s _ ) is a smooth function. Proof: From (2.12) we have [9] = (l + h(6+,0-,q+,q-)[F(q)})F[q]

(2.14)

By changing the sign + with — we know /i(0_,0 + , #_,#+) = —h(8+,6-,q+,q-), then h vanishes at [0] = [q] = 0 and (2.14) can be written as [ff] = ( l + /»i[F(«)] a )[F( g )] or

[6}2 = (l +

h2[F(q)f){F(q)f

where hi and h2 are suitable smooth functions vanishing at [0] = [q] = 0. Substituting r = 0 + F(g), 5 = 0 - F{q) yields [s +

r]2 =

( 1 +

^

[ s

_

r ]

2

) [ 5

_

r ]

2

In view of r_ = 0 we have r+([s-r}+r+) = ±h2[s-r}4

(2.15)

Denote z = r~^], 2; satisfies the quadratic equation z(l + z) = ^ / i 2 [ * - r ] 3 Notice that (2.12) implies z = -|[<9 + F(^)]/[F(g)] -> 1, when [q] tends to 0. Therefore, we should accept the root z — —\ — \\j\ + \h2[s — r ] 3 . Let r+ =

51

hs[s — r ] 3 , where J13 is a smooth function of r+, s+, s_, and take w = r + / [ s ] 3 , then [s - r] = [5] — w[s] 3 , (2.15) implies

Ma]3 = MM*]3,«+,*-)(M-M*]3)3 Namely w - /i 3 (^[s] 3 ,5+, s_)(l - w[s]2)3 = 0 Obviously, the derivative of the left side with respect to w equals 1 as [s] —> 0. Then applying the implicit theorem w can be solved as a function of s+ ans 5_ if [s] is small. 3

First Approximation

In order to simplify calculation we move the origin of the coordinate system to the cusp (xo,yo) of the envelope formed by the family of second characteristics and still denote the coordinate variables by (x,y). In the new coordinate system the solution of the original problem for x < 0 is known. Without loss of generality we assume that the second characteristics passing through the origin intersect with x = — 1 before they meet the surface of the wall , then the solution in a neighbourhood of the origin can be determined by the initial data on x = — 1. Therefore, to construct the solution near the origin we can consider the Cauchy problem of (2.2) with initial data s(-l,y)

= 80(y),

r(-l,j/)=0

(3.1)

instead. To solve the problem (2.2),(3.1) near the origin we construct a sequence of approximate solutions. By improving the degree of approximation of the sequence we can indicate the convergence of the sequence and confirm that its limit is the precise solution of the problem. The first approximation of the solution of (1.2) is chosen as r(x,y) = 0, and s(x1y) satisfies dxs(x, y) + \+dys(x, y) = 0 (3.2) s(-l,y)

= s0(y)

(3.3)

away from the shock. On shock curve the Rankine-Hugoniot condition a = — [u]/[v] and entropy condition are satisfied. Here the variable r in A+(r, s) is taken as 0. The equation (3.2) may correspond to many different forms of conservation law, but we seek a reasonable form such that the corresponding RankineHugoniot condition coincides with (2.9). Indeed, let e(u,v) — u — av(q2 —

52

a 2 ) - 1 / 2 , then eA+ = v + aw(g2 - a 2 ) - 1 / 2 . Denote by G the inverse function of F(q), then g = G ( ^ ) , and G"(F(g)) = aq(q2 - a 2 ) " 1 / 2 . Hence e = ?cos <9-ag sin %

2

- a2)~1/2 = (2G(^-^)sin ^ ^ )

eA+ = aqcosO(q2 - a2)~1/2

s

+ qsinO = - ( 2 G ( ^ - ^ ) cos ^ - ^ )

s

Multiplying (3.2) by e and letting r = 0 we obtain ( G ( - | ) sin | ) , + ( G ( - | ) cos | ) y = 0

(3.4)

Therefore, if we use (3.4) to determine the first approximation of discontinuous solution with a shock starting from the cusp of the envelope of A+ character­ istics, the Rankine-Hugoniot condition will be a

[2G(-j)cosj] [2G(-f)sinf]

~

(3 5)

-

which coincides with (2.9) obviously. The sequence {r^(x, y), s^(x, y)} is obtained from modification of the first approximation, so we need more precise information on the solution s(x, y) and its shock. To this end we introduce the result in [6] in this section. Here all notations are independent of those used in the other part of this paper. Consider the Cauchy problem du

df(u)

d- + ~ i r = 0

<3-6>

x

u(-l,y) = Mv) p

(3-7)

00

where f(u) <E C°°, u0(y) € C flL . Denote X(u) = f'(u),g(z) the family of characteristics of (3.6) is y = z + xg(z)

= X(uo{z)),

(3.8)

Assume that the family of characteristics forms an envelope with its cusp at the origin, through which the characteristics is y = z$ + xg(zo), and g(z) satisfies g(z0) = g"(z0) = 0,g'(zo) = -hg(3)(z0) we have the following propositions.

= 6,zg"(z) > 0

(3.9)

53

Lemma 3.1 There is a weak entropy solution u(x,y) to (3.6),(3.7), and a Cv'2 function y = <j>(x) defined in x > 0, such that u(x,y) is of Cp when y ^ (x). Besides, u(x,y) satisfies the following estimates: < C 0 (x 3 + 2/ 2 ) 1/6
\u(x,y)-u(Q,0)\ \dxu(x,y)\ yu(x,y)\ \d \dyyu(x,y)

(3.10) (6-W)


Lemma 3.2 The leftward characteristics starting any point (a, b) below shock does not intersect the shock. If we denote the characteristics by y = rj-(x,a,b)(resp.

y = r/+(x,a, &)),

then ±77±(0, a, 6) = v ^ + 0(a) + 01 (\b\^3) ±(v±(x,a, where 0\(b1^)

(3.11)

= Va(a - x) + Oi(|6| 1 / 3 (a - a;)) + 0(a{a - x))

(3.12)

stands for a positive bounded quantity no greater than

0(\b\1^).

b)-b)

Lemma 3.3 If (a,b) is in a neighbourhood of the origin, then x3+rj(x,a,b)2

Lemma 3.4 Let I — — tive, and

JQ(XU

> 4 ( a 3 + b2) lb

• uy)(x, n(x, a, b))dx then the integrand is posi­

\I\ < l n | + CV5 More over, if ((x,a,b)

(3.13)

is a function

(3.14)

satisfying

\C(x,a,b) — rj(x,a,b)\ < Ca(a - x)

(3.15)

then we also have fa | / (AM -Uy)(x,r)(x,a,b))dx\

3 < In- + Cy/a

(3.16)

54

Remark 3.1 Let us give some explanations on the assumption (3.9). g{zo) = 0 means zo — 0 and the characteristics through (—l,£o) coincides with the xaxis, g'(zo) — — 1 means that the characteristics is tangential to the envelope at the origin, g"(zo) — 0 means that the cusp of the envelope is just at the ori­ gin. All these can be satisfied by a linear transformation of coordinate system. Besides, g^\z$) — 6 is an assumption for simplifying calculation. In general case we should add a factor (-7377—7)2 before the first term in the right side of the estimate (3.11) and (3.12). But the estimates (3.13), (3.14) hold without changing constants. In order to use the result listed above, we reduce (3.4) to the standard form (3.6). Set m = G(— f) sin | , then G{— f) cos | can be denoted by / ( m ) , we notice that both G{s) and sin | are monotonic increasing functions. In fact, sin I is a monotonic increasing function for small s , and G ( | ) is also monotonic increasing, because G is the inverse function of monotonic increas­ ing function F(q). Hence m(s) is monotonic decreasing, and then its inverse function h(m) and / ( m ) = [G(—|) cos %]s=h(m) a r e wen< defined. Namely, the problem (3.4),(3.3) can be rewritten as mx + {f(m))y

=0

(3.17)

m(-l,y)=m(s0(y))

(3.18)

Applying Lemma 3.1 - 3.3 to the problem (3.17),(3.18), we obtain the existence of the solution m(x,y) with shock y = (#), which satisfies the estimates shown in these lemma. Correspondingly, the solution s(x,y) of the problem (3.4),(3.3) is also obtained, it also satisfies the estimates (3.10) to (3.14) near the origin. 4

Iteration Scheme

Starting from the first approximation we will graduately improve the degree of approximation by a suitable iteration scheme. Since the shock front also varies in the iterative process, in order to compare the approximation in different step, we introduce a transformation, whichfixthe location of the shock in the whole process of iteration. Assume that at v~th step the shock is y — ^(x), the coordinate transformation is

&*:

*i=», y1 = \y-^)^

IJ ^ °


(4.1)

[ax if — 1 < x < 0 where a is the slope of the characteristics of second class passing through the origin. Obviously, (4.1) transforms the shock front into yi — 0. Furthermore,

55 we define a^

by

a(v) = { )y(x)

if if

{f

x >0 - 1< x < 0

(4.2)

To simplify notations we denote (#i,2/i) by (x,y) again in what follows. Since we only consider the local existence of the solution in the neighbourhood of the origin , our discussion can be restricted in the domain Q = n 0 un_|_Uft_, where ft0 = {(x, y), - 1 < x < 0, - e < y < e} ft+ = {(x,y),0 < x < r/,0 < 2/ < e - x) ft_ = {(x,y),0<x
=0

(4.3)

For v > 0 we define < j ^ \ r ^ + 1 \ s^ +1 ^ inductively as follows. (") = -

[G(

[G(

.<")_ 5 —)cos r (")- s (") )sin ui 2

v)

/

(v)

+ 5'( " > 1

(4.4)

T(v)+8{v)}

"

2

J

+1

^ ^ " ^ ( a r . v ) + (AL -a )dyA» Xx,y) =0 5 x s^ +1 Ha;,2/) + (A^ ) -<7 (,/) )5 y s^ +1 )(a;,2/) = 0 r ( " + i ) ( _ l , y ) = o, s ( " + D ( - i , y ) = s0(y)

(4.5)

where A ^ = A±(r(^,s<")). Our task is to prove the sequences a^ ,r^"\s^ are well defined and convergent. To this end we are going to prove the following facts F^ in«("> = «(°) in n 0 U Sl+

( ) ductively: F{ V)

s^ec^n-

(0,0))

\s(-,/'>(x,y)-s^(x,y)\ = 0 in fl 0 Uft+ „(„) I r M e C H f i - U O . O ) ) 2 | \r^{x,y)\
56

5

Boundedness of the Sequence of Approximate Solutions

To confirm the validity of proposition F^ for all v inductively, we prove the following lemmas. In the sequel all constant C is uniform with respect to u, and it may take different value in different inequalities. Lemma 5.1 Under the assumptions of validity of

F^

\aM-a(°)\
(5.1)

Proof: Notice that r^v\s^ in fi+is independent of v, and from (4.3) we can write o~^ = H(r,s)\r-r(V)i8=s(V),a^ = H(r,s)\r=r(o)s=s(o). Therefore, \aM - a (o)| < <7(| r H | + | a M _ s (o)|) < Denote by y = rj(u\x,a,b) through (a, b), we have

Cx

the characteristics of equation (4.4)2 passing

Lemma 5.2 Under the assumptions of validity of \r}^(x,a,b)-r)W(x,a,b)\

F^


(5.2)

Proof: In view of b — rj^ (a, a, b) b - 77M (x, a, 6) = T ( A + } ( 0 , VM (a, a, 6)) - a M (a))da

(5.3)

To estimate rj^(x,a,b), we introduce another iterative process. Tem­ porarily fix the index 1/, and let y = n^(x,a,b) be the characteristics of the equation (4.3), which satisfies i/°>(*,a,b) = b-

r(A^0)(a,T/(°)(a,a,b))

-

a™(a))da

JX

Take Co(x,a,b)=^°\x,a,b)

(5.4)

C„+i(ar,o,6) = 6 - /"°(A^ ) (a,Cn(a,o,6)) - <7(a))da ./x

Since Co(^,a, 6) satisfies Co(ar,a,6) = 6 - f\x^\a,(o(a,a,b)) ./x

- ^°>(a))da

(5.5)

57

we have pa

Co(x,a,b) -Cn+i(x,a,b)

= /

X^\a,

(n(a,a,b)

- A^c^CoO^a^da

JX

JX

Assume that Q(x,a,b)

satisfies the estimate

\Ci(x,a,b) - £o(x,a,b)\ < Ca(a - x)

(5.6)

for i < n, we confirm that the estimate (5.6) also holds for i = n -f-1. In fact (o(x,a,b) -Cn+i(x,a,b) = f" A ^ (a, Cn(a, a, 6) - A^_0) (a, Co (a, a, b))da

+ £Ai%,C»(a,a,&))-A^ (5.7) By using Lemma 5.1 and the assumption F^ \<jW(a)-aW(a)\ |A^)(a,Cn(a,a,6))-A^(a,Cn(a,o,6)| < CHVAHioo ( | T » ( a , C„(a, a, 6))| + \s^(a,
we have < Ca

(5.8)

C„(a, a, 6)) - *«»(<*, C„(a, a, 6))|) (5.9)

On the other hand pa

| /

A^o^CnO^M) -

X^(aXo{oL,a,b))da\

JX

< I /"(A+i -4 0 ) )(a,(Co + 0(Cn - Co))(a,o,6) • (<„(a,a,&) -
(aXn(a,a,b))-X(°\a,(o(*,a,b))da\

< (In | + Cyfr)a(a-x)

(5.10)

Substituting (5.8)-(5.10) into (5.7) we know that (5.6) is also true for i = n+1. Therefore, (5.6) is true for all i by induction.

58

By using similar method we can prove the convergence of the sequence {(n(a,a,b)}. Indeed |C n +i(a;,a,6)

-Cn{x,a,b)\

pa

= | / (\{+\a,(n(a,a,b))

\^](a,Cn-i(<^,a,b))da\

-

JX

< I / " ( a ^ X M C n - l JX

(Cn(<x,a,b)

+0((n

-(n-l)(a,a,b))

■

-Cn-i(a,a,b))da

< I ri(^A^)(a,(Cn-i+^(Cn-Cn-i))(a,a,&)|da. JO

||Cn(a, a, 6) ~ Cn-i(a, a, 6)||L~ The integral on the right side is no more than / " |(<9yA^0)(a, (Cn-i + #(Cn - Cn-i))(a, a, 6)|da

is then dominated by In | + Cy/a by virtue of Lemma 3

Since (Cn-i + #(Cn — Cn-i))(^ 5 «, 6) satisfies (3.15), the first integral in (5.11)


Hence for small a we have

(C„_i + 0(C„ - Cn-i))(a,o,6)|da| < In | + C^

< \

Substituting it into (5.11) we establish the contractivity of the sequence {Cn}: ||Cn+l "" CnlU 0 0 < xllCn ~ Cn-l||l,«>

(5.H)

which leads the convergence of {Cn}- Obviously, the limit ((x,y) of {(n(x,y)} is the unique solution of (5.3), while the estimate (5.2) is just the limit of (5.6).

.4.

59 Remark 5.1

Combining Lemma 5.2 with (3.12), we have

rfg (x, a,b)-b=

±{y/a{a - x) + Oi(|6| 1 / 3 (a - a:))) + 0(a(a - x))

(5.12)

The estimate means that the leftward X± characteristics starting from the point (a,b) with a > 0,6 > 0(b < 0) can leave ft± only at the side x = 0. The fact allows us to use characteristic method to estimate s^ + 1 ^ and its derivatives. Lemma 5.3 Under the assumptions of validity of

F^

|(*<" +1 >-a< 0 >)(x,2,)|
= {-\{?

(5.13)

then v satisfies

+ A f +ff(">- aM)dysM

,, K

M, LV

°-

Integrating along the characteristics we have \v(x,y)\ < I* \(-\{;} + \f Using the assumptions F[v , F 2

+/»

-(T<°))(a w *W)(a ) r,(a,o,6))|«ia

we have

|A<*} - A f | < CHVAIU-dlrMlU- + | | # > - s ^ l U - ) < Ca Meanwhile, Lemma 3.1 implies |a tf s(°)(a,T/(a,x,j/))| < C(a 3 + 7 / 2 ) " 1 / 3 < C a " 1 and Lemma 5.1 implies |a^-(7(0)|


Therefore, ^(x,?/)| < C / a • a _ 1 d a < Cx ./o Lemma 5.4 Under the assumptions of validity of |r^+1)|
F^ (5.15)

60

Proof: r ^ + 1 ) = 0 in the domain ft0 U ft+ . In domian ft_ the function r (H-i) i s determined by dxr^+1)

+ ( A ^ - a))
( 5 - 16 )

satisfying r ^ + 1 ) = 0 on x = 0 and the Rankine-Hugoniot condition on y — 0. Therefore, in f]_ the solution r^ + 1 ^ can be determined by the characteristics method with data on x = 0 or y — 0. Denote l- \ y — rj-(x,a,b) as the A_ characteristics line passing through (a,b). Assume the line intersects with x axis at (£,0), then I- can also be denoted by y — n-(x,£,0). At the point (£, 0) we have |r("+i)|17) according to Lemma 2.2. On the other hand Lemma 3.2 implies [s^] < C^' Combining with Lemma 5.3 we have

.

[s(v+1)] < C&/2 Substituting it into (5.18) we obtain (5.16) on y = 0. Then by integrating (4.4) 2 along A_ characteristics with initial data on y axis or x axis as we did in the proof of Lemma 5.3, we obtain (5.16) in the whole domain f&_ Now let turn to the estimate of the derivatives of s^ + 1 ) and r^v+l\ Lemma 5.5 Under the assumptions of validity of \dy(s^

-sW)(x,y)\

Proof: Set v = dy(s^+x>> - s^),

F^

< C(x3 +y2)-l'«

(5.18)

then v satisfies

dxv + (X1? - aM)d„v = ( - A ^ + Xf + _ „ ( o ) ) ^ s « » - dyX^v

+ dy(\{;} -

\f)dys^ (5.19)

By integrating we obtain v(a, b) = f [ ( - A f + A f + a<"> - ^)dyys^ Jo +dy(X(;} -

-

dy\^v

\W)dvsW]da

where the variables in the functions s^\s^°K (a,ri(a,a,b)).

Under the integral sign are

61

From Lemma 3.1 and the assumptions

F[V\F^V\

we have

Jo

< r Ca- (a3 + r? 2 )- 5 / 6 < C r ada ■ (a3 + b2)~5/6 Jo Jo < Ca2(a3 + 6 2 )- 5 / 6 < C(a3 + b2)'1/6

| f dy(\M Jo < f \{^]ryv) Jo

X^dyS^da] + AM#> - A<°>«<°>)| ■ \dysW\da

< r K ^ r M + A<">(#> - 4°)) + (A<"> - A(°))S(°))| • Jo
r \(a3 + ry 2 )- 1 / 6 + a" 1 / 2 + a(a3 + r ? 2 ) - 1 ^ . ( a 3 ./o

< C I" a-1'2da

+

\dys^\da ^-1/3^

• (a 3 + 6 2 )" 1 / 3 < C(a 3 + o 2 )" 1 / 6

Therefore, we obtain an inequality M < /

g{a)\v\da + h(a)

(5.20)

where h(a) < C(a3 - f o 2 ) - 1 / 6 and JQa p(a)da < In f + C^/a according to Lemma 3.4. Then, by using Gronwall's inequality, we establish Lemma 5.5. Remark 5.2 The similar argument yields the estimate of derivatives ofdxs^ and dx(s^ — s^). To this end we only need to replace the estimate for \dyu\ by \dxu\ in (3.10), and replace (3.14) by

I dx\{^da
Jo

Jo

Then we obtain \dx(sC+1)

~ s{0))(x,y)\

< C{x3+y2)-ll*

(5.21)

and \dxs^\x,y)\
(5.22)

62

Lemma 5.6 Under the assumptions of validity of F^ \dx(r{"+1))(x,y)\
(5.23)

\dy(r^)(x,y)\
(5.24)

Proof: Let us prove (5.25) first. By using Lemma 2.2 we have r%+1)(x)

=

f(s{:+1)(x),s^+1)(x))[s^}3

Differentiating with respect to x we obtain \dxrl"+1Hx)\

< C(3[s^}2\[dxs^}\ + \d*S{;+1)(x)\))

+

[s^}3(\dxs^+1)(x)\

< Cy/i-

on y — 0 according to Lemma 5.5. Moreover, by using similar method of integrating along A_ characteristics of (4.4) with initial data given on y axis as we did in the proof of Lemma 5.5, we can establish the estimate (5.25) of <9 x r^ +1 ) in the whole domain ft_. Because r ^ + 1 ) satisfies (4.4)i and A_ — a^ is bounded away from 0, so (5.26) can be derived from (5.25) immediately. Evidently, according to Lemma 5.1 - 5.6 we obtain the validity of F± , F 2 for all v by induction. 6

Convergence of the Sequence of Approximate Solutions

Based on the estimates established in Section 5, we can prove the convergence of the sequence {r^} and {s^} now. Lemma 6.1 |(TH_CT(-i)|
^-^IIL-)

(6.2)

To prove more precise estimate (6.1) we need more calculation. Because on the upper side of x axis rv', sv are independent of v, we simply denote the right

63 side of (4.3) by H{r^\s^).

Hence H(r^-l\s(v-V)\

!<,(") _ f f ( " - i ) | = |ff(,•<">,»<">) < C|r<"> - r*"" 1 '! + \H(rM,sM)

-

ff^^s*"-1')!

< C|r<"> - r ^ - 1 ' | + |[ff (r<">, s("») - ff(r(">, s^" 1 *)] - [ff (0, «<">) - ff (0, s ( l / _ 1 ) )]| + |ff(0, «M) - ff(0, s'^- 1 ')! By using the notations in (3.17) we have

v

v

'

y

ds[m(s)})s=s.{S dm r

[m^"-1))]

[m(s("))] j

[ml J m_m,

ds

,

_ / H-/ (m)-[/(m)]>\

H2

*"/,(„) _

Jy-i)

y m = m . rfs

where s* = a*""1) + 0(s("> - a*""1)) with 0 < 0 < l,m* = m(**). Hence |ff(0,*<">) - H(0,8^-^)\ = (\x'+(so)

< \\f"(m0)+O([m})\ + 0(\s^

• | ^ | • |*(") -

s ^ \

- aW|))|5<") - aC- 1 )!

Substituting it into (6.2) we obtain | a (") -a^-1)]


+ (^|AV(*o)| + 0(8M

-/"-V]

- s (0) ) + 0(r<">))|s<"> -

< C||r<"> - r ^ - ^ I U - + \\X'+(s0)\

s^'l

\\sM - 8
L e m m a 6.2

||

gM

_ s <"-D|| Lo . < A ( | | r M _ r^-^IUco + ||«M - S ^ H L - )

(6

64

Proof: Let v = s^ + 1 ) - s^u\ then v satisfies dxU

+ ( A M - - ^ - ^ J f l y ^ M

Integrating along the characteristics yields |v| < / 0 a | ( ^ - 1 ) - A ^ + a<"> - ff("-1))as(*M - s(°))(a,ry(a,a,6))| da + / 0 a |((A^- 1 > - A ^ ) ^ - 1 ) -a^)dys(°HaM^a,b)))\ da (6.4) In (6.5) the estimate of dys{-°\dy{s^ - S<°)),(TM - CT^-1) has been es­ tablished above. Furthermore, jAM _ ^ - D j <

(|5

^o)|

+ 0(a.))(|rM

_ r(-D|

+

|SM _

S(-D|)

Then combining with Lemma 6.1 and Lemma 3.3 we obtain If I < ( f In f + CV5)(||rM - r ^ U -

+ ||,<"> -

a^U-)

Hence (6.4) holds if a is small enough. Lemma 6.3 || r ("+i) _ r M | | L o o < ± ( | | r ( " ) _ r ^ l U - + || S W - s^-^IUoo)

(6.5)

Proof: Let v = r^v+l^ — r^v\ then v satisfies dxv + (ALV) - a^)dyv

= (X^-V

- X^ + a<") -

1 M ff('- >)V

(6.6)

and «(0,») = 0

(6.7)

Let us integrate (6.7) along A_ characteristics. In the case when the left­ ward characteristics intersect with z-axis before its meeting y axis, we have |v| < | r ^ + 1 ) ( x , 0 _ ) - r W ( a ; , 0 _ ) | + / : ; | ( A L l ' " 1 ) - A L t ' ) H r M ( a , 7 ? ( a , a , 6 ) ) | d a + XT l ^ ( " _ 1 ) - °M)dyrM(a,r)(a,a, b))\da (6.8) Otherwise, the first two terms in the left side of (6.9) vanish and the lower limit of the integral is replaced by 0. Evidently, it is enough to derive the estimate of v from (6.9). In view of \dyr^\ < Cy/a the integral in (6.9) is dominated by

65

Besides, (2.13) implies | r ^ + 1 ) ( x , 0 _ ) - r<">(x,0-)| < C\[s^^]3

- [ s ^ ] 3 | < Ca\\s^^

- s (z/) || L ~

Combining all these facts we obtain |(r^+i)_rM)(ajfc)| < C(a\\s^

_ 5 H|| L O O + a 3/2|| r .) _ r ( - D | | L o o + || S M _ 5 ( - 1 ) | | L o o )

Therefore, the estimate (6.6) holds for small a. The proof of Theorem 1.1: Now we can prove the main theorem in this paper without difficulty. Adding the estimates in (6.4) and (6.6), we have || r ("+D _ r (") || + ||S("+D _ 8M|| < ^ ( | | r M - r ^ " 1 ) || + | k H - s^~^||)

(6.9)

which indicates the uniform convergence of the sequence {r^ (x, y)1 s^ (#, y)}. Then the convergence of the sequence {a^} can also be derived readily. De­ note the limit of these sequences by r(x, y), s(x, y) and a(x), then r(x, y), s(x, y) are continuous in fi_ and a(x) is continuous for x > 0. Let v tends to infinity in the integral expression of (4.5), we confirm (r(x,y),s(x,y)) is the solution of corresponding integral system, hence it satisfies the system f dxr(x, y) + (A_ - a)dyr(x, y) = 0 \ dxs{x, y) + (A+ -
(6.10)

Besides, on the shock ?/ = (x) the Rankine-Hugoniot condition [ G r(r=i)cosi±i]

[G(^)sin

r+sl

(6.11)

which is the limit of (4.3). Coming back to the original coordinate system we obtain the weak solution of (2.2). The remark under Lemma 5.2 indicates the A+ characteristics of the solution on both sides of the shock is directional to the shock from left side, while the A_ characteristic passes through the shock from left-above to right-below. Therefore, the solution r(x,y),s(x,y) is a weak entropy solution. Meanwhile, From the estimate of s^(x,y) and the limit of the estimate as shown in F^ we have estimates on r(x,y) and s(x,y) in (1.4). Moreover, from the limit of (5.1) we have a(x) = a^(x) + 0(x — XQ) — a + 0(x — x0) where a is the slope of A+ characteristics at (xo,yo). Because <j>(x) is the integral of a(x), we are led to the first estimate in (1.4). Hence the proof of Theorem 1.1 is complete.

66

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

P. D. Lax, J. Math. Phys. 5, 611 (1964) T. P. Liu, J. Diff. Eqs. 33, 92 (1979) S. X. Chen and Z. B. Zhang, Fudan Journal (Natural Science) , 13 (1963) J. Smoller, Shock Waves and Reaction-Diffusion Equations, (SpringerVerlag, New York Heiderberg Berlin, 1982). A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984). M. P. Lebaud, J. Math. Pures Appl. 73, 523 (1994) R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, (Springer-Verlag, New York, 1948). A. Majda, IMA , 217 (1990) C. S. Morawetz, Comm. Pure Appl. Math. 47, 593 (1994) S. X. Chen, Science in China 4 1 , 39 (1998)

67 I N D E F I N I T E ELLIPTIC P R O B L E M S W I T H CRITICAL EXPONENTS WENXIONG CHEN * Department of Mathematics Southwest Missouri State University CONGMING LI + Department of Applied Mathematics University of Colorado at Boulder

Dedicated to Professor Xiaqi Ding on the Occasion of His 70th Birthday A b s t r a c t . In this paper, we study the elliptic boundary value problem in a bounded star-shaped domain Q, in Rn, with smooth boundary: f -Au = R(x)up, \ u(x) = 0,

u > 0,

x G ft, x e dQ,

where R(x) is a smooth function that is allowed to change signs. We obtain a priori estimates on the solutions for all exponent p up to the critical number, that is, 1 < P < ^ f - This generalized a result of Berestycki, Dolcetta, and Nirenberg [3].

1

Introduction

Let ft be a bounded star-shaped domain in Rn (n > 3) with smooth bound­ ary. let R(x) be a smooth function defined on A. We consider the following semilinear elliptic boundary value problem f -Au

= R{x)up,

\ u(x) = 0,

u>0,

xeTL,

x e on.

m

^

In the case when R changes signs, it becomes more difficult. The problem is then called one with indefinite nonlinearity (see [1, 3, 4] and the references * Partially supported by NSF Grant DMS-9704861 tPartially supported by NSF Grant DMS-9623390 and NSFC 19471082

}

68

there in). In [3], Berestycki, Dolcetta, and Nirenberg considered the following more general problem with indefinite R(x):

-Lu = R(x)g(u), u > 0, xGfi,

Bu{x) = o,

x e on

(2)

where L is a linear elliptic operator and Bu = 0 imposes a Dirichlet or an oblique boundary condition. Under the assumption that the nonlinearity g(u) grows essentially like up with 1 < p < ^ 3 j , they established a priori estimates on the solutions of (2) and hence proved the existence of a solution. Note that the upper bound for p there is a little bit away from the usual critical Sobolev exponent ^ | . This was due to the difficulty in a neighborhood of R(x) = 0. In this part of 17, the elliptic theory does not work, nor does the standard blowing up argument. In [3], Berestycki, Dolcetta, and Nirenberg introduced a new blowing -up analysis. If a sequence of solutions went to infinity at R(x) = 0, by a proper rescaling, they were able to obtain a limiting equation in Rn. This would contradict to the Liouville Theorem (non-existence of solutions) established in that paper [3]. The restriction on the exponent p was due to the Liouville Theorem. Then Nirenberg asked: If one can fill the gap of the exponent up to the critical number 11^;1 To answer this question, in article [10], we used the method of moving planes to compare the values of the solutions near R(x) — 0, and thus obtain an a priori estimate. This method works for all exponent p in a neighborhood of R(x) — 0. In the region where R is positively bounded away from zero, we used a standard blowing-up argument similar to that in [3] and thus established an a priori bound on the solutions for all 1 < p < ^ | . In the remaining critical exponent case where p = ^ | , the major difficulty one encounters is the boundary estimate. We will deal with it in this paper. To better illustrate the idea, we concentrate our attention on the essential issue, the range of the exponent, and only demonstrate our proof on the model problem (1). Interested readers may make some kinds of generalizations to problem (2). Concerning the indefinite function -R(x), our first three assumptions are the same as those in [3]:

(R1)R(x)ec2((i), (R2) ft+ := {x G 0 : R(x) > 0} and Q~ := {x G ft : R(x) < 0} are nonempty, and {R3) r := n+ n TF c fi, with S?R(X) / o Vx G r. Since we are dealing with the critical exponent case, the following 'flatness condition' is needed: (-R4) For any positive critical point x° of i?, there exists a = a(x°) > n — 2,

69 such that, in a small neighborhood of x°, n

R(x) = R(x°) + J^ ai\xi ~ XT

+ o(\x -

x°\a),

2=1

where a{ = ai{x°) ^ 0, YJUi a* # °Roughly speaking, the 'flatness condition' requires that, at every positive critical point of R, the derivatives of R up to the order n — 2 vanish while some higher order derivative is nonzero. A similar condition has been used by several authors (see [11, 12, 15, 17, 19]) to guarantee the boundedness of the solutions for the prescribing scalar curvature equation on Sn. What we listed is a typical one. For generalizations of this condition, please see [15]. Here due to the boundary effect, we do not require that a < n. Our main result is Theorem 1 Let p — ^ | . Assume that the function R satisfies (R\), (Rs) and (R4). Then the solutions of (1) are uniformly bounded on ft.

(R2),

Remark 1 For the subcritical exponent case 1 < p < ^ | , we obtained the same estimates without requiring the domain to be start shaped and (R4). Cer­ tain estimates have been obtained in the critical exponent case for different boundary conditions (see [14], [16], [20] and references therein). To obtain the a priori estimate, we divide ft into two parts: ftj := {x G ft : R(x) < 6} and ft^ := {x G ft : R(x) > 5}. In [10], we have obtained estimates in ftj for some 6 > 0 and for all p > 1. (See sections 2 and 3 in [10].) What left is to show that the solutions are also uniformly bounded in ft~$ where R is positively bounded away from zero. To this end, we use a blowing up analysis. A similar kind of argument has been used by Li [15] and Schoen and Zhang [19] when they considered prescribing scalar curvature equation for positive functions R(x)on Sn. Their results can be used to infer that, in the interior of fl^ the set of blow up points of the solutions of our equation are discrete. To obtain the boundary estimate, and finally the estimate on the whole 0, we need to employ some new approaches. From the condition (-R3), one can see that, near dft, it is either (i) R(x) < -c < 0, or (ii) R(x) > c > 0. In the first case, by our estimate in ftj (see [10]) and the interior estimates of Li [15] or of Schoen and Zhang [19], one immediately obtains an a priori bound for the solutions on ft. Therefore, in the rest of the article, we assume that R(x) > c > 0 near Oft, and establish a boundary estimate.

70

2

The Blowing Up Analysis Near the Boundary

In this section, we estimate the solutions in a neighborhood of the boundary dCt assuming that R(x) is positively bounded away from zero in this region. Before illustrating our main approach, we need to introduce a basic defini­ tion and a Lemma. Let {ui} be a sequence of solutions of (1) with p — T :— ^ | . Definition 2.1 A point x° G Rn is called a blow up point if there exists a sequence xl tending to x°, such that Ui(x1)—too. A blow up point x° is called isolated, if there exists C, ro > 0, and a sequence {x1} tending to x°, such that u x

i( )

<

C

-l^T

\x — Xl\

VX G

2

Bro(x°).

Let Si = {x | x is a local maximum of Ui in Hj~}. If {ui} is not bounded, then it may blow up along some of the points in Si. Similar to [7] (see also [15, 19]) we are able to select a subset Si consisting of those 'bad' points. While away from the 'bad' points, the solutions are bounded. More precisely, we have Lemma 2.1 For every constant C, there is a subset S{(C) consisting of points p G Si with Ui(p) > C and a constant K(C) Si(C), we have

depending on C, such that for any two points p,q G Ui{p)[d{p,q)]^ Ui(x)[d(x,Si(C)]^

>C,

(3)


(4)

Inequality (3) shows that the distance between any two points in Si(C) can not be too close (but may still tend to 0), and (4) indicates that away from Si(C), the solutions are bounded, that is, Si(C) contains all the possible blow up points. To obtain an a priori estimate, one only need to show that, for some fixed C, Si(C) are empty as i gets large. Outline of the proof of Theorem 1. The interior estimates of Li [15] or of Schoen and Zhang [19] imply that Si(C) are discrete in the interior of Qf. we will show that Si(C) are bounded away from the boundary dft for some fixed C. To this end, we first show that §i(C) are also discrete in a neighborhood of dft.

71

For convenience, fix C — 1 and denote Si = §i(l). distance between the two points p and q. Define G{ - d(p\ql) := min d(p,q).

Let d(p,q) be the

We want to show that o~i > c0 > 0, for all i.

(5) l

i

Suppose in the contrary, ai~>0. Assume that d{ \— d(p ,dfi,) < d{q ^dQ) and p{->p° e dQ. Let There are three possible cases: (i) Xi = o(di) and u{ - 0(di) , (ii) Xi - o(di) and di = o(or») . (iii) di = O(Xi) , Case (i) is an essentially interior blow up situation. We will easily eliminate it in subsection 2.1. The main difficulty lies in case (ii). After rescaling, one obtains a sequence {vi}, which blows up at one and only one point in a half space. In this situation, the arguments in [15] or in [19] (which were based on two blow up points ) can not be applied. Instead, we will use Pohozaev identities in a large half ball to derive a contradiction. This will be carry on in subsection 2.2. Based on the result in subsection 2.2, we can make a proper rescaling in case (iii) and arrive at a limiting solution of the similar equation in a half space. This will contradict to the well-known classification result. After proving the discreteness of Si, an entirely similar argument will show that Si is bounded away from dft. Therefore, the solutions can only blow up at finitely many points in the interior of fl^. Finally, in Section 3, using a Pohozaev identity and under the star-shape assumption on <9H, we establish a uniform bound on the solutions. The following lemmas are basic tools in the proof. Lemma 2.2 (Pohozaev type identities) Let u be a solution of (1) in D and v the outward normal vector on dD, then I x-\/RuTJrl JD

= / [ -{—X'\ju--X'v\\ju\2)+nu JdD n — 2 ov 2

— +x-vRuTJrl}, ov

(6)

and f s7Rur+1 JD

= f

[^(pVu

J dD n — z OV

- J | V « l M + RuT+H z

(7)

72

The proof of these identities are standard and can be found, for example in [8] or [15]. Lemma 2.3 Assume that B2(0) C Q} and R satisfies (Ri). Let xl^0 be an isolated blow up point in B2(0) Then there exists a positive constant C, such that i( *! —)^ CK\? + ki\x-x*\2J

< Ui(x) < , .Nl °—r—r ~ V ; - tii(x*)k-»,n"2

for all \x - xl\ < 1,

(8) where A* = u^x1)'^^ and ki = ^J2)' Furthermore, for some harmonic function h(x) in Bi(0), we have, possibly passing to a subsequence, that Ui(xi)Ui{x)^-^

+ h{x),

in

Cfoc(Bi\{0}),

(9)

\x\ where n-2

a = lim ki

2

.

i—^oo

This lemma indicates that, near an isolated blow up point, {ui} behaves almost like a family of standard solutions. For the proof, please see Proposition 2.2 and 2.3 in [15].

2.1

Eliminating case (i).

In case (i), make a rescaling Vi(x) - a{

n-2 2

Ui(aiX

+pl).

Obviously, Vi has only isolated blow up points. If Oi ~ Xi (here ~ means 'in the same order as'), then a standard argument implies that vi converges weakly in Hl(Rn) to a function v0 satisfying -Av0=R(p°)vT0,

xeRn,

v o (0) = l.

By the well-known classification result, v0 has only one local maximum. On the other hand, by Lemma 2.1, vi converges to v0 in C2(.B2(0)) and each Vi has at least two local maxima in # i ( 0 ) . This contradicts with the above classification result. Now assume that Aj = o(ai). In this situation, Vi blow up at two interior points with finite distance between them. Similar to [15] or [19], one can derive a contradiction.

73

2.2

Deriving a contradiction in case (ii).

Outline of our approaches. Make a rescaling. Let Vi(x) - di

n-2 2

Ui(diX 4-p 2 ).

After the rescaling, Vi is defined in a domain Di which converges to the half space H := {x = (xi, ■•-,£„) e Rn :xn> - 1 } . Notice that A; = o(di) implies Vi(0)->oo while di = o(ai) indicates that the images of ql are pushed to infinity. Hence the sequence {v^ is blowing up at one and only one point in the half space H. First we will show that

^ ( ^ ( 0 ) - { ^ -

R T

^ +^ }

do)

w i t h a ^ l i m t ^ ^ ] 1 1 ^ and e = ((),••-,0,2). Then we will apply Pohozaev identities in a large half ball in H to derive a contradiction. Step 1. Deriving (10) . First, by Lemma 2.3, for any r < 1, Vi(x)vi(0) < - ^ - , Vx G Br(0).

(11)

One may regard Vi as the solution of the linear problem J — Avi = Ci(x)vi x G Di i Vi(x) =0 x G dD.

(12)

Applying the well-known Harnack inequality (see [5]), we conclude that Vi satisfies (11) in any bounded set in Di and it follows that Vi(x)vi(0)-+w(x)

:= p r ^ + h(x) xeH \x\

(13)

hat Vi(x) — 0 on dDi, we write

where h(x) is some harmonic function. Due to the fact t

74

Here b{x) is another harmonic function on H with zero boundary value on dH. By the wellknown classification result on harmonic functions on half spaces, we must have b(x) = c(xn + 1) (14) for some constant c. More precisely, we must also have c — ^ r - Otherwise, h(0) ^ 0, and we can apply the similar argument as in [15] to derive a contradiction as follows: Apply the Pohozaev identity (6) to V{ on a small ball Bp(0), and multiply both sides by v2(0). Then the left hand side tends to 0, while the right hand side converges to _( n ~ 2 ) \Sn~l\h(0). This is impossible. To summerize, we have

~a

h{x)= v ;

2+%±i>.

\x + e\n~2

(15)

2n~2

v

}

Step 2. Applying the Pohozaev identities to derive a contradiction. Now we use the Pohozaev identities to V{ on G*, the intersection of some large ball Br(0) with D{. First, Vi satisfies -Avi

= RidiX+p^vJix)

x e d.

(16)

If we apply (6) to v\ and multiply both sides by v2 (0), then the left hand side becomes II = divUO) I x • \/R{d{x +pi)vT+1, (17) JGi

while by (13), the right hand side converges to the integral _

f r 2n (dw 1 . l2x / ^ ( ^ ~ x ' V^ - ox " v V^ ) + JdGn-2 vv 2 We prove the following IR

Claim 1.

=

IL~+0,

as

dw. nw-^-]. ov

(18)

i^oo.

Claim 2. IR becomes very negative as r gets large. Obviously, these two claims contradict with each other. Proof of Claim 1. Case (a). S7R{p°) ^ 0. First apply the second Pohozaev identity (7) to vi on B1/2(0) and multiply both sides by v2(Q). By (13), the right hand side is bounded. And consequently, by (8), there is some constant C, such that div2{0) < C

(19)

75

On the other hand, through a straight forward calculation, one can easily verify that

" MV+1~7f,

(20)

/.

where D is any bounded domain in Di and 7* = [v{(0)] n~2. Now Claim 1 follows from (19), (20) and the boundedness of \/R. Case(b). V#(p°) = 0 . In this case, the 'flatness condition' (i?4) applies. Let x1 — pl — p°. If \xl\ = 0(7i), then the 'flatness condition', the bound­ edness of \/R, a n d (20) imply immediately that

\IL\ < CdivU0h?^0. Notice here a > n — 2. Therefore, we may assume that there is a sequence of numbers {K{}, with \x{\ > Km

and K^oo.

(21)

On one hand, applying the second Pohozaev identity (7) and by a straight forward calculation, one can verify that divlmxY-1
(22)

On the other hand, applying the first identity (6) and by the 'flatness condition' (R4), one arrives at \IL\
(23)

Now (22) and (23) imply that / L - ^ 0 , since |x*|-+0. This proves Claim 1. Proof of Claim 2. We evaluate IR on dG for the function w(x) = , 1-* + h(x) with h

]_

, x n -f-1

^) = ~^ x +, e\, w _ 2 +

9n_2

Here for convenience, we set a = 1 and this will obviously not alter the sign of IR.

Divide dG into two parts, the curve part <9iG:=<9B r (0)n#, and the flat part d2G:=Br(0)ndH.

76

On <9iG, the integral becomes fdiG[-n(n

- 2)r*-»h - n r » - g ] + ^

fdlG[r(§)2

- ||V^|2 +

*?h%]

:=h+h. (24) On ^ G , taking into account that xn = — 1 and w = 0, we reduce the integral into

•kofelV"!2+««£] = (25) Applying the Pohozaev identity (6) to the harmonic function h (with R = 0), we find h + I 4 = 0. (26) We will show that h < 0, for sufficiently large r;

(27)

and 73—> — 00, as r-»oo. (28) To see (27), we only need to consider the highest order terms in r. Use a spherical coordinate, write xn = rcosd. Then for sufficiently large r, we have, on d\G,

* ~ - i ^ +! : ^ > ° , f o and

dh o

0«>
(29)

n —2 cosO n „ ^ ^ -K T + T;—o>0 f o r O < 0 < - .

, x (30)

r

Here in (29) and (30), for § < 6 < § + e ( e is very small for r large), one has cosO < 0, however, by symmetry, this negativeness is overcome by the positiveness of cosO in ( | — e, f ) . This verifies (27). To derive (28), we notice that the integral of the first four terms in 73 is bounded. For the fifth term, we have 2_ndh

|x|2-"

It follows that Is—> — 00 as r—>oo. This completes the proof of Claim 2 and thus derives a contradiction in Case(ii).

77

2.3

Eliminating case (iii) .

Let Vi(x) — \

n-2 2

Ui(XiX + pl)

Lemma 2.1 guarantees that Vi converges in #1/2(0) with Vi(0) = 1. This in fact implies the following Claim 3. vi converges weakly in H1 to a function v in some half space H := {x = (xi, • • •, xn) E Rn : xn > c] satisfying

{

-Av = R(p°)vT t/(0) = 1

xeH (31)

v(x) = 0 xedH. However, due to the wellknown classification result, problem (31) has only trivial solution. This is a contradiction. Now what left is to justify Claim 3. Suppose V{ does not converge weakly, then it would blow up at some point zi with d(zl,dDi)—>0. Let Z{ be the preimage of z\. Let r; = d(zl, dft). Make a rescaling n-2

Wi(x) - ri

2

Uifcx + z1).

Then W{ has only one possible blow up point in H. Similar arguments as in Case (i) and (ii) will lead to a contradiction.

3

Completing t h e Proof of Theorem 1

So far we have established (5), which indicates that Si are discrete. Further­ more, we can show that Si are bounded away from dft. In fact, suppose in the contrary, there exists pl E Si with d(pl, d£l)-*0. Now there are only two possi­ bilities: Xi = o(di) or di = O(Xi). One can use an entirely similar argument as in subsection 2.2 or 2.3 to derive contradictions. Therefore, the solutions are uniformly bounded in a neighborhood of dft. It follows that the solutions {ui} can only blow up at finitely many points in the interior of f£^~. Assume that

are those isolated blow up points. Again we will use the Pohozaev identity to derive a contradiction. First apply (6) to equation (1) in fi \ (Be(pl), • • • ,Be{pm)) for some small e and multiply both sides by uf(pu). By Lemma 2.4, we have

ii,(p»)«i(*H«,(*) :=

|g_%|n.a

+ • • • + | g _ ^ | , - 8 + M*)

(32)

78

with some harmonic function h(x) in ft. It follows that -™ E

f

r

2n ,dw

1

.

l9,

dw.

f

:r 2 [^^(^7 -V^-^^-HV^| ) + ^ —V ] = / du l

"=i /

k n 2

JdBe{p ) - '

°

n

.

l2

2 -—-x-i/lvH n Z

JdQ -

(33) where z/ is the unit outward normal vector. Since ft is star-shaped, the right hand side is positive. Hence at least one of the integrals on the left is positive, say f

. 2n ,dw

/

h—o^x

1

-^w-ox-

. v

l9x

dw,

„

\VH ) + nwar] > °-

/rtJX

(34)

On the other hand, applying (6) on Be(p1), we find that the left hand side u2i(pU) [

x-vRuT+i-tO,

however (34) infers that the right hand side must be positive. A contradiction. Therefore, {ui} are uniformly bounded in ft. This completes the proof of Theorem 1.

References [1] S. Alama, G. Tarantello, On semilinear elliptic equations with indefi­ nite nonlinearities, Calculus of Variations and PDE, 1(1993), 439-475. [2] A. Bahri, J. Coron, The scalar curvature problem on three-dimensional sphere, J. Funct. Anal., 95(1991) 106-172. [3] H. Berestycki, I. C. Dolcetta, L. Nirenberg, Superlinear indefinite el­ liptic problems and nonlinear Liouville theorems, Topological methods in Nonlinear Analysis, 4(1994), 59-78. [4] H. Berestycki, I. C. Dolcetta, L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, to appear. [5] L. Caffarelli, G. Fabes, E. Mortola, S. Salsa, Boundary behavior of nonnegative solution of elliptic operator in divergent form, Indiana Univ. Math. J. 30(1981), 621-640. [6] W. Chen, C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145(1997), 547-564. [7] W. Chen, C. Li,

Prescribing scalar curvature on 5 3 ,

preprint, 1996.

79 [8] W. Chen, C. Li, A note on Kazdan-Warner type conditions, Geom. Vol. 41, No. 2(1995) 259-268.

J. Diff.

[9] W. Chen, C. Li A priori estimates for solutions to nonlinear elliptic equations, Arch. Rat. Mech. Anal., 122(1993) 145-157. 10] W. Chen, C. Li Indefinite elliptic problems in a domain, and Continuous Dynamical Systems, 3 (1997), 333-340.

Discrete

11] A. Chang, P. Yang, A perturbation result in prescribing scalar curva­ ture on 5 n , Duke Math. J., 64(1991) 27-69. 12] A. Chang, M. Gursky, P. Yang, The scalar curvature equation on 2and 3-spheres, Calc. Var., 1(1993) 205-229. 13] J. Escobar, R. Schoen, Conformal metric with prescribed scalar cur­ vature, Invent. Math. 86(1986) 243-254. [14] Z. H. Han, Y. Y. Li, The Yamabe problem on manifolds with bound­ ary: existence and compactness results, preprint. [15] Y. Y. Li, Prescribing scalar curvature on Sn and related problems, J. Differential Equations 120 (1995), 319-410. [16] Y. Y. Li, The Nirenberg problem in a domain with boundary, Topological methods in nonlinear analysis 6(1995), 309-329. [17] Y. Y. Li, Prescribing scalar curvature on Sn and related problems, Part II: Existence and compactness, Comm. Pure and Appl. Math. 49 (1996) 541-597. [18] P. Rabinowitz, Minimax methods in critical point theory with appli­ cations to differential equations, CBMS Lecture No. 65, 1986. [19] R. Schoen, D. Zhang, Prescribed scalar curvature on the n-spheres, Calculus of Variations and Partial Differential Equations, to appear. [20] M. Zhu, Uniqueness results through a priori estimates, Part I: A 3dimensional Neuman problem, preprint.

80

A N EXPLICIT E X A M P L E OF STABLE A N D INSTABLE M O T I O N S IN FLUID M E C H A N I C S

Department

Institute

ZHI-MIN C H E N of Mathematics, Tianjin University, Tianjin 300072, P.R. China E-mail: [email protected]

of Applied Mathematics, Beijing 100080,

Chinese Academy P.R. China

of

Sciences

Dedicated to Professor Xiaqi Ding on the Occasion of His 70th Birthday A b s t r a c t . The dynamical behavior of incompressible viscous fluid motions excited by an external sinusoidal force is studied. This forcing gives rise to the occurrence of the basic steady flow wo = (sin 22,0,0). It is shown that there exist exactly twelve global branches of different steady flows bifurcated from uo, which, on the other hand, is local stable in an inifinte-dimensional flow invariant subspace irrespective of the magnitude of the Reynolds number.

1

Introduction

From the well known criteria of Landau [15] and Ruelle and Takens [25], Much of the flow behavior at the verge of transition from laminar to the turbulent state can be understood through a series of steps of bifurcations, yet it is diffi­ cult even in examining primary bifurcations. For example, the hydrodynamic instabilities of the parallel shear fluid flows have generally been studied by con­ sidering the Orr-Sommerfeld equations reduced from the linearized equations around a unidirectional basic steady flow (^i(z),0,0)(see Lin [16]). Although many more approaches have been provided (cf. [3,4,11]), there is still lacking exactly analysis for observed primary bifurcations of this basic flow. In 1959, Kolmogorov (see [1]) presented a simpler model describing the two-dimensional spatially periodic fluid motion excited by the sinusoidal ex­ ternal force k2(sinky,0). This forcing give rise to the unidirectional basic steady flow (sinky,0). The first analysis on this model is due to Mishalkin and Sinai [21] showing the global stability when k = 1 (see also [20] for an alternative approach to this result). Instabilities arise when k > 2 and some critical Reynolds numbers were observed by Iudovich [12](see also [2,18] for the lower bound estimates on Hausdorff dimensions of the associated global attractors), and each of the numbers is connected with a real eigenfunction of the governed linearized Navier-Stokes equations around the basic flow. In fact,

81

the eigenspaces are even-dimensional, and thus more careful examination is necessary for the bifurcation study. To overcome this difficulty, a flow invari­ ant structure was found for the Kolmogorov's model, which helps to give the existence of steady-state bifurcations and Hopf bifurcations (see [7,8]). For the study of this model from viewpoints of numerical experiment and applied mathematics, one refers to the investigations [22,23] and [10]. Moreover, the well known stability result due to Mishalkin and Sinai [21] has been extended to the three-dimensional case in [17]. One can also refer to Fujita et. al [9] for an interest study on the stability of the two-dimensional Navier-Stokes flows in Annuli. Recently, we obtained some stability and instability results for the threedimensional fluid motion driven by the external force k2(sinky,0,0) for any integer k. In this note, we shall take k — 2 as an special example to describe the nonlinear phenomena we observed recently. More precisely, we consider the three-dimensional spatially periodic fluid motion driven by the external force 4(sin2z,0,0). The hydrodynamic behavior of this fluid motion defined in terms of the velocity u and pressure p is described by the Navier-Stokes equations: dtu - Au + \(u-V)u + AVp = 4(sin2z, 0,0), m where A > 0 represents the Reynolds number defining the viscous fluid motion, and the velocity u satisfies the spatially periodic condition: u(t, x, y, z) = u(t, x + 27r, y, z) — u(t, x, y + 27r, Z) = u(t, x,y,z

+

2TT).

(2)

To ensure the uniqueness of the associated Stokes problem, we require the additional condition: /•27T

/

P2TT

/

/>2TT

/

udxdydz = 0.

(3)

Jo Jo Jo Similar to the shear fluid flows, this problem admits a unidirectional basic steady flow u0 = (sin 2z, 0,0). We intend to specify all of the steady-state solutions bifurcated from I/Q, and on the other hand, to present an infinite-dimensional flow invariant space, in which u 0 is stable irrespective of the magnitude of the Reynolds number. As a by-product of this stability observation, we obtain an example for the global existence of large strong solutions to three-dimensional Navier-Stokes equations. To help formulate the problem, it is convenient to introduce the Hilbert spaces: H2 = {ue L2([0,27r]3;,R3)| Au €

L 2 ([0,2TT] 3 ;

i? 3 ),

82

V • u — 0, u satisfies Eqs. (2-3)} , Hi = lueH2\ + ]C

u=

^(^n,r/n,Cn)sin(2nz)

] C (^n,m,^n,m,Cn,m)C0s(2ny +

2mz)

neN mez + ^2

^^n,m,f}n,rnXn,rn)sm(2ny

+

2mz)

neN mez + 5Z leN

Y2 (€l,n,m,m,n,mXl,n,rn)cOs(2lx n,meZ

+ 2ny +

2mz)

+ 5 Z ] C (6,n,m, ^/,n,m, 6,n,m) sin(2/x + 2ri?/-f 2mz) > , leN n,meZ

J

where TV and Z denote respectively the positive integer set and the integer set. The main results of this paper read as follows: Theorem 1.1 Let A > 0, and let ui e Hi with ||A(^i -

UQ)\\2

< 6,

for some constant 6 > 0. Then Eqs. (1-3) admit a unique solution u in the space C ( [ 0 , O O ) ; # Q )

suc

^

^ai

u(0) = ui and \\A(u(t) - u0)\\2 < ce~3\ where and in what follows \\-\\r denotes the Lr-norm, andc represents a generic constant. Theorem 1.2 There exist two critical Reynolds numbers and twelve branches of steady-state solutions 0 < A0 < Ai and {unA\ A > 0} C H2, n = 1,..., 12 with respect to Eqs. (1-3) such that un,\ = ^o for 0 < A < A o , l < n < 4 , Un,\ 7^ ^m,A ^ U0 for X0 < X,

1 < U / 777, < 4,

un,\ = u0 for 0 < A < Ai, 5 < n < 12, wn,A ^ um,\ ^ u0 for Ai < A, l < n / m < 12, and l|Vtxn>A||2 < ||Vti 0 ||2 =

4TT 3/ 2

for X > 0,1 < n < 12.

(4)

Additionally, there are no other steady-state bifurcation points along the half line {(A,UQ)| A > 0} ; and no other steady-state solutions branching offuo.

83

In fact, there also exist time-dependent periodic solutions branch off the steady state solution u0 (i.e., Hopf bifurcation). For convenience to understand the instability of the fluid motion, we now present in the following such a Hopf bifurcation result, of which the proof is rather lengthy and is to be given elsewhere. Theorem 1.3 Eqs. 1-3 admit two critical Reynolds numbers XH1, XH2 > 0 and six different time-dependent periodic solutions *>I,A, V2,\

for XHl

< X < XHl

+ e

and V3, A, • ■ •, ve,\ for

XH2

< X<

XH2

+e

such that lim vn\

-

UQ

for n = 1,2,

lim vn,\ —

UQ

for 3 < n < 6,

A->AHl

A—>AH2

uo y^ vi,\ ^ v2,\ ^ u0b for XHl < X< XHl H- e, 1

vn,\ i Vm,\ ^ u0 for 3
XH2

<X<

XH2

+ e,

where e > 0 is a small constant. Additionally, there are no other timedependent periodic solutions branching off UQ. Note that the exponential stability for small steady flows in bounded do­ mains is a simple problem. However, Theorem 1.1 gives a conditionally expo­ nential stability of the steady flow ?/o, which is not small and may lose stable. As an immediate application, it implies that Eqs. 1-3 admit some large global strong solutions, which are initially from the functions in HQ and near UQ. It still remains unknown that whether a three-dimensional local solution can be smoothly extended for all time, although partial regularity of weak solutions has been obtained (see [5,26,28]). For special examples of global existence results, we refer to [14,29] for axially symmetric solutions and [19] for helically symmetric solutions. As is well known, for a three-dimensional parallel fluid motion, the gov­ erned linearized Navier-Stokes system may be reduced to a two-dimensional one by using the Squire transformation (see, [16,27]). Thus three-dimensional in­ stability phenomenon near a steady flow may be observed from a two-dimensional problem (see [16] for the parallel shear fluid motion and [18] for the spatially periodic fluid motion). However, the Squire transformation is, in fact, not invertible, and thus is not suitable for bifurcation investigation. Instead of us­ ing Squire transformation, we shall study directly the spectral behavior of the

84

three-dimensional linearized problem by developing a technique from Mishalkin and Sinai [21] on continued fractions. Our examination is essentially based on the flow invariance observation extended from [6, 7,8]. It should be noted that exact analysis on the bifurcation study for a nondegenerate three-dimensional Navier-Stokes motion is rather limited. Theorem 1.2 and Theorem 1.3 seem the first contribution to this kind of investigation. This paper is organized as follows: Section 2 contains some basic lemmas on the spectral behavior of the linearized Navier-Stokes operator. The proof of Theorem 1.1 will be completed in Section 3. To give additional preparations for the proof of Theorem 1.2, Section 4 describes the vorticity formulation for Eqs. (l)-(3). Finally, Theorem 1.2 is to be shown in Section 5. This paper is dedicated to Professor DING Xiaqi on the occasion of his 70th birthday to express our admiration for his scientific work in the various fields of the mathematical sciences 2

Spectral Problem

Denoting by du/dt — Au + XAu = 0 the linearization of Eqs. (1-3) in H2 around the steady-state solution uo, we have the linear operator Au — XAu = Au — Xuo • Vu — Xu • Vi^o — AVp with p = - A _ 1 V • (u • Vu 0 +

UQ

• Vu).

To give preliminary preparations for the proof of the main results, we shall specify all positive eigenfunctions of the operator A - XA. Without lose of generality, H2 is supposed to be a complex space in this section. For integers / > 0, j G Z and k = 0,1, we define the subspaces of H2: H2\u=

^(£ n ,77 n ,Cn)sin(Zx+ jy + kz + 2nz) > , nez J

Ei,j,k =
^(£ n ,77n,Cn)cos(Za; + jy + kz + 2nz) > . nez J

Ei,j,k = lue I

We see that these subspaces are invariant with respect to A — XA, and this operator is defined by its spectral behavior in these subspaces.

85

Le

mma 2.1 For I G N, j G Z and k = 0,l,

with

0 n = sin(Z:r + jy + kz + 2nz) or cos(lx + jy + kz + 2nz) satisfy the spectral problem Au — XAu — pu = 0 with Rep > — 1. Then {£n} and {r)n} are uniquely determined by {Cn}Proof. After an elementary calculation, the spectral problem Au — XAu — pu = 0 can be rewritten in the form: ] > ^ f ( £ n + P)£n + y ( f n - l

~ f n + l ) + A ( C n - l + C n + l ) J (t>n=~Xdxp,

(5)

+ p)Vn+ir(Vn-l

(6)

nez '52[(Pn

~ Vn+l) ) n=-Xdyp,

(?) XI

SUmneZ

( {Pn + P)(n + y ( C n - 1 ~ C n + l ) J n=-Xdzp,

(8)

^ (f£n + jr/ n + (A: + 2n)Cn)0n=O, nez

(9)

where A* = i8n (l,h k) = l2+ j 2 + (2n + A:)2. From Eqs. (5-6) it follows that Yl

( (Pn + P)(!£n+jr)ny-

nez^

+ I T ( i f n - 1 + j ? M - l ~ i f n + 1 ~ i ^ n + l ) ) <\>n Z

J

= - ^2 A/(Cn-l+Cn+l)0n ~ lX8xp - jXdyp. nez Appllying the operator dz to this equation and the operator — ldx — jdy to Eq. (8) respectively, and summing the resultant equations, we have, for n £ Z, {Pn + p)(l£n+jrin)

+ -J(/£n-l

+ j ^ n - 1 ~ *&i+l ~ J>M+l) J ( 2 n + k)

= - A Z ( C n - l + C n + l ) ( 2 n + *) + ( (/3 n + p)Cn + y ( C n - 1 " Cn+l) ) (Z2 + j 2 ) .

let

86

This together with Eq. (9) implies, for n £ Z,

/3n(^n + p ) C n + y ( C n - l - C n + l ) ( / 2 + i 2 )

= j((2n

+ 2 + fc)C„+i - (2n - 2 + *)Cn-i + 2(Cn-i + Cn+i))(2n + *)

- j((2n

+ 2 + A:)2Cn+i - (2n - 2 + A;)2Cn-i + 4Cn-i ~ 4Cn+i)

From Eqs. (5-6), we may also derive the following equations without the pressure term involved: (Pn + P)U

+ y ( f n - l " f n + l ) + A(Cn-l + Cn+l) J j

= ((Ai+P>7n + y ( 7 7 n _ i - 7 7 n + i ) j /, U E Z. Thus Eqs. (5-9) become the couple set of algebraic equations, for n £ Z, 2/3 n (/? n +p) -Cn A 2(/? n +p)

titn-lVn)

+ l{j£,n-l-lr]n-l)

+ J ( / ? n - l ~ 4 ) C n - l = l{Pn+l

" 4)Cn+l,

(10)

~ / ( i f n + l - f y n + l ) = ~ 2 j ( C n - l + Cn+l), (11)

f f n + j * 7 n + ( 2 n + A;)Cn = 0.

(12)

On the other hand, we shall follow a technique from [21] to show the absence of nontrivial solutions to the couple set of equations

2

xt

n + Tn_1 Tn+1 = n ez

"

°'

'

(13)

Here {r n } is subject to the sumability condition J2nez l r ^l 2 < °°We may suppose rn ^ 0 for all n £ Z, since X ^ G Z l r ^l 2 < °° anc * Eq. (13) with r n o = 0 imply either rn = 0 or -, ^ i r n 0 +n+l | ^ 2(/? nQ+n + Rep) 1> I 1 > —

> oo as n -* oo for n 0 > 0,

87 I T~n0— n—1

1 > |—

T~no — n

,1 ^> 2 ( / 3 n o-—— _ n M+ Rep)

» oo as n -> oo for no < 0,

Dividing Eq. (13) by r n gives - ^ 2 -

=

^

= 0, n > 0,

XI

(14)

r±n

and so 0 as n —> oo.

T

±(n-1)

Applying Eq. (14) repeatedly gives T±n

=Fl

T±(n-1) — 2(P±n + p) A/

r

\ n

for n > 0.

1 2Q9±(w+1)+p) XI

1

Since 0 _ n = fin w h e n A: = 0 a n d (3-n = / 3 n - i w h e n A: = 1, we have T_i

Ti

r0

To

1

2(^1 + p) A/

-, when k = 0,

1 2(02 + p) XI

+

J_

and r_i r0

r0 r_i

1 2(/3o + p) A/

yhen A: = 1.

2(/9i+p) XI

1

Observing that 2(A) + p)

[

A/

T-i

Ti

T0

To

= Q

we have ^

+ i

(

A

+

r t XI

'

I

2(/3 2 + p) A/

= 0 . whe„* = 0, 1

(15)

and 2(A)+ P) + TTTQ \ XI 2(^+p) XI

i

— h when k — 1,

(16)

+

2(fo+p) A/

1

This leads to a contradiction, since the real parts of the right hand sides of Eqs. (15-16) are positive. Thus Eq. (13) has no nontrivial solution. Applying this criterion on Eq. (13) and using the Riesz-Schauder theory, we readily seen from Eq. (11) that {j£n — lr]n} is uniquely determined by {£ n } This together with Eq. (12) implies the desired assertion. The proof is complete. From the proof of Lemma 2.1, we deduce readily the following. Corollary 2.1 There holds, for integers j > 0 and k = 0 , 1 , dim{u e Eo,j,k U ^o,j,fc| Au — XAu — pu = 0} = 0. Now we proceed to study on the spectral behavior of the operator A — XA reduced in Eij^ U Eij^ with / > 1. L e m m a 2.2 Let I G N, j G Z, X > 0 and Rep > -I2 - j 2 . Then dim{u e EtJA

U Ettjil \ Au - XAu - pu = 0} = 0 if Imp = 0,

(17)

dim{u e

U Eljji0\ Au - XAu - pu = 0} = 0 if Imp ^ 0,

(18)

and when (l,j)^

EIJJ0

(1,-1),

(1,0), (1,1),

dim{u € Eiiji0 U Eltji0\ Au - XAu - pu = 0} = 0.

(19)

Proof. With the use of Lemma 2.1, we see that it suffices to deduce the desired assertion by examining Eq. (10). Following the derivation of Eqs. (15-16) from Eq. (13), we deduce from Eq. (10) that, for n ^ 0, (^~4K ±n _ (/?±(n-l)—4)C±(n-l)

=F1 AJ(#fc„-4)

20Mn+1)(0±(n+1)+p) A/(/3±(„+i)-4)

(20)

1

89

Deviding Eq. (10) by (/?„ - 4)£„ implies 2/?n(/3n+/>)

(/?„-! - 4 ) C n - l

AZ(/3„-4) Thus Eq. (10) becomes 2 W O + P) +

M(P0-4)

(P„+l - 4)Cn+l

=

(iS»-4)C„

,

(/3n-4)Cn

l 2A(A+p) AZ(A-4)

+

'

. , . . = . , w h e n * = l,

1 2lh(Jh+p) A/(/3 2 -4)

,01, (21)

1 ..

and

T77^

7T + o/a //a

A/(/3 0 -4)

■—^

2/?i(/?i+p) A/(A~4)

=

1

1 2/?2(/?2 + p) A/(/? 2 -4)

°>

wnen

* = 0-

(22)

1 ..

+

Obviously, Eq. (21) implies Eq. (17). In showing Eq. (18), we follow a technique from [17] to argue by contra­ diction. Suppose that p with Imp ^ 0 satisfies Eq. (22). We see, for n > 0, |ar

H Ai(fem-4) ; i = i" g (^ ±m +p)i I

, 2/J±(m+i)(/3±(TO+i) + />) A/(/? ± ( m + 1 ) - 4)

and thus, from Eq. (15), ar lars(-T77^—rr)l s( 2/?±i(/?± 1 +p) A/(/?0 - 4) " = 'l 6V

A/(/5±i - 4)

|arg

^ A/(/3±i - 4)

2/? ±2 (l±2 + p) AZ(/9±2-4) +

2/3 ±2 (/3 ±2 + P ) A/(/3 ± 2 -4)

1

1

90 which leads to a contradiction, and arrives at Eq. (18). Moreover, we see that (IJ) f£ { ( 1 , - 1 ) , (1,0), (1,1)} gives/3 0 -4 > 0, which implies that the real part for right hand side of Eq. (22) is positive and thus Eq. (19) is valid. As an immediate application of Lemmas 2.1 and 2.2, we have Corollary 2.2 For A > 0 and Rep > — 4, there holds dim{u e HQ I Au - XAu - pu = 0} = 0. Finally, we present a result on the existence of an eigenvalue p(X) increasing across the imaginary line. Lemma 2.3 Let j = —1,0,1. There exists a function pj : (0, oo) -> R such that dpj(X)/dX > 0, lim pj(X) = - 1 — j 2 , A->-0+

lim Pj(X) — oo, A->-oo

dim{u G #i,j,o| Aw — XAu — pu — 0} < 1, and dim{u G -Ei,.7,01 Au — XAu — pu — 0} < 1 2

for Rep > —1—j and X > 0, where the equalities hold if and only if p — Pj{X). Proof. It follows from Lemma 2.2 that it suffices to consider the existence of the real eigenvalues in the half line (—1 — j 2 , oo). By the proofs of Lemmas 2.1 and 2.2, we see that the spectral problem A — A A — p = 0 reduced in Eijto U Eij,o is equivalent to Eq. (22), that is,

A(/9b-4)

+

j2+4n2.Multiplyingthisequationby

2(3^ + p) A(/3x - 4)

Recall that n (3= /3n(l,j,0) = 1 + -(Po-WPoWo + p))-1 yields

+

2(32(p2+p) A(/? 2 -4)

92{p)

1

1

+

J_ ..

-U"

91

where , x WnM0n+p)(0O+p) 9n{p) = —y- A ^~\nS 7\—>

,

.

,, >

w h e n n 1S o d d

, wQ8w + p ) ( 4 - A ) ) , when # n (p) = -2/?— n is even. Po(Pn - 4 ) ( / % + p)

dg2n(p) ^n i ,o

Denoting by G(A, p) the right hand side of Eq. (24), and observing for n E N, d#2n-i(p) ^n

,

— — > 0 and dp we obtain

—^- < 0, when p > - / 3 0 , dp

Hence the observation lim G(A, p) = oo and lim G(A, p) = 0 implies the uniqueness and existence of p = Pj(A) > — /?o = — 1 — j 2 satisfying | = G(A, Pi (A)).

(26)

Note that d(XG(\,p)) dX Eq. (26) gives for p = Pj(X), d(XG(X,p)) dA

=

> 0 for A > 0 and p > -/30-

d(XG(X,p)) dX

8G(X,p) dp dp dX

dG(X,p) dp dp dX'

This shows, by Eq. (25), dpj/dX > 0. Thus , multiplying Eq. (23) by A gives

Po(J3o + PiW) _ A, - 4 201(Jh+PiW) A 2 (/3i-4)

1

+

2/32(/g2 + Pi (A)) A - 4

1

92 Passing the limits A —> 0 + and A —► oo respectively, we obtain immediately lim pj(X) — — 1 — j

2

and

A—►()+

lim p.-(A) = oo. A—>oo

Now it remains to determine the eigenfunction u = S n 6 Z ( £ n , rjn, Cn)0n to the problem A - XAu - pjU — 0. On setting =fl l±r

2(3n(Pn+Pj(\)) A(/? n -4)

|

1 2/9 (w+1) Q3 (n+1) +p i (A)) A(/3 ( n + 1 ) -4)

1

we have, by Eq. (20),

and thus

(Pn ~ 4)C±n , . n 7±n = 7^ T^T forn>0, (Pn-1 ~ 4)C±(n-l) Co

= c,

where c is an arbitrary constant. From Lemma 2.1, we see that all the eigenfunctions with respect to the eigenvalue pj form a one-dimensional subspace contained in EIJJ0 and £ij,o respectively. The proof is complete. 3

Proof of Theorem 1.1

For convenience, we set A\ = —A + A A Note that A\ is an unbounded operator mapping Hfi into itself and the domain D{A\) is dense in H§. From Corollary 2.2, it not difficult to verify the resolvent estimates \P\ 1Mb + M 1 / 2 ||V U || 2 + ||Au|| 2 < c\\Axu + pu\\2 for u e HQ and Rep > —7/2, where c is independent of u and p. This im­ plies that the operator A\ generates a strongly continuous analytic semigroup e~tAx, t > 0 in Hi and there holds the following estimates: | | e - ^ | | 2 + ^/2||Ve-M^||2 + *||Ae-MH|2 <

ce-n/2\\u\\2,

93

l|Au|| 2 < c\\Axu\\2 < c||Au|| 2l | | 4 / 4 u | | 2 < c||Vu|| 2 for u G HQ . Here A% denotes the fractional power of the operator A\ in H§. On substitution of u — v + ^o G HQ into Eqs. (1-3), we have dtv + AA*; + \B(v)

C([0, oo); # 0 2 )

=0,ve

associated with the initial condition v(0) = v0 = ui -u0

£

HQ,

where B(v) = (v • V)v + Vp = (v • V)v - V A " 1 V • ((v • V)v). Thus on considering the solution v in the space C([0, OO);HQ) with v(0) = ^o, the integral equation

Jo

is equivalent to the above Cauchy problem. Therefore it remains to show the uniqueness and existence of solutions to this integral equation in the space C([0,<x>);tf02). Given a constants S > 0, we define

Ws = {ve C([0,oo);F 0 2 )| v(0) = v 0 , ||Av0||2 < *\ IHk,=supe3t||At;(*)||2<4,

Mv(t) = e~tAxv0 -X [ Jo

e-^-^Bivis^ds.

For v G Ws, we have

| | A e - M ^ o | | 2 < c | | ^ e - M * « o | | 2 < ce" 3 *||^ A ^o|| 2 < ce- 3 *||A«o|| 2 ,

94

e-{t-s)AxB(v(s))ds\\

||A f Jo
\\Axe-^-^B(v(s))\\2da

< c f\t - *)-3/4e-7<*-*>/2||4;/4£(i;(*))||2d* Jo

s)-3^e-7^-s^2\\VB(v(s))\\2ds


s)-3/4e-7^-s^2\\V((v(s)


-

S)-

• V)v(s))\\2ds

3 4

/ e-7^s)/2(||Vi;(S)||i +

\\v(s)\U\Av(s)\\2)ds

Jo

< c f\t

- s r ^ e - ^ - ^ I I A ^ H ^ < ct-3t\\v\\*w„

Jo

and hence ll-Mv||w, < c||Avo||2 + c\\v\\ws < c62 < 5, by setting cS < 1, where the use is made of the Holder inequalities, the Sobolev immedding in­ equalities and L2 estimates. Similarly, for v, v' G Ws, \\Mv-Mv'\\W6
\\v'\\ws)\\v-v'\\Ws

< c6\\v-v'\\w6 < (l/2)||v - v'\\w6,

by setting c6 < 1/2.

Moreover, note V-B(v) = 0 for v G H§. From the definition of iJg, it is readily seen that e~tAxv,

e-tA*B(v)

G H2 for v G H2.

This yields Mv G HQ for v G Ws, and by the strong continuity of e~tAx, it readily seen that M is a contraction operator mapping Ws into itself, provided that 5 is sufficiently small. Therefore, the contraction mapping principle gives the uniqueness and existence of the solution v to the integral equation in Ws. The proof is complete.

95 4

Vorticity Formulation

It has been obtained in Lemma 2.3 that the operator S — XA admits the real eigenvalue pj increasing across the origin. However, pj is not simple in the whole space H2. This leads to a difficulty in studying the steady-state bifurca­ tion problem in H2. Fortunately, we may find a flow invariant subspace other than HQ SO that pj is simple when the fluid problem is reduced in the subspace. For this purpose, it is convenient to consider the vorticity formulation of the stationary equations with respect to Eqs. (1-3): -AUJ

+ \B(LJ)

= 8 ( 0 , C O S 2 Z , 0 ) in H2,

(27)

where B(LJ) = -(A~ 1 Vxcj)-Vu;-hcc;-V(A~ 1 VxcL;). The steady-state solution u0 — (sin 2z, 0,0) now becomes UJQ — V x u0 — 2(0, cos 2z, 0) a solution of Eq. (27). Let us introduce the Sobolev space W2'2([0,2n)3;R)

= tyl 1>, M

e L2([0,27T]3;JR)} .

By the Sobolev embedding theorem, the multiplication (p-i/; G W 2 ' 2 ([0,2n] 3 ; R) whenever 0, I/J G W 2 ' 2 ( [ 0 , 2 T T ] 3 ; R). This allows W 2 ' 2 ( [ 0 , 2 T T ] 3 ; R) to be a Banach algebra. Setting ^2([0,2TT]3;JR)

=

W2>2([0,2ir]3;R)/R,

we see that iJ 2 ([0,27r] 3 ;i?) is also a Banach algebra for the induced multipli­ cation. For / G TV and j G Z, we introduce the Hilbert spaces: Hftj([0,2?r]3; i?)4he Banach subalgebra of # 2 ([0, 2TT]3; R) generated by the three modes cos2z, cos(/x -h jy) and cos(lx + jy + 2z), Hij([0,

Banach subalgebra of # 2 ([0, 2TT]3; R) generated by the three modes cos 2z, sin(/x + jy) and sin(/x + jy + 2z),

2TT]3; R}the

H2j = {u=

(LJUUJ2,UJ3)\

UULJ2,U3

e J^.([0,27r] 3 ;fl), V • u = 0} ,

96

Let us present several basic results on this vorticity formulation. Lemma 4.1 Let LJ G H2 be a solution to Eq. (27). Then ||Aw|| 2 < c(A4 + 1) for some constant c independent of X and LJ. Proof. Taking the inner product of Eq. (27) with parts, we have

A~XLJ,

and integrating by

IMIa < IMI2 = 4;r3/2, and hence, by Eq. (27),

||Au|| 2 < A d l A - ^ x w||6||Vu;||3 + I M U I V A ^ V x w || 2 ) + 4||wo||2 < c A | | W | | ^ 4 | | A W | | ^ 4 + 4||u;o||2
-Au

+ XALJ + XB(LJ) = 0 in

H 2,

where AUJ =

-(A"1VXCC;O>VCJ+CC;O-V(A"1VXCJ)-(A"1VXCJ>VCJO4-^V(A~1VX^

With the use of Holder and Sobolev inequalities, we readily see that for LJ G H2 \\Au\\2 < c H A ^ b ,

\\B(u)\\2
and V • ALJ = 0. Thus from the definition of H2 •, it is not difficult to verify the following.

97 Lemma 4.2 A 1A and A lB are compact and continuous operators mapping Hf ■ (resp. Hf-) into itself, and ||£(u,)|| 2 = o(||Ao;||i), u> € Hi ( resp. u € H?j), where I E N and j E Z. Additionally, note that UJ = V x u and AUJ = V x A A _ 1 V x a;. As an immediate consequence of Lemmas 2.2 and 2.3 and the definition of Hf^ we have the following result for the spectral behavior of the operator A. Lemma 4.3 For j — — 1,0,1, Rep > - 1 - j 2 and X > 0, we have dim{uj E HfJ

AUJ - XAUJ - puj = 0} < 1,

dim{uj E H2j\

AUJ - XAUJ - pu = 0} < 1,

and where the equalities hold if and only if p — Pj(X) with pj the function given in Lemma 2.3. 5

Proof of Theorem 1.2

Let us now begin with a bifurcation result to specify the location of bifurcated steady-state solutions. Lemma 5.1 For j — —1,0,1, denote by Xj the critical Reynolds number such that pj{Xj) = 0, where Pj(X) is given in Lemma J^.S or Lemma 2.3. Then the equations -AUJ

+ XB(UJ) = 8(0, cos 2z, 0)

admit two branches of solutions Witjt\,uj2j,\

(28)

£ Hi ■ such that

^i,j,A — ^2j,A — ^o 5 when X < Xj, Wn,j,\ 7^ ^m,j,A 7^ ^o, when X > Xj and 0 < m ^ n < 1. Proof. In order to use the Leray-Schauder degree method in the space H2j by following the proof of Krasnoselskii's theorem (see [13]), let us adopt the following notation: FX(UJ) = u - XA^AUJ

0 j ? s = {u E E\j

- AA _1 B(a;) for UJ E

| ||Ao;|| < s} for s > 0,

Hfj,

98

fijfr,e = {^ € Hfj

I e < ||A^|| < r} for 0 < e < r.

Now our purpose becomes to study the bifurcation of the equation F\(u) = 0 around the trivial solution. Let So > 0 be a constant sufficiently small. By Lemma 4.1, we may choose r such that 0 J > r contains all solutions of the equation F\ (LJ) — 0 in R\ ■ for 0 < A < Xj; + So- By Lemma 4.3, there exists a constant e = e(S) so that Fx(u;) ^OforO^ue

Qj,2e and Xj + S < X < Xj + 60.

-1

In view of Lemma 4.2, A A and A _ 1 i? are compact in H\ •. Hence, it is well defined the following Larey-Schauder degrees of F A over ®j,r,@j,e a n d fij.r.e with respect to 0: deg(F A , 0 ^ , 0) for 0 < A < Xj + £0, deg(F A , 0 j ? e , 0) and deg(F A , fiJ>jC, 0) for A,- + (5 < A < Xj + £0On the other hand, note that I — XA~XA in iJf • is invertible for 0 < A < Xj + $o and X ^ Xj, and due to Lemma 4.3, FX(CJ) # 0 for 0 ^ a; e 0 ^

and 0 < A < A,-.

We have, deg(F A , 0 j j r , 0) = deg(J - XA, ©,>, 0) = 1, 0 < A < AJ5

deg(F A , 0 />e , 0) - deg(7 - A A " M , 0/, e , 0) = - 1 , A, + S < X < Xj + 50, and so, by the continuity property and the choice of the ball 0 j , r , deg(F A , e i i r , 0 ) = 1, 0 < A < A, + * 0 . Hence deg(F A ,% r , e ,0) = d e g ( F A , 0 i i r , O ) - d e g ( F A , 0 j j e , O ) = 2 for Xj + S < X < Xj + So- This shows that Eq. (28) admits two different solutions ujijt\ and LJ2,j,\ in H\ ■ when A > Xj is close to Xj. From Lemma 4.3 it follows that (Xj,uo) is the unique bifurcation point on the half line {(A,CJ 0 )| A > 0} with respect to the space Hfj. Thus by the global bifurcation result of Rabinowitz [24], the bifurcated solutions continuously exist for all A > Xj and

99 ^I,J,A ^ 0 /

^2,i,A for A >

Aj.

Moreover we see that LJIJ^X and U;2,J,A are always separated by the stable manifold of LJ0 in H2j. This gives u>ij,\ ^ u2,j,\ for all A > A,. The proof is complete. Following the proof of Lemma 5.1, we have the following. Lemma 5.2 Let Xj with j = - 1 , 0 , 1 be given in Lemma 5.1. Then Eq. (28) admits two branches of solutions U ^ A ^ ^ A € Hfj such that ^3,j,A = ^4,j,\ = ^o, when A < A^^n,j,A 7^ ^m,j,A 7^ ^o 5 wAen A > Aj and 3 < m ^ n < 4. With these preparations, we can now proceed to the proof of Theorem 1.2. Proof of Theorem 1.2. By Corollary 2.1, Lemmas 2.1, 2.2, 2.3,4.3, 5.1 and 5.2, we see that Eq. (28) has and only has twelve branches of solutions bifurcated from LU0. TO obtain the desired properties, we recall from Eq. (22) that Xj satisfies the equation Pl 2 A j ;(A)-4) + - ^ 2/3? + A,-(A-4) '

"

;

= 0,

2%

1

A,-032-4)

..

(29)

where (3n — /? n (l, j , 0) = 1 + j 2 + 4n 2 . This equation yields A_i = Ai. To show the monotonicity AQ < Ai, we rewrite Eq.(29) in the form

1

9i U) + A

i

92(j) +

1 1

fls 0') ,

<74(j) +

1

where ...

2/%/?02

2/%fl,

pn - An2

**<>) = (4-^)^-4) = m

• i^r'

.

when n 1S odd

'

100

p ...0) =2/%(4-ft>) =

2/3„ fl> + 4n 2

4 - fl,

- u^m ^' Har-' IT'when n 1S even-

We see that # n (0) <
(XO,LJO)

for n = 1,2,3,4,

and eight branches of solutions (A,u; n) _i ?/ \), {\wn,i,\)

bifurcated from

(AI,CJO)

for n — 1>2,3,4.

Especially, wnj,\

€ Hij for n = 1,2 and j — — 1,0,1,

Vnj,\ £ Hij for n = 3,4 and j = —1,0,1. Since U\ ■ fi Hfj — H^2-, with the help of Lemma 2.2 it is not difficult to verify that Eq. (28) admits no solution other than t^o in the subspace H\^y This shows Wnj,A 7^^m,i,A for U = 1,2,771 = 3 , 4 , j = - 1 , 0 , 1 , A > A j .

Similarly, since

Hitjntf2^

= Hljntf2,-, = Hlfi for - 1 < j? j ' < 1,

Corollary 2.1 implies the nonexistence of the solution other than UQ to Eq. (28) in £T£0.This gives Vnj,\ # w mji A for 1 < n ^ m < 4, j = - 1 , 0 , 1 , A > Ai. Finally, observing that the bound presented in Eq. (4) is implied in the proof of Lemma 4.1, we thus obtain the desired assertions and complete the proof. A c k n o w l e d g m e n t s . The author would like to thank Professor S. Wang for helpful discussions. This research was partially supported by the National Natural Science Foundation of China.

101

1. Arnold, V.I. and Meshalkin, L.D., Kolmogorov's seminar on selected problems of analysis (1958-1959), Russ. Math. Surv. 15 (1), 247-250 (1960). 2. Babin, A.V. and Vishik, M.I., Attractors of partial differential evolution equations and estimates of their dimension, Russ. Math. Surv. 38, 151213 (1983). 3. Baggett J.S., Driscoll T.A. and Trefethen L.N., A mostly linear model of transition to turbulence, Phys. Fluids 7, 833-838 (1995). 4. Benney, D.J., The evolution of disturbances in shear flows at high Reynolds numbers, Stud. Appl. Math. 70, 1-19 (1984). 5. Caffarelli, L., Kohn, R. and Nirenberg, L., Partial reqularity of suitable weak solution of the Navier-Stokes equations. Comm. Pure. Appl. Math. 35, 771-837 (1982). 6. Chen, Z.M., A remark on the flow invariant problem of the semilinear parabolic equations, Isreal J. Math. 158, 293-303 (1991). 7. Chen, Z.M. and Price, W.G., Time-dependent periodic Navier-Stokes flows in a two-dimensional torus, Commun. Math. Phys. 179, 577-597 (1996). 8. Chen, Z.M. and Price, W.G., Long time behaviour of Navier-Stokes flows on a two-dimensional torus excited by a sinusoidal force, J. Stat. Phys. 86, 301-335 (1997). 9. Fujita, H., Morimoto, H. and Okamoto, H., Stability analysis of NavierStokes flows in Annuli, Math. Meth. Appl Sci. 20, 959-978(1997). 10. Green, J.S.A., Two-dimensional turbulence near the viscous limit, J. fluid. Mech. 62, 273-287 (1974). 11. Hamilton, J.M., Kim, J. and Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech. 287, 317-348 (1995). 12. Iudovich, V.I., Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech. 29(4), 587-603 (1965). 13. Krasnoselskii, A., Topological Methods in the Theory of Non-Linear Integral Equations, New York; Pergamon, 1964. 14. Ladyzhenskaya, O.A., Unique solvability in the large of threedimensioanl Cauchy problem for the Navier-Stokes equations in the pres­ ence of axial symmetry, Seminar in Mathematics, Steklov Mathematical Institute 7, 70-79 (1970). 15. Landau, L., On the problem of turbulence, Comptes Rend. Acad. Sci. USSR 44, 311-316 (1944). 16. Lin, C.C., The Theory of Hydrodynamic Stability, Cambridge; Cam-

102

bridge University Press, 1955. 17. Liu, V.X., Instability for the Navier-Stokes equations on the 2dimensional torus and a lower bound for the Hausdorff dimension of their global attractors, Commun. Math. Phys. 147: 217-230(1990); A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations, Commun. Math. Phys. 158,327-339 (1993). 18. Liu, V.X., Remarks on the Navier-Stokes equations on the two and three dimensional torus, Commun. Partial Diff. Eqs. 19, 873-900 (1994). 19. Mahalov, A., Titi, E.S. and Leibovich, S., Invariant helical subspaces for the Navier-Stokes equations, Arch. Rational Mech. Anal. 112, 19322 2(1990). 20. Marchioro, C , An example of absence of turbulence for any Reynolds number, Commun. Math. Phys. 105, 99-106 (1986). 21. Meshalkin, L.D. and Sinai, Ya.G., Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid, J. Appl. Math. Mech. 19(9), 1700-1705 (1961). 22. Okamoto, H. and Shoji, M., Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori, Kyoto Univ. Res. Inst. Math. Sci. 767, 29-51 (1991). 23. Platt, N., Sirovich, L. and Fitzmaurice, N., An investigation of chaotic Kolmogorov flows, Phys. Fluids A 3, 681-696 (1991). 24. Rabinowitz, P.H., Some global results for nonlinear eigenvalue prob­ lems, J. Functional Anal. 7, 487-513 (1971). 25. Ruelle, D. and Takens, F., On the nuture of turbulence, Commun. Math. Phys. 20 , 167-192 (1971). 26. Scheffer, V., Partial regularity of solutions to the Navier-Stokes equa­ tions, Pacific J. Math. 66, 535-552 (1976). 27. Squire, H.B., On the stability of the three-dimensional disturbulences of viscous flow between parallel walls, Proc. Roy. Soc. A 142, 621-628 (1933). 28. Tian, G. and Xin, Z., Gradient estimation on Navier-Stokes equations, preprint. 29. Ukhovskii, M.R. and Iudovich, V.I., Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech. 32, 52-62 (1968).

103

N O N L I N E A R D I F F U S I V E - D I S P E R S I V E LIMITS FOR MULTIDIMENSIONAL CONSERVATION LAWS

Centre

J O A Q U I M M. C. C O R R E I A de Mathematiques Appliquees, Ecole Poly technique, 91128 Palaiseau France & Departamento de Matemdtica, Instituto Superior Tecnico, 1096 Lisboa Codex, Portugal E-mail: [email protected]

Cedex,

P H I L I P P E G. L E F L O C H Centre de Mathematiques Appliquees & Centre National de la Recherche Scientifique, UA 756, Ecole Poly technique, 91128 Palaiseau Cedex, France E-mail: [email protected]

A b s t r a c t . We consider a class of multidimensional conservation laws with vanish­ ing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we establish that the diffusive-dispersive solutions are uniformly bounded in a space Lp (p arbitrarily large, depending on the nonlinearity of the diffusion) and converge to the classical entropy solution of the corresponding multidimensional hyperbolic conservation law. Previous results were restricted to one-dimensional equations and specific spaces Lp. Our proof is based on DiPerna's uniqueness theorem in the class of entropy measure-valued solutions.

1

Introduction

Nonlinear hyperbolic conservation laws arise in the modeling of many prob­ lems from continuum mechanics, physics, chemistry, etc. The equations be­ come parabolic when additional small scale dissipation mechanisms are taken into account: diffusion, heat conduction, capillarity in fluids, Hall effect in magnet ohydrodynamics, etc. From a general standpoint, hyperbolic equations admit discontinuous solutions while parabolic equations have smooth solutions. Discontinuous solutions, understood in the generalized sense of the distribu­ tion theory, are usually non-unique. It is therefore fundamental to understand which solutions are selected by a specific zero diffusion-dispersion limit. In this paper, we address this issue for multidimensional scalar conservation laws, and review previous work on the subject restricted to one-dimensional equations. Consider the Cauchy problem dtu + div/(ix) = 0, u(x,0)=uo(x),

(x, t) G JRd x IR+, xGlRd,

(1) (2)

104

where the unknown function u = u(x, t) is scalar-valued and the flux / : IR -» lRd is a given function. Smooth solutions to (1) also satisfy an infinite list of additional conservation laws: dtri(u)+divq(u)

= 0,


(3)

where 77 is a convex function of u. For discontinuous solutions, Kruzkov [5] shows that (3) should be replaced by the set of inequalities dtr)(u) + div q(u) < 0,

(4)

which select physically meaningful, discontinuous solutions. The condition (4) is called an entropy inequality; it is motivated by the second law of thermo­ dynamics, in the context of gas dynamics. By definition, an entropy solution of the Cauchy problem satisfies (l)-(2) in the sense of distributions, and addi­ tionally (4) for any entropy pair (77, q) with convex 77. Consider the following approximation of (l)-(2) obtained by adding a non­ linear diffusion, b : JRd —> IRd, and a linear dispersion to the right hand side of (1): dtu + div/(u) = div (e&,-(Vu) + 6d£.u) 3

V

u(x,0) =u£/(x),

xeJRd.

,

(x,t) 6 IRd x IR+, (5)

'l<j
(6)

Let u£>6 : JRd x [0, T] -> IR be smooth solutions defined on an interval [0, T] with a uniform T independent of e,6. In (6), u£0' is an approximation of the initial condition UQ in (2). Our main objective is to derive conditions under which, as e > 0 and 6 tend to zero, the solutions u£,s converge in a strong topology to the entropy solution of (l)-(2). When e = 0, equation (5) is a generalized version of the well-known Korteweg-de Vries (KdV) equation, and the solutions become more and more oscillatory as S -» 0: the approximate solutions do not converge to zero; see Lax and Levermore [6]. When S = 0, (5) reduces to a nonlinear parabolic equation, resembling the pseudo-viscosity approximation of von Neumann and Richtmyer [8]; in that regime, the solution converges strongly to the entropy solution. Therefore, to ensure the convergence of the zero diffusion-dispersion approximation (5)-(6), it is necessary that diffusion dominate dispersion. In­ deed the main result of the present paper establishes that, under rather broad assumptions (see Theorems 3.1-3.3 below), the solution of (5)-(6) tends to the entropy solution of (l)-(2) when e,5 -> 0 with \S\ « e. For clarity, the main assumptions made in this paper are collected here. First concerning the flux function we shall assume

105

( # i ) for some CUC[ u e IR.

> 0 and m > 1,

|/'(ti)| < d + C[ | u | m _ 1

for all

For the diffusion term, we fix r > 0 and assume (H2) for some C 2 ,C 3 > 0,

C 2 |A| r + 1 < A- 6(A) < C 3 |A| r + 1

for all A G IRd.

In the case 0 < r < 2, we will need also (Hz) D6(A) is a positive definite matrix uniformly in A € IRd. We remark that the diffusion bj(Vu) = dXju satisfies (#3). The case d = 1 of one-dimensional equations was treated in the important paper by Schonbek [9], where, in particular, the concept of Lp Young mea­ sures is introduced together with an extension of the compensated compact­ ness method for conservation laws. We follow here LeFloch and Natalini [7] who, also for one-dimensional equations, developed another approach based on DiPerna's uniqueness theorem for entropy measure-valued solutions [2] (see Section 2 for a review). Specifically one uses a generalization of DiPerna's result to Lp functions, due to Szepessy [10]. The present paper relies on a method of proof that was successful first in proving convergence of finite dif­ ference schemes. We refer to Szepessy ([10] and the references therein by Szepessy and co-authors) and Coquel and LeFloch [1]. Recent work by Hayes and LeFloch (see [3,4]) treats the transitional case where both terms, the diffusion and the dispersion, are in balance. Conver­ gence results in this regime cannot be obtained by the measure-valued solutions approach. 2

Entropy Measure-Valued Solutions

We include here basic material on Young measures and entropy measure-valued (e.m.-v.) solutions. First of all we will need Schonbek's representation theorem for the Young measures associated with a sequence of uniformly bounded in Lq. The corresponding setting in L°° was first established by Tartar [11]. In the whole of this subsection, q € (1, 00) and T < 00 are fixed. Lemma 2.1. (See [9]) Let {uk} be a bounded sequence in L°°((0,T); Lq{lRd)). Then there exists a subsequence still denoted by {uk} and a weakly-* measurable mapping v : IRd x (0, T) —> Prob(IR) taking its values in the space of nonnegative measures with unit total mass (probability measures) such that, for all functions g € C(IR) satisfying g(u) — 0{\u\s)

as \u\ -)• 00,

for some s E [Q,q),

(7)

106

(v(x,t),g) belongs to L°°((0,T); L^ c s (lR d )) and the following limit representa­ tion holds lim / / fc

g(uk(x,t))(j)(x,t)

->°° J JjRd

x

dxdt = / /

(v(x,t)>9) 0 ( M ) dxdt

J JJRd x (0,T)

(0,T)

(8) for all (j) G Lx(TRd x (0,T)) nL°°(IR d x (0,T)). Conversely, given v, there exists a sequence {uk} satisfying the same con­ ditions as above such that (8) holds for any g satisfying (7). We use the notation (^(Xj£), #) := JJR
in Ls((0,T);Lploc(]Rd)),

for all s < oo and p e [l,q);

for a.e. (x,t) G IRd x (0,T).

In (ii) above, the notation 5u(x,t) is used for the Dirac mass denned by ($u(x,t),g) = 9(u(x,t)),

for all g G C(H) satisfying (7).

Following DiPerna [2] and Szepessy [10], we define a very weak notion of solution to the first order Cauchy problem (l)-(2). Definition 2.1. Assume that / G C(JR)d satisfies the growth condition (7) and u0 G L 1 ^ ) n L*(IRd). A Young measure v associated with a sequence {uk}, which is assumed to be bounded in L°°((0,T);Lq(JRd)), is called an entropy measure-valued (e.m.-v.) solution to (l)-(2) if a t (i/ ( . ) ,|w-A;|)-fdiv(z/ ( . ) , sgn(u - k)(f(u) - f(k))) < 0,

for all k G IR, (9)

in the sense of distributions on ]Rd x (0, T) and lim i_>o+ tJ0JK

(i/(x s ), \u '

UQ(X)\

) dxds — 0,

for all compact set K C IRd. (10)

107

A function u G L°°((0,T); L 1 ^ ) n Lq{md)) is an entropy weak solution to (1.1) in the sense of Kruzkov [5] and Volpert [12] if and only if the Dirac measure Su^ is an e.m.-v. solution. In the case q = +oo, existence and uniqueness of such solutions were proved in [5]. The following results on e.m.v. solutions were proved in [10]: Proposition 2.1 states that e.m.-v. solutions are actually Kruzkov's solutions. Proposition 2.2 states that the problem has a unique solution in Lq. Proposition 2.1. Assume that f satisfies (7) and u0 G Lx(TRd) n Lq(JRd). Suppose that v is an e.m.-v. solution to (l)-(2). Then there exists a function u G Loo((0,T);L1(lRd)nLq(JRd)) such that f°r

"(x,t) = <**(*,*),

ae

-

(M) € ^

x (0,T).

(11)

Proposition 2.2. Assume that f satisfies (7) and u0 G L 1 (H d ) n Lq(JRd). Then there exists a unique entropy solution u G L o o ((0,T);L 1 (IR d ) DLq(lRd)) to (l)-(7) which, moreover, satisfies \\u(t)\\Lp(JRd) <

II^OILPCIR^),

for a.e. t G (0,T) and allp G [l,q].

(12)

T/ie measure-valued mapping i/(xj) — ^u(x,t) *5 ^ e unique e.m.-v. solution of the same problem. Combining Propositions 2.1 and 2.2 and Lemma 2.2, we obtain the main convergence tool: Corollary 2.1. Assume that f satisfies (2.1) and u0 G L1^) Pi Lq(JRd) for q d q > 1. Let {uk} be a sequence bounded in L°°((0,T); L (lR )) and let v be a Young measure associated with this sequence. If v is an e.m.-v. solution to (l)-(2), then lim uk = u in Ls((0,T);Lf(JRd)),

for all s < oo and all p G [l,q),

(13)

k—>oo

where u G L°°((0,T); L x (lR d ) n L 9 (IR d )) is tte unique entropy solution to (l)-(2). 3

Convergence Results

Throughout it is assumed u0 G L^IR/*) D Lq(lRd) and the initial data in (6) are smooth functions with compact support and are uniformly bounded in L1 (JRd) n Lq(JRd) for some q > 2. While in previous works [9, 7], a single value of q was treated, we can here handle arbitrary large values of q. For simplicity in the presentation, we will always consider exponents q of the form q = 2 + n(r-

1),

108

where n > 0 is any integer. Therefore, when the diffusion is superlinear, in the sense that (H2) holds with r > 1, then arbitrary large values of q are obtained. Restricting attention to the diffusion-dominant regime we regard 6 = S(e) and we suppose that UQ approaches the initial condition UQ of (2) in the sense that: limc_>0+ u/

= uo

in L1^)

f) L«(IRrf),

IK^H^M') < IMIL'OR")-

(14)

The following three convergence theorems concern a sequence u£,s of smooth solutions to problem (5)-(6), defined on lRd x [0,T] and decaying rapidly at infinity. First consider the hypothesis (H2) with r > 2, that is the case of diffusions with (at least) quadratic growth. Theorem 3.1. Suppose that the flux f satisfies (Hi) with m < q (which is always possible when r > 1 by choosing q large enough). Suppose that the diffusion b satisfies (H2) with r > 2. If 6 = o(ev+T), then the sequence u£'S converges in Ls ((0,T);Lfoc(IRd) L for all s < oo and p < q, to a function u G I/°° ( ( O j T ) ; ! / 1 ^ ^ ) C\Lq(JRd)),

which is the unique entropy solution to

(l)-(2). Observe that m and q can be arbitrarily large in Theorem 3.1. To treat the case r < 2, we need the additional condition (H3) on the diffusion. First for diffusion with linear growth (r = 1), we obtain a result in the space L2 : Theorem 3.2. Suppose that f satisfies (Hi) with m — 1, and b satisfies (#2)-(#3) with r — 1. If S = o(e2), then the sequence u£,s converges in Ls ((0,T);Lfoc(lRd)J; for all s < 00 and p < 2, to a function u G L^^0,T);L1(JRd)nL2(JRd)y

which is the unique entropy solution

to

(l)-(2). In particular Theorem 3.2 covers the interesting case of a linear diffusion and a linear dispersion with an (at most) linear flux at infinity. The condition S = o(e2) is sharp, since for S = As2 (A fixed) the functions may converge to "nonclassical" entropy solutions; see Hayes-LeFloch [3,4]. More generally, for general r > 1 we establish that: Theorem 3.3. Suppose that f satisfies (Hi) with m < ^ - < q, and b satis­ fies (H2)-(Hs)

for some r > 1. If S = o f e ^ ) then the sequence u£>5 con­

verges in Ls ((0,T);Lfoc(JRd)),

for all s < 00 and p < q, to a function

109

u G L°° f(0,T);L 1 (]R )f)Lq(JR

)), which is the unique entropy solution to

(1)- (2). Our results can be extended to more general diffusions of the form b(u, Vu, D 2 u). 4

Convergence Proofs

The superscripts e and 8 are omitted in this section, except when emphasis is necessary. In the proof, we make frequent use of the following computation. Multiply (5) by rj'(u) where 77 : IR -» IR is a sufficiently smooth function and define q : IR -> JRd by q'3: = 7/ / j . We have dtrj{u) = -V'(u)divf(u)

+

£]T<9X.(T/(U)

bj(Vu)) - dXjV'(u)

bj(Vu)

3

d2Xju)-dXiV'(u)

+8Y,d*i(v»

d2xu

3

= -divqM+e^dxtfiu)

bj(Vu))-eri"(u)^2dx.u

3

+ 52 £

2d

bj{Vu)

3

*t{l'(u)

d

lP) ~ »?"(«) dXj{dXjuf ,

3

thus dtrj(u) + div q{u) = ediv(r)'(u) 6(Vw)) - er)"(u) Vu • 6(Vix)

5

(15)

3

When 77 is convex, the term containing rj"(u) has a favorable sign: the dif­ fusion dissipates the entropy rj. The last two terms in the right hand side of (15) take also the form 6

-£

„ ' » {dXjuf

- 3d Xi (v"(u) (dXju)2)

+ 2 d2Xj(r,'(u) dXju) .

(16)

3

We begin by collecting fundamental energy estimates in several lemma. Lemma 4.1. Let a > 1 be any real. Any solution of (5) satisfies, fort G [0,T], /

M^°

dx + ae [ f

\u\a-lVu-b(Vu)dxds

= /" ^ ^ d x - ^ - 8 f I lur'Td^d^u)2 J]Rd Ct + 1 2 J0 Jjftd *-?

(17) dxds.

110

For a > 2, the last term in the above identity also equals a (a — 1) 5u [I [I

Jo JWLd

a 2 3 sgn(u) dxds. sgii^a; \u\ \u\ ~ V (dXju)

Proof. Integrate (15) over the whole of IRd with rj(u) = 1

/»

I I

(18)

3

^+ i

/»

Q-|~l

/ E! dx = -ae / lu^Vu dt J^a a + 1 JjRd

- b(Vu) dx

—7T / y ' l ^ r - 1 ^ ^ ^ ) 2 dx, which yields (17) after integration over [0,i\. One may use (16), instead, to obtain (18). □ Choosing a = 1 in Lemma 4.1, we deduce immediately a uniform bound for u in L°°((0,r); L2(lRd)) together with a control for both Vu • b(Vu) in Lx(IRrf x (0,T)) and Vu in L r + 1 ( R d x (0,T)). Proposition 4.1. For an?/ solution of (5) ana7 £ G [0,T], we /m?;e /

u(t)2 dx + 2e

./IRd

Vu • 6(Vix) dxds = / JO JJRd

u^ dx

(19)

JjRd

and, assuming (H2), \Vu\r+1 dxds < C f

[ [ JO JJRd

u2dx.

(20)

JJRd

To derive additional a priori estimates, we use another value of a, mo­ tivated by controling the dispersive term in (18) with Holder inequality, as follows: /

/

JO J]R

d

sgn(ti) \u\a~2 V (dXju)3 dxds\ < f

\u\a~2 \Vu\3 dxds

f

(21)

d

t

Jo JjR

<

[ [ \u\(a-Vp dxds\' Jo JjRd J

\[ [ \Vu\3p' dxds d [Jo JlR

To take advantage of (20), we can choose 3p' = r + 1 provided r > 2. Then p = £*§, so (a - 2)p = (r + l j ^ f f . Therefore it is rather natural to take

111

the exponent a = r for the entropy, where r is given by the diffusion term. Thus we deduce from Lemma 4.1 a natural estimate for | ^ ( t ) | r + 1 , involving the combination S e~^+^ of 5 and e. Proposition 4.2. Assume that (H2) holds with r > 2 and u0 G L r + 1 (IR d ). Fort e [0,T], we have f

\u(t)\r+1

dx + (r + l)re

J1SRd

\u\r~lVu-b{Vu)dxds

[ [ JO

< Ci(uo) (l + Se'^

(22)

J]Rd

m a x j l , [tCi(tio) ( l + S e " * ) ]

~ j )

ang

£ / f l u r - 1 | V u | r + 1 dxds < , C , , g i f j g - ^ r ) , Jo Jm* - (r + 1 ) r V / where C > 0 is some fixed constant and d M

:=max{||«o||£} 1 ( R «<)' (?" + ^

"

1}

(23)

(c|Wo||| 2 ( I R d ) ) ^ } •

In particular Proposition 4.2 shows that, if u0 G L2f)Lr+1 and S = ( ^ ( e ^ ) , then u(t) e L r + 1 uniformly for all t > 0. To motivate the forthcoming derivation, let us consider the special case r = 2. Then (22) gives us an L 3 estimate. Returning to the original inequality (21), but now with the new value a — 3, we now can estimate the dispersive term in (18) directly in view of the estimate (23). In this fashion, we deduce an L 4 estimate from Lemma 4.1. This argument can be continued inductively to reach any space Lq. Actually Propositions 4.1 and 4.2 are the first two cases of a general result derived now. We define, for n > 1, H0(6 e~^J

= C0(u0) := Huoll^jRd);

Hn(s

:=Cn(u0)

e~^^j

(l + 5

e~^x

x max j l , [* C n (tio) (l + 6 e " * ) ] ~ n ( \ / n nn(r-i)+2 Cn(u0) : = m a x j | M L ^ _ 1 H

W

,

_

}) 5

n(r - 1) + 2 _ ^ + ^

(24)

x

112

n(r - 1) + 1

_,L_I( CB , M ( i£ -A))*j.

* [(„ - l)(r - 1) + lp

Here C > 0 is some fixed constant. Note that Hn and Cn are uniformly bounded in e, 6 provided u0 € L2 P\ Ln<^r~^+2 and 8 = 0(e^+T). Proposition 4.3. Assume that (#2) holds with r > 2 and uo G L ? (]R d ). For t € [0, T] and n > 0 swc/i #ia£ n(r — 1) + 2 < g, we k » e f

| u ( t ) | n ( r - 1 ) + 2
| « r ( r _ 1 ) Vu • 6(Vu)dxds < Hn(s

/

(25) e~^)

,

rt

f I M * * - 1 ) | V u | r + 1 dzds < — rrS-r T^-TT^nf5 «"*) . Jo Jn.' - (n(r - 1) + 2)(n(r - 1) + 1) "V / (26) Proo/ 0/ Propositions 1^.2 and J^.S. Note first that (26) is an immediate conse­ quence of (25) and the hypothesis (H2). If n = 0, (25) coincides with (19) in Proposition 4.1. For n = 1, the estimate is Proposition 4.2. To estimate the term in (18), with a = r, we use (21): £

\u(t)\r+1dx

/

+ (r + l ) r e /

JJRd

/

M * - 1 Vu • 6(Vti)dxds

(27)

JO JlRd

± 1 ' uo|

|r+i

, , (r + l)r(r - 1) . da; + — Sx

JjRd

r-2

r+1

\f [ \u\ dxdsV*1 \f [ \Vu\r+1 dxds d L/0 JJR J Uo JJRd By (H2) the second term in the left hand side of (27) is positive. Integrate (27) over [0,t] and use (20):

+ (r + 1 ) f " 1 } ' ( c | K l l i , ( R - , ) * Se-^h |NrL7+2I(IR,x(0,T)) < t d M ^ + tfe-* ( N | ^ 1 ( I R d x ( 0 , T ) ) ) ^ ) . Observe that the inequality

0 < X < K(I +

AX^Y

113

where 0 < 0 < r -h 1 and K > 0, implies X < maxjl, [K(1 + A ) ] ^ ^ J .

(28)

Thus we deduce HL"+1(]R/IX(O,T)) -

m a x

| l , [tCi(tio)(l + * e - * ) ]

H

and, returning to (27): \u(t)\r+1

/

dx + (r + l)re

d

f

\u\r~lVu-b{Vu)dxds

[ d

JjR

JO JJR

< Ci(no) (l + Se~^

m a x j l , [t d(uo)

(l + <J*T^)]

~ j )

:= f r i ( < J e - * ) . This completes the proof of (22). This argument can be iterated. We return to the dispersive term and make an estimate similar to (21), but now having in view to apply (26), already established for n = 1: /

sgn(u)\u\a-2

f

\Jo JJR

d

V (dXju)3 dxdsl < f ■

\u\a~2 \Vu\3 dxds

f

(29)

d

JO JjR

<\ f f \u\{a-2~l)p d Uo JlR

\ [ [ \u\™' \\7u\3p' dxdsY UO JjRd J

dxdsV J

where we choose 3p' = r + 1 and jp'

,

= r — 1, so (a — 2 — 7)/} =

( a - 2 - 3 ^ i ) £ § . Then (18) gives / \u(t)\a+1 J]Rd

f f \u\a-1 V u • b{V u) dxds Jo JjRd ^ _L < / |w 0ia+i r + 1 da; + (tt + l ) a ( a - 1 ) d JjR /iR 2 [(r + 1) r] dx + (a + l)ae

(30)

=?(°*("-*)r

rt

xSe'^lf

[ UO JjRd

2

p

r-2 r+1

\u\^- ~^ dxdt] J

We choose a so that a + 1 = (a — 2 — 7)/?, i.e., a = 2r — 1

114

Integrating (30) over the interval [0,t], we obtain II

II2T"

\\U\\L^{TRdx(0,T))

^

A. II

-

* ll U 0|| L 2r( ] R d)

II2T

r (2r - 1) (2r - 2) , / „ , , / , . [(r + 1 ) r]-+

1

v

s_ /„

__2_\\T+T,

v

/y

\

ll2r

v

^

'

(l|w|li r 2, ( 1 R J x ( o,T))) r + 1 )'

< t C2(uo) (l + 6 e-&

(C H*

with C2(u0) := m a x l l l n o l l ^ ^ , ' ^ f f i

* " * ) ) * } •

By (28), we obtain again I ll^ r l^llL 2 -(IR d x(0,T))

<maxi 1, [t C2(u0) (l + i r * ) ]

3

>.

Then (30) gives K * ) | 2 r dx + 2r(2r-l)e

/ JJR

d

M 2 ( r - 1 } Vu • b(Vu) dxds

[ [ d

JO JlR

< C2(u0) (l + 5 e-&

max j l , [t C2(u0) ( l + <J £ " * ) ] ~

})

:= tf2 ( < * £ " * ) . This proves (25) for n = 2. The general case follows by induction on n.

□

We are now concerned with the case where the diffusion exponent in (#2) satisfies r < 2. In this situation, we require the assumption (#3), which for instance is satisfied by bj(Vu) — dXju. P r o p o s i t i o n 4.4. Suppose that (Hi)-(Hs) ^ y and r > 1. Forte [0, T], we have r+3

\Vu(t)\2 dx + £ ^

[ J]R

[ \u{t)\2+r-^ JjRd

d

1

hold with m and r such that m <

rT

f JO

\D2u\2 dxdt < C,

f JjR

dx +e [ [ d \u\^\Vu\r+ldxdt< Jo JjR

(31)

d

c (Vl +

8 ^ 6 ' ^ ) . ' (32)

115

Proof. We differentiate (5) with respect to the space variable x: dtVu + div (f'(u).Vu)

d

=eVj2

xj (MVti)) + S J2 93Xj (Vw).

3

3

Then we multiply by Vu and integrate in R . After integration by parts, we obtain IT- I \Vu{t)\2 dx- [ * dt JjRd JjRd

Auf'(u)-\7udx

fc

3

\

rZ

/

Thus, integrating on [0,£] using (Hi) yields / \Vu(t)\2 dx +2e \ d JjRd J]R ^ < [ d

y^VdXkwDb(Vu)-VdXkudx

\Vu0\2 dx + 2d

JJR

JO 22

Jm.

+ C4e I I \D'2u\* /O J]Rd JO

and so, using (#3), \Vu(t)\2 dx +C5e

f JJR

JO

dxdt,

JJR

\D2u

[ [

d

JjR

dxdt

d

< [ |Vu0|2 dx+-[[ JjR

JJR

d

dx + -dx+~[[ [ I \u\2m~2 \Vu\2 dxdt £ Jo Jm.

< f d |Vuo| |Vuo|

fl

\D2u\ \u\m-1 \Vu\ dxdt

[ [

d

£

d

\u\2 m _ 2

|Vu| 2 dxdt.

Jo J]R

By Holder inequality and for m < ^ j , / \Vu(t)\2 dx +C5e [ [ \D2u\2 dxdt < [ |Vu 0 | dx JlRd Jo JjRd JlRd 1

+ Ce- \[

f

Uo JjRd

|Vu|

r+1

and now (31) follows from (19)-(20).

dxdt

fl

JO

JiRd

u\

dxdt

r-1 r+1

116

To prove (32) we use (17) for a > 1: \u(t)\a+l dx + Ce I f

/ d

{u^1

\Vu)\r^

dxdt

d

JjR

JO JlR

<[ JjR

d

\u0\a+1dx

\u\a-lVu\\D2u\dxdt.

+ C'8 f f JO JjRd

We evaluate the last term using (H2):

6

[ f Jo JjRd

\u\a~1Vu\\D2u\dxdt

< ^ / n i d H a - 1 ( 7(rr+^1)T6l v < + 1 +

r

( S

r+ 1

\C2e

\D2u\

r

dxdt

< £- [ [ M"-1 |V< + 1 dxdt * Jo JjRd + C"5r^Le-1r

[ [ \u\a-l\D2u[^ Jo JjRd

dxdt.

So we have / \u(t)\a+1 dx + Ce [ [ In]*-1 | V u | r + 1 dxdt JjRd Jo JlRd < [ luol^dx + CS^e'i [ [ \u\a-x \D2uf^ d JjR Jo JjRd Taking a = 1 + L^1 we deduce

/ J]Rd

\u(t)\2+z^

\u\^ \Vu\r+l dxdt 2d+ r + l _ 1. Jo JjR dx + C 5 r e r d

dxdt.

dx + Ce [ [

< f K|

^

JjR

f I M: dxdt

7 0 JWLd

x [ [* [ \D2u

dxdt\

VJo JjRd

The conclusion follows now easily.

D

117

Proof of Theorem 3.1. We first prove (9), based on the conservation law (16) with an arbitrary convex function, rj, where we assume rj' ,rj",77'" bounded functions on IR. We claim that there exists a bounded measure /i < 0 such that dtr)(u) + divq{u) — ^ in T>\JRd x (0,T)). From (16), we obtain dtrj(u) + div q(u) =ediw('q,(u) b(Vu)) -er)"(u) Vu • b(Vu) 5 j E ^ ' M {dXiu)3-3dXj(p"(u) {dXju)2)+2d2Xj{r1'(u)

+

dxu)

with obvious notation. For each positive 6 G Co°(]Rd x (0,T)) we evaluate (Hi,0) for i = 1,2,3. To treat /zi, we use Holder inequality with the exponent L ^. In view of (H2) and (20) of Proposition 4.1 and assumption (14), we get K^i,0)| < e [

[

Y] |7/(ti)fy(Vu) dXj6\ dxdt

JO JJRd


[ JO

j

f

|V0| |6(Vtx)| dxdt

JjRd

SO

\(Hi,O)\
1

(JRdx(0,T))

If

ivur1 G?£(i£

y ./ s u p p 0


i(M3,0)i < \ I j z

+

^0

JlR

d

J2\6"'»(a^)3+3??»(^u)2^ t

'

2T)'(u)dXjU%0 dxdt

118

e |Vu|3 dxdt+ cs f [

< cs f f JO J]Rd

+

Jo JjRd

C5 T

[ [ IIM

JO JJR

d

Y" \dXju\2 \dXje\ dxdt j

dxdt

t

SO

|O*3,0>|< C<J||0|| r±i IX

" ~

"

H

L^(IR x(0,T))

"l,5^T(IR<'x(0 > T))

+ C5

|Vu| r + 1 dxdt

Iff

„ d

LJ J supp 6

r+1

If

|Vu|

2 r+1

dxdt

V «/ SUDD 9

jfu?i«>

dxdt

G?x
therefore

10*3,0)1 < cs U-^

+ e~^ + £ - ^ ) < C i r ^ T .

Finally the condition 6 = o(e7+I) is sufficient to imply the desired conclusion. Using a standard regularization of sgn(u) and \u — k\ (for k £ H ) , which fullfil the growth condition (7), we apply the limit representation (8) and con­ clude that v satisfies (9). To show (10) we follow DiPerna [2] and Szepessy [10]'s arguments. We have to check that, for each compact K of IRd, ,lim, 1 1 1 £->0+ t J0

(v(x,8),\u-u0(x)\)

dxds

JK

= lim lim - / / \u£,6(x. s) - uo(x)\ W | dxds = 0. *_>o+e->o+ t J0 JK\ By Jensen's inequality, where m{K) stands for Lebesgue measure of K, we have - / / \u£'6(x,s) * Jo JK

- uo(x)\ dxds < m{Kfl2

(\

l j

(u£'6(x,s)

- u0(x))2

dxds)

.

119

We will establish that lim lim - /

/

(u£,6(x,s)

— Unix))

dxds — 0.

Let K{ C ifi+i (i = 0,1,...) be an increasing sequence of compact sets such that Ko = K and Ui>o-Ki — IRd. We use the identity u2 — u\ — 2UQ(U — uo) — [u - u0)2 : (u£,s(-,s)

t Jo < 7 / < /

- uo)

dxds

JK

(/

\ue'6{-,s)\2dx-

u^dx + - I \ I d

JjR \Ki

f

u20dx-2f

uo (u£,s(-,s)

u0{u£'6(;s)-u0)

dx) ds

— uo) dx Ids

t Jo \JKi

I

for all i = 0,1,..., where we used (19)-(14). Since i

lim /

^°°

JjRd\Ki

u\ dx — 0,

we only consider the last term above. Take {#n}neiN cC§°(JRd) lim 9n =u0

in

such that

L2(JRd),

n—too

Cauchy-Schwarz inequality gives /

u0(us's(',s)

- u0) dx\ < /

JK{

\u0 - 8n\ \u£'s(',s)

- u0\ dx

JK{

+

I

6n (uy* - uo) + I

< \\U0 -0n\\L2{JRd)

6n (u£>6(; s) - u£/)

(\\U£> (•, s)\\L2{JRd)

dx

+ ||i/o||L2(lR d ))

f I endsu£J Jo

dxdr

JK{

In view of (19) and (14) I K - 0n\\L2(JR*) (IK'*(-, s)\\L2{JRd) + |ko||L2(IRd)) < 2ll^o||L2(IRd)II^O - ^n||L2(IRd)5

120 e,5

which tends to zero when n —► oo, and since lim£^o-f ll^o' ~~ uo\\L2{JRd) — 0 by (14), it remains only to see that rt

lim

|

lim - /

rs

/

*_>o+ e-+o+ t J0 \J0

/

JK.

# n ds?/'*5 dxdr ' = 0.

We have, by (5), \ fs f I / / 0n dsu dxdrl = \Jo JKi I

n n

6n ( - div/(w) + £ divfc -S^d^u)

(V0n • f(u) -eV0n-b

+

dxdr

5y2dl6n)udxdT

To deal with /ii, we use Holder inequality and (Hi) /

l V 6 U \f(u)\dxdr
/

f

JO JKi

f

\V6n\dxdr

JO JK{

|q-m dxdr

+c

/ ' / Hqdxdr JO JKi

< C . | | V < g | i ' < l l ' ) + C » 5 ^ l l ^ - H i i A ; , ^ , IMI".(R'x(0,T»For fi2, using (if 2 ) and once more Holder inequality with (20) and (14), we get e

f

f

\V0n\\b\dxdT
[ f

JO JKi

|V0„| |Vu| p dxdr

JO JKi

|V0nf+1 dxdr\


f'f iv<+1 dxdr JO JKi

< Ce1-^

s ^

\\V9n\\Lr+1{M,y

Finally, for fi3, we use Cauchy-Schwarz inequality with (19) and(14):

■I'f

«£^A

dxdr

<S\ff

\u\2 dxdr] \ j" I £ « 2 A J JO JK,

JO JKi

YJo JK{

< 5s\\V'ien\\L2{jRd)

||W0||L2(IR*'),

dxdr

121

thus lim 7 / -+°+ t J0

/ / 0 d u£>6dxdT\ds< \J0 JK. n a |

+c

I q^— +m 1 )

+C

\7+I

1

+1

lim ] ( § t2 ||V0 n || L i ( I Rv d ) J £^>0+ t \2

^+1 n w « i i ^ ( ^ , n«"'iir.(R-x(o.T,, e7TT

)

H W »lli' +1 (iR-)

r

+ 2* H V3 ^llL 2 (IR d ) llwo|L2(IRd) < Cn(^ + ^ h m

||^||-(IRax(0,T))

where we have used (25) in Proposition 4.3. The desired conclusion when t -> 0+ follows. □

Proof of Theorems 3.2 and 3.3. In the previous proof, to establish (9) we star­ ted with the identity (16) and the condition 6 = o(e7+T) as required, in par­ ticular to control the term in (16). We now keep the form (15) instead (16): the terms \i\ and H2 introduced in the previous proof do not change. We only need discuss \i^. It has now the form:

_5 ^ n!Mdxi {dxjUf + s £

dx. W(u)%.u).

The first term is bounded as follows i f f Y6V,,(u)dxudludxdt\ d J JO JjR j



f f 6\Vu\\D2u\ JO JjRd

f [ p\D2u\2 Jo JjRd


+ -(0\Vu\)2 V

le-**}

using (31) and (20), and we take ji = e and 8 — o(er+1).

dxdt

dxdt

122

The second term in /x3 behaves better:

/

0j2d*iW(u)d2x.u)

f

Jo

JTR

d

dxdt\<

j

+6 f I

JO JjR

I

Uo

d

Y,dXj0r1"(u)(dXju)2

JJR

i

dxdt

f I |V<+1

dxdt

C6 f f |Vu| 2 |V6>| dxdt Jo J}Rd + CS

f f ivnr1 JO

<

^2%9ri'(u)dXiudxdt d

j

< CS f f |V«| \D26\ dxdt+ Jo JjRd < CS

8 \f

dxdt

JJRd

CSe~^.

This completes the proof of Theorems 3.2 and 3.3.

□

Acknowledgments The authors were supported in parts by the French Ambassy in Portugal and Junta Nacional de Investigagao Cientifica e Tecnologica, Portugal. J.M.C.C. was partially supported by the Fundagao Calouste Gulbenkian. P.G.L. was supported in parts by the Centre National de la Recherche Scientifique and by a Faculty Early Career Development award (CAREER) from the National Science Foundation under grants DMS 94-01003 and DMS 95-02766. References 1. F. Coquel and P.G. LeFloch, Convergence of finite difference schemes for scalar conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math, of Comp. 57, 169 (1991). 2. R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88, 223 (1985). 3. B.T. Hayes and P.G. LeFloch, Nonclassical shock waves and kinetic re­ lations : scalar conservation laws, Arch. Rat. Mech. Anal. 39, 1 (1997). 4. B.T. Hayes and P.G. LeFloch, Nonclassical shock waves and kinetic rela­ tions : finite difference schemes, SI AM J. Numer. Anal. , (to appear). 5. S.N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sb. 8 1 , 285 (1970); Math. USSR Sb. 10, 217 (1970).

123

6. P.D. Lax and C D . Levermore, The small dispersion limit of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 36, 253 (1983). 7. P.G. LeFloch and R. Natalini, Conservation laws with vanishing nonlin­ ear diffusion and dispersion DEA Nonlinear Analysis , (1998). 8. J. Von Neumann and R.D. Richtmyer, A method for the numerical cal­ culation of hydrodynamical shocks, J. Appl. Phys. 2 1 , 380 (1950). 9. M.E. Schonbek, Convergence of solutions to nonlinear dispersive equa­ tions, Comm. Part. Diff. Equa. 7, 959 (1982). 10. A. Szepessy, An existence result for scalar conservation laws using measure-valued solutions, Comm. Part. Diff. Equa. 14, 1329 (1989). 11. L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equa­ tions, ed. J. M. Ball (NATO ASI Series, C. Reidel publishing Col.,p. 263-285, 1983). 12. A.I. Volpert, The space BV and quasilinear equations, Math. USSR Sb. 2, 257 (1967).

124 TWO-PRESSURE TWO-PHASE FLOW

J. G L I M M , D . S A L T Z of Applied Mathematics and Statistics University at Stony Brook Stony Brook, NY 11794-3600 [email protected], [email protected]

Department

Email:

D . H. S H A R P Theoretical Division, MS-B285 Los Alamos National Laboratory Los Alamos, NM 87545 Email: [email protected] A b s t r a c t . We analyze a recently proposed model for two-phase flow. This model has independent phase pressures, and for finite compressibility it has independent temperatures as well. It allows the combination of turbulence and multiphase flow modeling to describe the macroscopic structure of chaotic mixing layers formed in the late stages of interface instability growth. In this paper, we present three new results. The first is a new closure theory for all of the interface averages that couple the individual phases, generalizing a previous result for the average interface velocity. The major physics assumption underlying this closure theory is clearly identified: an absence of length scales be­ yond those contained in the primitive variables which define the interface average. The second result is a closed form solution of the two-phase flow model in the incompressible limit, which is valid for arbitrary trajectories of the mixing layer boundaries. The third result is an exact balance of forces relation for each such boundary, relating its acceleration to buoyancy, drag, and other forces. By replac­ ing certain terms in this relation with phenomenological laws, one can close the model and thus uniquely specify the two-phase flow.

1

Introduction

In this article we analyze closure relations and boundary conditions in a re­ cently proposed two-pressure, two-temperature model for two-phase flow [1,2,3] We also obtain a closed form solution for this model in the incompressible case. Up to the present time, the hypotheses of the model have been validated in the context of Rayleigh-Taylor mixing. We expect the model to have a wider range of applicability, but the extent of its validity is unknown at present. Although multipressure and multitemperature models have been proposed previously [4,5], they are less customary than single-pressure models for mul­ tiphase flow [6,7,8]. For the most part we confine our discussion to multi­ phase flow in the context of fluid instabilities, as the field of multiphase flow

125

is too broad (including pipe flow, fluidized beds, slug flow, spray, droplets, particulates, etc.) to be encompassed within the framework of a single set of modeling assumptions. The fluid instability which is the focus of this study is the Rayleigh-Taylor (RT) instability, which is the fluid instability of a density layer driven by a steady acceleration; the related case of a shock wave-driven instability is known as the Richtmyer-Meshkov (RM) instability. We contrast some of the principal differences between the single- and multipressure and temperature models of multiphase flow in this context. From a mathematical point of view, multipressure models are hyperbolic, and thus satisfy a necessary condition for well-posedness of the Cauchy problem, as has been shown by Ransom and Hicks [9]. Single-pressure models require additional regularizing assumptions which impose serious restrictions on the model's domain of validity and sometimes have an uncertain physical basis. For example, there exists a suitable closure for a hyperbolic single-pressure dispersed two-phase flow model [10], but extension of this closure to a nondispersed flow regime leads to a possibly unphysical constraint between the volume fraction and a model coefficient, if hyperbolicity is required [11]. This issue was less important when technological limitations mandated coarse-grid computations of multiphase flows, but with the advent of modern large-scale computing, single-pressure models introduce instabilities on fine grids, thus leading to the not very desirable practice of "convergence under mesh coars­ ening." From a physics perspective there are three problems with the single-pres­ sure and temperature models, which we summarize here. 1. Single-pressure or temperature models lack the required degrees of free­ dom to describe the relaxation process in the pressure or temperature fields, and do not apply to situations where this relaxation is not com­ plete. Fundamentally, instantaneous pressure or temperature equilibra­ tion is a quasi-equilibrium modeling assumption, used to close the equa­ tions of motion, and its validity or lack of validity should therefore be justified by reference to the physics of the specific problem. From this point of view, the burden of proof is on models which impose this restric­ tion, rather than on those which do not. 2. Single-pressure or temperature models for compressible fluids place a se­ rious constraint on equation of state (EOS) modeling. Fundamentally, they require a nonequilibrium EOS for a length-scale dependent chunk mix and, in practice, they use phenomenological equilibrium thermody­ namics of molecular mixed-fluid EOS models which lack a physical basis in the mixed-fluid situation, whenever the mixing does not occur at a

126

molecular level. 3. For constant acceleration RT instability, direct physical arguments, based on fluid flow considerations and analysis of the microphysics of single real­ izations, show that the pressures do not equilibrate [2,3]. Thus assuming that they vanish identically introduces potential modeling inconsistencies and equations with phenomenological parameters which lack a meaning outside of the data being modeled. Direct experimental confirmation of our model has been obtained by a zero-parameter prediction of the mixing zone edge expansion ratio in RT mix­ ing [3], compared to measured data [12,13]. This prediction improves upon that obtained from a single-pressure two-phase flow model for the same prob­ lem [8]. We also note that the nearly linear behavior of the volume fraction vs. height reproduces experimental data [13,14,15] at small to moderate Atwood number. For a discussion of the volume fraction in single-pressure models, see Ref. 8. In §2 we present a new constitutive theory for the interface averages in two-phase flow, and we characterize its physical basis precisely. In §3 we recast earlier results for the solution of the continuity and interface equations in a mathematically rigorous way. In the process, we clarify the nature of the problem presented by the nonconservative form of the interface equation. In §4 we present a closed form solution for the fluid pressures. The results of §3 and §4 comprise a closed form solution for incompressible two-phase flow in terms of arbitrary motion of the mixing layer edges. In §5 we propose and analyze a fractional linear model for the mixing coefficients in the constitutive law for the interface average. In §6 we derive Newton's law for the motion of these edges in terms of various forces, including buoyancy and drag. This law contains no modeling assumptions beyond those present in the two-phase flow equations and the model for the average interface pressure. The analytical solutions for the pressures, based on a constitutive law for the average interface pressure, are directly analogous to the solutions for the velocities, which are based on a constitutive law for the average interface velocity [3,16]. The pressures are coupled to the volume fraction and velocities through the dynamical laws for the mixing zone edges. Because these laws are derived from one-sided limits of the momentum equations, they do not supply independent information, and thus do not close the system. As was noted previously in the compressible case [1,3], additional closure hypotheses for the boundaries of the mixing layer are required to close the system, i.e., to uniquely specify the two-phase flow.

127

2

Closure of the Two-Pressure Model

The main steps in flow modeling, also called closure, can be described schemat­ ically through consideration of the nonlinear conservation law ^

+ V-F([/) = 0 .

(1)

Considering an ensemble of flows, and denoting an average by an overbar, we have -£■ + VF(U) = 0 .

(2)

This equation does not close, because F(U) ^ F(U). In other words, F(U) is not expressed as a function of the averaged dependent variables occurring in the time derivative of this equation, namely U. One can introduce new variables, requiring new dynamical equations, but ultimately a nonclosing expression as in the above equation must be addressed by a renormalization step, Fren(U) *F(U)

,

(3)

where F r e n is some function, to be determined, of the averaged dependent variables. Such an equation is never valid or even approximately valid for all microscopic (i.e., unaveraged) U. Equation (3) is a modeling assumption, and its validity is statistical, in that it holds for most, or typical U, relative to some ensemble and a probability measure defined on that ensemble. Validity thus refers to some choice of an ensemble and associated probability measure. In this paper, and in previous ones in this series, validity of our closure relations has been established for the RT flow regime. Validity beyond this regime is an open question at present. Mathematically, a necessary condition for closure is that the number of independent equations equal the number of unknowns. A test of independence is the unique solvability of the resulting system of equations. Dependent clo­ sure relations lead to insufficient equations, and thus to an under determined system with nonunique solutions. The analysis in this article will show to what extent our modeling assumptions provide a complete closure of the two-phase flow equations, and what information is still needed to complete the unique specification of the two-phase flow. It is also of interest to understand physically the information contained in the two-phase flow model. Before doing so, we introduce the model itself. The equations of motion are obtained by ensemble averaging of the Euler equations within each fluid, and of a kinematic equation for the motion of the material

128

interface [17]. Additional closure relations and symmetry assumptions are applied as described in Refs. 1 and 3. The two-phase equations resemble those of two independent fluids (phases), with additional interface coupling terms which transfer momentum and energy bewteen the two phases. Besides the issue of one vs. two pressures, the other important physics issue which accounts for the major differences among two-phase flow models concerns the modeling of the interface coupling terms. These terms are left in their general (unclosed) form in the equations below, and are modeled in the analysis which follows. In our notation, the fluids are distinguished by a subscript k, where k = 1 and k = 2 denote the light and heavy fluids, respectively, and the primed index k' denotes the fluid complementary to fluid &, i.e., k' = 3 — k. The dependent variables are /3k, Pk, Vk, Pk, and e^, which denote, respectively, the volume fraction, density, velocity, pressure, and specific internal energy of fluid (phase) k. We acceleration g = g(t) is time dependent. The equations of motion are

al ow h e r e t h e p o s i b i l t y t h a t a n e x t e r n a -df~+

dt

+

djPkPktk) dt

=

dz , d{/3kpkvkek) dz

+

°'

(5)

dz—+/3kPk9+p-dz->

(6)

=

dz

d(/3kvk) d(3k = - » - ^ - + CP") - 3 7 .

(7)

together with the constraint A + 0 2 = 1.

(8)

A single fluid EOS holds within each fluid. The quantities v*, p*, and (pv)* represent averages of microscopic quantities (products of primitive variables), which need to be modeled. Specifically, q* denotes the average of the fluid quantity q, conditioned on evaluation at the interface between the two mate­ rials; for example, p* is the average or expected interface pressure. Surface tension is neglected in this model, so that p* and (pv)* are well-defined quan­ tities. In the following, we shall establish a number of general properties of the averaged quantities q*, on the basis of a few basic assumptions. The funda­ mental physics assumption is that:

129 1. T h e only length scales in a constitutive law for q* come from t h e singlephase averages of the primitive variables which define q. We also make two mathematical assumptions concerning the regularity of t h e solution, namely: 2. q is a smooth function with respect t o these variables. 3. (For q — p or v) q* is nonnegative if b o t h q\ and q2 are nonnegative. As a consequence of physics assumption 1, i>* is a function only of t h e phase velocities v\ and v2 and additional variables of t h e problem which are spatially dimensionless (e.g., t and /?&). T h e same holds for p*, while (pv)* depends only on v\, v2, pi, p2, and spatially dimensionless variables. T h e following proposition will be used to show t h a t the above assumptions constrain q* t o be a convex linear combination of q\ and q2, when q — p or v. P r o p o s i t i o n 1. Let Q = Q ((71,(72) be a smooth function is both scale and translation invariant, i.e., (i) Q(aqi,aq2)

= aQ(qi,q2)

for all

(n) Q(qi + b,q2 + b) = Q(qx,q2) Assume

also that Q is nonnegative

where the coefficients

which

a>0,

+ b for all b G E 1 . if both q\ and q2 are

Then Q is a convex linear combination Q(qi,q2)

from E 2 to E 1

of qi and q2,

nonnegative.

i.e.,

= VU2 + Vq2qi ,

iiqk are nonnegative

and satisfy [i\ H- [i\ — 1.

Proof. Applying first translation invariance and t h e n scale invariance, we ob­ tain 0 ( 9 i ^ 2 ) = 92 + Q(q\ ~ 92,0) = q2 + \qx - 92|Q(sgn(^i - q2),0) T h u s Q is (9) in t h e dQ/dqi = boundary therefore

uniquely determined by the two numbers Q ( ± 1 , 0 ) . two regions qi > q2 and q\ < q2 we obtain dQ/dqi — Q(—1,0), respectively. Enforcing smoothness of Q q\ = q2 of these two regions, we obtain — Q(—1,0)

Q(<7i,<72) = <72 + (<71-<72)<2(1,0).

.

(9)

Differentiating = <3(1,0) and on the common = (5(1,0), and

(10)

130

It follows that Q is a linear combination of q\ and q2, and the coefficients | i | = <2(1,0) and n\ = 1 - <2(1,0) sum to unity. From the nonnegativity assumption, we see that ji\ and p\ are each nonnegative, so that the linear combination is convex. □ In the application of Prop. 1 to the interface velocity t>*, smoothness and nonnegativity of v* are assumptions 2 and 3, respectively. Scale invariance fol­ lows from dimensional reasoning. Translation invariance follows from Galilean frame invariance of the microphysical equations. In other words, an arbitrary change of inertial reference frame changes fluid velocities by a fixed amount. Because this transformation is linear, the average fluid velocities Vk, as well as the average interface velocity v*, must also change by the same amount. This property is precisely that of translation invariance as defined in Prop. 1. Applying Prop. 1, we obtain a model giving v* as a convex linear combi­ nation of v\ and v2, V* = /l^Vi + Ml t>2 ,

(11)

where \i\ > 0, \i\ + \i\ — 1, and pvk depends only on spatially dimensionless quantities (assumption 1). Consistency of v* with the microphysical equa­ tions requires that v* = Vk in the limit of vanishing /?*., which translates into boundary conditions on [x\,

to

= 0,

iivk\

= 1.

(12)

For the interface pressure, smoothness and nonnegativity of p* are assump­ tions 2 and 3, respectively. Scale invariance again follows from dimensional reasoning. In the incompressible limit, the microphysical equations are invari­ ant under an arbitrary translation of both pressure fields. Carried through the averaging, this translation amounts to an identical translation of p*, so that p* admits the property of translation invariance as defined in Prop. 1. For compressible fluids, this symmetry is only approximate, and its use requires further justification. The transformation that adds an arbitrary constant P0 to each pressure leaves the form of the compressible Euler equations unchanged if P0 is simulta­ neously subtracted from the energy density pe within each material. However, this transformation modifies the EOS of each material, and is thus not a sym­ metry of the compressible Euler equations. To illustrate this point, consider a stiffened polytropic gas [18], defined by the relation P + lPs pe =

7-1

131

This EOS is characterized by the two parameters 7 and Ps. Applying the transformation (p -> p + P0, pe -» pe - P 0 ) to this EOS leaves 7 unchanged, but maps P s onto Ps — P0. In the context of two-phase flow, where Ps,k is the value of Ps in the EOS of fluid k, p* can depend on the dimensionless quantity Ps,i/Ps,2, which changes under this transformation. Any weakly compressible fluid can be described by a stiffened polytropic EOS with a large value of Ps (the sound speed c2 = j(p+Ps)/p). In this regime, the ratio Ps,i/Ps,2 is approximately invariant under the translation of both numerator and denominator by the same relatively small (but finite) amount. Thus translation invariance of the pressures is an approximate symmetry of the weakly compressible microphysical equations, which when carried through the averaging process implies translation invariance of p*. Applying Prop. 1, we obtain p* as a convex linear combination of pi and P2, P* = P2P1 + M1P2 ,

(13)

where nvk > 0 and ji\ + 11% = 1, and ppk depends only on spatially dimensionless quantities (assumption 1). As in Eq. (12), consistency with the microphysical equations leads to the boundary conditions

^

(14)

= ° > V\

Vll |/3fc=0

^

A model for p* of the form of Eq. (13) has been demonstrated to give nearly perfect agreement with two-dimensional RT simulation data over a broad range of compressibilities and density ratios [1,19]. We therefore propose that Eq. (13) be applied to the fully compressible case, with the understanding that this hypothesis does not follow from the present modeling assumptions. Proposition 2. Let Q(q\,q2) and R(ri,r2) Q = V1Q2 + P2Q1 >

be functions of the form R

= Kr2 + /4ri »

where /j,qk and \x\ are nonnegative and p\ + \x\ — /i[ + /i£ = 1. Let S — S(qi,q2,ri,r2) be a smooth function from M4 to E 1 which satisfies the scale and translation properties (i) S(aqi,aq2, br^br 2) = abS(qi,q2,r1,r2) (ii) S(q1+a,q2+a,r1+b,r2+b) for all a, b € M1.

=

for all a>0,

b> 0,

S(qi,q2,r1,r2)+aR(r1,r2)+bQ(q1,q2)+ab

132

Then S is a linear combination of q\T\, q\r2l q2T\, and q2r2, given by S =^\q

2r2 + |(/xf + M2 - Mi - M D ^ I + \{[i\ + /ii - / i | - /if )^ir 2 + M ^ m ,

/ lg v

where £/ae coefficients fisk satisfy the constraint M* - A**- = A** - A*J, -

(16)

Proof Applying in order properties (i) and (ii), we have S{qi,q2,r1,r2)

= \qi - g 2 | | r i - r 2 | 5 ( s g n ( g i - g 2 ),0,sgn(ri - r 2 ) , 0 ) -f q2R + r 2 Q - 02 r 2 •

Thus 5 is uniquely determined by the four numbers 5 ( ± 1 , 0 , ±1,0). Consider the case 7*1 > r2. Differentiating this expression in the two regions q\ > q2 and qi < Q2 and using the formulas above for Q and R, we obtain dS/dqi — {r\ — r 2 ) 5 ( l , 0 , l , 0 ) + r 2 / / | and <95/<9<7i = —(ri—r 2 )5(—1,0, l,0)+r 2 //^, respectively. Enforcing smoothness of 5 on the common boundary qi = q2 of these two regions, we obtain 5(1,0,1,0) = —5(—1,0,1,0). Similar reasoning for the case 7*1 < r2 leads to 5(1,0, - 1 , 0 ) = —5(—1,0, —1,0). Repeating these steps with q and r interchanged leads to 5 ( - l , 0,1,0) = - 5 ( - l , 0, - 1 , 0 ) and 5(1,0,1,0) = —5(1,0,-1,0). From these facts, we conclude that S(qi,q2,r1,r2)

= (qi -tf 2 )(ri - r 2 ) 5 ( l , 0 , 1 , 0 ) + q2R + r2Q - q2r2 .

Expanding Q and R in the linear combinations given above and re-arranging terms, we obtain 5 = \i\q\Tx + (M£ - M D ^ n + (i4 ~ M2)^ir2 + (M2 + Mi + Mi ~ 1)^2^2 , with \i\ — 5(1,0,1,0). Finally, we repeat the derivation of this expression with the indices 1 and 2 interchanged to derive the equivalent expression 5 = fi{q2r2 + (fzj - nl)qir2 + (M? ~ M i ) ^ n + (M* + M2 + M2 ~ 1)(liri • These two expressions together imply Eqs. (15) and (16).

□

In the application of Prop. 2 to the model for (pv)*, we again assume weakly compressible mixing. Therefore, q — p and r = v, with Q and R given by the models (13) and (11), respectively. Condition (i) follows from dimensional reasoning. Condition (ii) follows from the translation invariance

133

of both the pressure and velocity fields (as explained above), carried through the averaging of the product pv. The resulting model for (pv)* is (PV)* = tfVp2V2 + K/iJ + A*2 - MlV ~ lAV)P2Vl

(17)

where the coefficients /ij™ satisfy the constraint

PT - / # = MX - ^ • As before, consistency with the microphysical equations leads to the boundary conditions

Mr 3

0k=O

pv "

= 1•

(18)

Solution of the Incompressible Continuity Equations

The constitutive law for v* contains information describing the microscopic kinematic constraints between the two phases not contained in the averaged continuity equations. In the incompressible limit, the continuity equations assume the form [2]

sibility condition Vv = 0

which results from ensemble averaging the incompres within phase k. The solution for the velocities as a function of volume fraction in incompressible two-phase mixing was derived in Refs. 3 and 16 and has recently been extended to arbitrary mixing zone edge trajectories [20]. Let Zk = Zk(t) denote the position of mixing zone edge k, defined as the location of vanishing /3k (z, t). Then 14 = Zk is the velocity of edge k. Here we are considering a mixing layer occupying a planar strip Z\ < z < Z2, with the fluid below the strip purely phase 2 (heavy) and above the strip purely phase 1 (light).

Theorem 1. Consider an incompressible flow in a finite interval I satisfying the closure condition (11) and its associated boundary conditions (12). Choose an inertial frame in which the boundaries of I are at rest. Assume that f3k is a C1 function of z, nvk depends only on (3k > and f^l/Pk is continuous on 0<&<1.

134

Then Eqs. (19) for k = 1,2, with the boundary conditions Vk = Vk at z — Zk, have solutions for the velocities as a function of volume fraction of the form Vk=Vkpk,e-

F

"(t^

,

(20)

where

f JO

dk ■ I

k

(21)

<\>k>

In the above integration, the relation (f>k + (j>k' = 1 holds. Substitution of f3k = (jk(z,t) in (20) solves Eq. (19) uniquely for Vk = Vk{z,t). Proof. Summing Eq. (19) over k and using Eq. (8) gives ^GSit;i+/32i;2) = 0 .

(22)

The choice of inertial reference frame requires that the fluid velocity vanish at the end walls of the domain I. Because fluid k is incompressible, Vk has a zero divergence in the phase k region (/?*. = 1). It follows that Vk = 0 everywhere in this region, and hence by continuity Vk = 0 at edge k', for A: = 1,2. The unique solution of the ODE (22) with these boundary conditions is therefore /?1 Vl + 0 2 * 2 = 0 .

(23)

We now use (23) and the closure relation (11) to eliminate one of the velocities from Eq. (19) and derive the equivalent equation dvk dz

=

_

d/3k dz

0k

0k'

0k'

(24)

We see that Eqs. (19) for k — 1,2 decouple into separate linear ODEs for v\ and v2 as functions of z. Given the boundary condition v\—V\^Xz — Z\, the C 1 v assumption for /31? and continuity dl\i\jfi\ and \x 2j'fa, the ODE (24) for k = 1 has a unique solution v\ = vi(z,t) on the half-open interval Z\ < z < Z2, and similarly for v2 — v2(z,t) on Z\ < z < Z2. Eq. (24) has a solution for Vk as a function of (3k on 0 < @k < 1 given by Eqs. (20) and (21), as one can easily check. From continuity of nl/Pk on 0 < Pk < 1, which ensures that the integral in (21) is bounded, it follows that Vk{t,flk) given by (20) is C 1 in /?*. on the closed interval 0 < ^ < 1. By the chain rule and the C 1 property of 0k(z), the function Vk(z,t) obtained by substitution of f3k = Pk(z,t) in (20) is C1 in z on the closed interval Z\ <

135

z < Z2. Moreover, vk(z,t) satisfies the ODE (24) on Z\ < z < Z2 because Vk/Pk' = Vk exp[—Ffc(£, (3k)} is continuous, hence the RHS of (24) is continuous, on this closed interval. We have shown that Eq. (20) with the substitution (3k = (3k(z,t) gives a solution of (24) on Z\ < z < Z2 which is unique for the given boundary data on the half-open z-interval. Two such solutions must be identical on the half-open z-interval and smooth on the closed z-interval; thus they must also coincide on the closed z-interval. □ Corollary 1.1. The coefficients //j! satisfy the constraint

^W = - „-*(*,!) e-F«>» .

(25)

Vi(t)

Proof From Eqs. (19) and (24) we have v*

= [pk,rtt(t,pk,)-pkrt(t,(3k)}^

.

Pk' The RHS of this expression must give the same v* for both k = 1 and k = 2. A simple calculation shows that this requirement is satisfied unconditionally in the interior of the mixing zone, and it is satisfied at the edges if and only if (25) holds. □ Remarks. If (3k is monotone as a function of z, the assertion that [ivk is a single-valued function of ftk at any t is reasonable. If (3k is not monotone in z, then the conclusions of Theorem 1, and hence the assumptions (e.g., omission of other spatially dimensionless arguments in the assumed form of //£), are too strong. Another consequence of Theorem 1 which may lack generality is the requirement that V\ and V2 have opposite signs, which is apparent by inspection of Eq. (25). The analysis of Ref. 16 relates the profiles of v\, v2, and v* to the vol­ ume fraction profile, which is directly measurable, and is a statistically stable quantity commonly presented in experimental data analysis of the RT prob­ lem [13,14,15]. That this statistically measured quantity can be related to closure relations and can in effect determine them experimentally indicates that the form of the closure (11) has significant physical content. Corollary 1.2. Eq. (5), with the substitution of vk given by (20), is a scalar hyperbolic conservation law for the evolution of fik, with flux term Fk(t,Pk) = Vkpkf3k,e-F^t>'3«K

(26)

136

Remark.

The passage from the conservation law

eak solution o

to (4)-(5) for weak solutions requires further study, as the w the nonconservative equation (4) does not fall within the unique regularization theory of conservation laws.

Theorem 2. Assume the conditions of Theorem 1, in the case where the mix­ ing layer expands outward, i.e., V\ < 0 < V2 for all t, and both V\ and V2 are piecewise continuous in t. Further, assume that the given initial data ft(z,0) satisfies 8Pi(z,0)/dz > 0 on Z\ < z < Z2, and that fi\ is a C1 nondecreasing function of ft for all t. Then Eqs. (4) and (5) have solutions Pi = Pi(z,t), v\ — vi(z,t), V2(z,t) which are C1 nondecreasing functions of z for all t.

and V2 =

Proof We first show that dv* /d/3i > 0 on 0 < Pi < 1 for all t. Differentiating (11) with respect to Pi, and recalling that 5/<9ft = —<9/<9ft, dv*

d[i\

dfi%

vdv2

vdvi

(28)

By Eq. (20), Vk retains the same sign as Vk on 0 < Pk < 1, and is zero at Pk — 1. Hence, by the assumptions of this theorem, the sum of the first two terms in (28) is nonnegative on 0 < ft < 1. Differentiating (20) with respect to Pk and simplifying,

** =-&vke-™** OPh

.

(29)

Pk

Therefore, each of the last two terms in (28) is nonnegative on 0 < ft < 1. It follows that dv*/dz = {dv*/<9ft)(dft/<9z) > 0 on Zx < z < Z2 at t = 0. We now apply the method of characteristics to construct ft for each strip over which both V\ and V2 are continuous in t, and we extend this construction from one strip to the next by a finite induction. In each strip, the characteristics are non-focusing and diverge at a bounded rate, since d2J7i/dPi2 = dv*/dPi > 0 and hence V\ < v* < V2. Thus dPi/dz is bounded above by its values at t = 0 and below by 0. From these facts, we can extend the C 1 and nondecreasing properties of ft to all t > 0. It follows from Theorem 1 and Eq. (29) that Vi and V2 are also C 1 nondecreasing functions. □

f

137

Corollary 2.1. The solution for fa at any t is given implicitly by the formula z(fa,t)

= z(fa,0)

+ f v*(s,pk)ds Jo

,

(30)

with v* = (fai — ii\)Vk exp[—Fk(t,/3k)], and it is unique within the class of C1 solutions to (27). Proof The fact that (30) satisfies each of Eqs. (4) and (27) follows directly from the method of characteristics, noting that v* = dTkjdfa. The form for v* as a function of t and fa given above is implied by conservation of mass (5) and is derived from the formulas given in this section. Uniqueness in the class of C1 solutions to (27) is given in standard treatments of hyperbolic conservation law theory. □ 4

Solution of the Incompressible Pressure Equations

In this section, we solve the momentum equations (6) for the pressures p\ and P2 in the incompressible limit. The solution is based on the closure theory for p* proposed in §2. As in our earlier analysis of the continuity equations, the solution of the pressure (momentum) equations assumes an arbitrarily specified motion of the mixing zone boundaries. By the discussion of §2, we regard fa and Vk as known functions of z and t so that (6) is a pair of ODEs for pi and p2, which we now solve explicitly up to quadratures. After some manipulation Eqs. (6) can be written as

^

+ ( - i > * ^ 7 T = M*,t),

oz

fa

(3D

dz

where hk(z,t)

= pkg-

d(fapkVk) dt

— Pk

d{fapkvkvk) dz

It is convenient to introduce the phase k convective derivative Dt -

dt+Vk8z

and use it to write hk in the form

{z,t) kh

= pk(g-^j

.

(32)

138

With the notation p± = p2 ± p\ and the closure relation (13), we have the identity Pjk -p*

= (-l)Vfr-.

(33)

Forming sums and differences of (31), we find dp± _ £ dp « =f±jTP-+h±, oz oz

(34)

where

f± = f±f, P2

h± = h2±h1.

Pi

The following theorem uses the function

In this definition and in the theorem below, we omit the fixed argument t. T h e o r e m 3 . Assume the conditions of Theorem 2 and the p* closure rela­ tion (13) with the associated boundary conditions (14). Also assume that, at a fixed time t, pFk depends only onfa?f^/Pk and {dfJ^/dt) /fa are continuous in (3k on 0 < fa < 1, and that the edge accelerations Vk are continuous at this time. Then Eqs. (5) satisfy a Fredholm alternative: A. If the degeneracy condition l + 9(Z2) = j

2 z

f+(z)^g(z)dz

(35)

(z) +

is not satisfied, then Eqs. (6) can be solved uniquely given either pi(Z2) or P2{Zl). B. If Eq. (35) is satisfied, then there is a solvability condition, namely fZ2

CZ2 h

= -g(Z -g(Z2)2) /f 2 -=ffdz 2 /' \fah (ft/*! + p2h2)dz 1+fah 2)dz = z JZi

JZx

9\ )

139

/ / Eq. (36) is satisfied, then there is a one-parameter family of solutions to the pressure equations, so that an additional pressure (boundary) condition is needed to close the system. If Eq. (36) is not satisfied, then Eqs. (6) have no solution. Proof. Given continuity of fJ%{0k)/0k and smoothness of /3\(z), f±df3\/dz is continuous in z on Z\ < z < Z In view of Eq. (32), to prove that hk is continuous in z it suffices to show that dvk/dt is continuous in z. Let Vk denote Vk expressed as a function of /?&, i.e., Vk((3k) — Vk{z). Then, applying the chain rule and Eq. (4), dvk _ dvk_ _ * dpk dvk V dt " dt dz dpk ' so that continuity of dvk/dt follows from continuity of dvk/dt. Eq. (20),

Differentiating

§r = Vk0k,e~Fk - ^k ■ By our assumption on Vk, the first term on the RHS of this expression is con­ tinuous, so that it remains to show that dFk/dt is continuous. From Eq. (21),

dFk dt

-n

rk f l f l / i j

J0

1 dfil,

[(f)k dt

(j)k> dt

d(j>k

Continuity of dFk/dt follows from our assumption of continuous (djipkldt) //3k. We have now established that hk is continuous in z, hence the RHS of the ODE (34) is continuous in z, on Z\ < z < Z2. Next, we solve (34), first for p_ and then for p + . The solution for p- is obtained by the method of variation of parameters, while the solution for p + follows by direct integration:

^

w=j{i)

p+(z) =p+(Z2) - I

k

)+

\f+(z')^p-(z')

{^ +h+(z') dz' .

(37)

(38)

To complete the solution, we consider pressure boundary conditions at z — Z\,Z2. Combining (37) and (38) with the identities P+(Z2)

= 2Pl(Z2)

+p-(Z2)

,

p-{Zl)

= 2p2(Z1) - p + ( Z i ) ,

140

we obtain P+(Z2)

- g(Z2)P-(Z1)

= 2pi(Z 2 ) + g(Z2) f

' ^ y ^ '

(39)

,

p+(Z2) +p-(Z1) = 2p2{Zx) + I * [/+(*') ^ r M * ' ) + h+(z'dz'

.

(40)

Substituting the solution (37) in (40) and rearranging terms, P+(Z2)

+ \l- JZJ f+(z')^rg(z')dz'\p-(Z1)

(41)

r-Z?

+ JZi

= 2p2(Z1)

h+(z') +

.,M f+(z')

h

-^±dz»

9(z")

dz'

Equations (39) and (41) comprise a linear system of equations in p+(Z2) a n d p - ( Z i ) . We now show that p\{Z2) and p2{Z\) are not independent bound­ ary data by evaluating an exact integral of the momentum equations. Summing the two momentum equations (6), we obtain dtfipi

+ P2P2) = Pihi+l32h2 dz

(42)

.

Integrating this identity across the mixing zone and using the known values of Pk at the edges, we have Pi(Z2) -MZi)

= / '(Ahi

+foh2)dz

,

(43)

JZx

as claimed. Alternatives A and B now follow directly from Eqs. (39), (41), and (43).

□ Remark. The pressure equations have a unique solution whenever boundary values for p\ and p2 are specified at the same z. In other words, the degeneracy in the pressure equations, as described in Theorem 3, is a direct consequence of having mixed pressure boundary conditions. The mixed boundary data that we consider in Theorem 3, pi(Z2) and p2(Zi), are suggested by the geometry of the mixing layer. For example, the solution for the pressures outside the mixing layer are pi(z) for z > Z2 and p2(z) for z < Z\. By continuity, these functions give precisely the boundary data that are assumed in Theorem 3.

141

5

Interface Closure Relations

In this section we propose and analyze a fractional linear form for the mixing coefficients fiqk in the constitutive law for the interface average q*. As in pre­ vious theorems we assume that jiqk depends only on t and /?&, but we do not assume incompressibility unless otherwise stated. At each edge of the mixing zone, q* is required to coincide with the q of the phase which is vanishing there. This condition is translated into the requirements that fiqk(t,/3k = 0) = 0 and iiqk{t,f3k = 1) = 1. On the basis of these boundary conditions on \x\ and the freedom to choose a common scaling factor for both the numerator and denominator, we can restrict the fractional linear form to the following:

Pk +cl{t)(Jk>

time-dependent

For each interface quantity q, there are two undetermined coefficients, c\ and c\. The determination of cqk is specific to the definition of q-

Velocity. Imposing the condition \x\ + ii\ = 1 leads to the requirement that c\cv2 — 1. Thus the velocity closure v* contains one free time-dependent func­ tion, namely c\ (or c^). The A — 0 problem associated with Rayleigh-Taylor mixing has a discrete symmetry about the mixing zone center line, under the operation of interchang­ ing the two fluids and sending z into — z (assuming, of course, that the initial data has the same symmetry). For this case we require the additional sym­ metry fj,l(t,P) = /J>2(t,P), which, within the fractional linear context, implies that c\ = c\ = 1. We thus recover the model l*l(t,fik) = Pk

(45)

that was proposed in Ref. 1. At this stage, nothing more is known about ck(t) except when the fluids are incompressible. In this case, one can use Eq. (25) as a further constraint, and it is easy to show [16] that ck(t) = \Vk>/Vk\ for all A. Pressure. The interface pressure is determined by arguments that are anal­ ogous to those used for the velocity, except that there does not appear to be a complete closure in the incompressible limit. As above, we have n\ + /i£ = 1, hence c\c\ — 1, leaving one coefficient undetermined.

142

At A — 0, symmetry implies that /j%(t,0) = //f^,/?), hence (?x — cf = 1, and as above we recover the model jipk = fik that was proposed earlier[1]. In the incompressible limit, we have /_ = 0, /+ = 2, and g = 1, using this closure. Thus the degeneracy condition (35) is satisfied, as was discussed in Ref. 2. The extra condition needed to assure uniqueness of the pressure difference p_ results from the problem symmetry for the ^4 = 0 case, namely p~(z — 0) = 0. Next consider the limit A — 1, in which case fluid 1 is a vacuum, at zero pressure, so that p* = p\ = 0 and thus /u,p = 0, /J,P = 1. The A — 1 limit is realized in the fractional linear model (44) by setting cf = 0. Recalling that (?2 — 1 at A — 0, we propose that, for RT mixing, c?2 is a time-independent decreasing function of A, with c^O) = 1 and c^(l) = 0. In the case of incompressible flow at A = 1, the conditions on fipk necessary for the application of Theorem 3 are not satisfied, but in this limit there is only one pressure and one momentum equation to consider. The A = 1 limit of Eq. (6) for k = 2 reduces to df32p2 a 7 —Q^~ = /?2/*2 ,

,Af* (46)

which is obvious by inspection of (42), recallling that p\ — 0, hence h\ — 0, at A = 1. This ODE is easily integrated from Z\ to z to yield the unique solution

KW

= s p ( ^ i ) + // 2

AM*7

Because / _ = l//3 2 and thus ^ = l / ^ 2 at A — 1, it is evident that the solution given above is the A — 1 limit of the solution given by (37). The boundary value pi^Zi) is already determined because we assumed that the vacuum is at zero pressure, as one can see by integrating (46) from z to Z 2 to yield the equivalent solution

{z) = ~jj

2 2h(5 2dz'

,

(47)

which specifies P2(z) independent of any boundary pressure. It may not be immediately evident that ^2(^2) = 0, as required by the boundary condition that P2 = P* — 0 at edge 2, but one can confirm that it is zero by evaluating the z = Z2 limit of (47) using L'Hopital's Rule, and recalling the assumption of Theorem 2 that df32 /dz < 0 on Zx < z < Z2. Consider the degeneracy condition (35) evaluated at A = 1. Both sides of this equation are singular, so we evaluate their ratio for z < Z2 and take the limit z —> Z2. This ratio is 1, hence degeneracy holds at both A — 0 and A = 1.

143

Numerical evidence, not presented here, suggests that the fractional linear model (44) satisfies both the degeneracy (35) and solvability (36) conditions to a high degree of accuracy for cf between 0 and 1. Work. The coefficients pF£ satisfy the constraint /j%v - p\v — \x\ — p\. In­ serting the fractional linear form (44) for q — pv into this expression, we obtain

ft ft + c f f t

ft ft

ft

=

+ cfft

ft

+ c^ft

ft ft

+ c?ft •

This constraint is equivalent to a linear combination of the 4 th -degree products /?2~zft, z = 0 , 1 , . . . ,4. The fractional linear form (44) yields a compatible model for (pv)* if and only if every term in this sum vanishes. There are five terms, hence five coefficients that must vanish, and combined they yield two independent constraints on c%v and c%v, namely d^d^ — d[x;c£ and d[v (1 + ^ c f ) - cf(1 + c f o f ) (or cf2v(l + cld[) = cv2(l + <%'<%'))• There are two solutions: either (a) c f = \jc\ and c f = 1/cf or (b) d[v = cj and c f = eg. At A = 0, either solution is valid and gives d[v = cP2v = 1, and we again recover the earlier model pP^ = ft. At A = 1, we require (pv)* = 0, hence pPy — 0, and only solution (b) is compatible (recalling that c^ = oo at A — 1). It seems reasonable to assume that solution (b) holds at all A. To summarize, the fractional linear form provides a complete closure for v* and p* up to a single undetermined time-dependent coefficient in each model. Both coefficients are 1 in the symmetric (A — 0) limit. The coefficient in the p* model is known at A — 1. The coefficient in the v* model is known at all A when the fluids are incompressible. The model for (pv)* contains no undetermined coefficients, but it does have two branches, only one of which is compatible at all A. 6

Dynamical Laws for the Mixing Zone Edges

We were able to solve the continuity equations in the incompressible limit because they decouple from the momentum (pressure) equations. Boundary conditions for the velocity equations were specified in terms of the motion of the mixing zone edges z — Zk(t), where Zk(t) denotes the location at which phase A: goes to zero volume fraction. In §4, analogous results were achieved for the pressure equations. Thus the two-phase flow equations can be viewed as giving a solution for volume fractions, velocities, and pressures in terms of arbitrary edge trajectories Zk(t). Here we evaluate the momentum equations to express the complete balance of forces at the mixing zone edges. The result is an identity relating the edge acceleration Zk(t) to various forcing terms,

144

including buoyancy and drag. This identity, because it allows arbitrary edge motion, does not close the system. However, it does specify exactly what forces arise in a fundamental fashion. Any further phenomenological or problemdependent information, which imposes an additional relationship among the coefficients in the equation derived below, will uniquely determine Zk(t), and thus close the system (up to a possible additional pressure boundary condition if case B holds in Theorem 3). For example, in some closures certain forces are set equal to zero (e.g., pressure drag in a single-pressure model), while the coefficients of other forces are adjusted on a phenomenological basis. The equations which we derive below contain no new modeling assumptions. For the theorem and proof which follow, we use the identity

z=Zk

We also introduce the notation

^,...)=lim/<'f'->

(48)

and recall that lim/?fc_^i p%(t,j3k, •.. )/Pk — 1Theorem 4. Assume incompressible flow and the p* closure relation (13). Then ( 1^„ 7 (n n\„ (-1) pkZk = (p2 - pi)9 -Pk.V^

d(ff2-Pl) 5

(49)

dz

+

(-l)\vk-l){p,-Pl)d-^,

where the RHS of this equation is evaluated along the trajectory z — Zk (t). Remark. Because the closure relation (13) has a general basis of validity (cf. §2), we see that the identity (49) is universal for incompressible two-phase flow, in the sense that it is independent of physics-based, flow regime-specific, modeling assumptions. Proof. We rewrite the momentum equations (6) as DkVk

dpk ,

, P*-Pkd(3k

/

K

m

145

Consider the inertial term of the ambient phase at mixing zone edge k. Eval­ uating along the curve z = Zk(t), we have vk> — 0 and vk - Vk and therefore Dk'Vk' _ dvk> Dt ~ dt '

_ Dkvk> _ dvk> Dt ~ dt

k

dvk> dz '

It follows that Dk>Vk>

Dt along z — Zk(t). tion (22),

T,

dvk>

-Vk-

dz

Expanding the derivative in the incompressiblity condi­ dvk oz

dpk oz

dvk> dz

Evaluating this expression along z — Zk(t), vk> = 0 , we obtain dvk>

z

dpk> dz where (3k = 0, vk — Vk, and

d/3k = —Vk-

oz

so that Dk'Vk'

~~DT

_v290k

-Vk~dz~'

r.

v

(51)

Subtracting the k' from the k momentum equation (50), recalling (33), evaluating along z — Zk(t), and using (51), we obtain (49). □ We now analyze the terms appearing on the RHS of Eq. (49). Recall the labeling conventions that p2 > Pi and (—l)kZk > 0. A positive value for the inertial term on the LHS of (49) indicates an outward acceleration. We use the terminology that the frontier portion of heavy fluid that determines mixing zone edge 2 is a spike, while on the other side of the mixing zone, the leading structure is a bubble. First, observe that d(3i/dz, an increasing function of z near the mixing zone edges, defines an inverse longitudinal length scale, representing a surfaceto-volume ratio for the leading spike or bubble. The terms in Eq. (49) are force densities, and one should therefore multiply by a bubble or spike volume to obtain an expression involving actual forces. Doing so shows that the terms proportional to d(3i/dz are surface forces, and that all other terms are bulk (volume) forces.

146

The first term, (p2 — pi) 0. Therefore the pressure difference surface force (—l)k(yk — 1)(P2 — Pi)d/3i/dz inhibits the growth of the bubble or spike when p2 < Pi, and reinforces it when p p\. This force vanishes when v^ = 1, which occurs in the limit ^4—^0. Since the A = 0 closure (45) fits simulation data well for small to moderate A [1], we conclude that the pressure difference surface force in this range of A is small. The added mass effect conventionally appears as a correction to the inertial force, but no such term is present in (49). Because added mass is a volume force, this effect must be accounted for by the only remaining volume force in (49), d(pi —p2)/dz. The sign of the added mass term follows the sign of Z&, but there is no basis for drawing a conclusion concerning the sign of d(pi — p2)jdz, as this force may include other effects besides added mass. The coefficient of every force term in Eq. (49) is completely determined from first principles, i.e., from averaging the microphysical equations and the p* closure relation (13). Prop. 1. (Inclusion of the single-phase Reynolds stress in the two-phase flow equations would add new terms to Eq. (49), but would not alter any of the existing terms.) Furthermore, there are bulk and surface forces that depend on the pressure difference p2 — Pi- They are obviously neglected in single-pressure models, which must instead account for their absence by a phenomenological adjustment of the other coefficients, specifically the ones appearing in the inertial, buoyancy, and form drag terms (see, for example, Refs. 8 and 21). Acknowledgments The work of JG and DS is supported by the Applied Mathematics Subpro­ gram of the U.S. Department of Energy, DE-FG02-90ER25084. JG is also supported by the Army Research Office, grant DAAH04-95-10414, and the National Science Foundation, grant DMS-95-00568. DHS is supported by the U.S. Department of Energy. The authors also thank Brad Plohr and Folkert

147

Tangerman for helpful comments.

References 1. Y. Chen, J. Glimm, D. H. Sharp, and Q. Zhang. A two-phase flow model of the Rayleigh-Taylor mixing zone. Phys. Fluids, 8(3):816-825, 1996. 2. J. Glimm, D. Saltz, and D. H. Sharp. Renormalization group solution of two-phase flow equations for Rayleigh-Taylor mixing. Phys. Lett. A, 222:171-176, 1996. 3. J. Glimm, D. Saltz, and D. H. Sharp. Two-phase modeling of a fluid mixing layer. Submitted to J. Fluid Mech. 4. H. B. Stewart and B. Wendroff. Two-phase flow: Models and methods. J. Comp. Phys., 56:363-409, 1984. 5. D. Holm and B. Kupershmidt. Multipressure regularization for multi­ phase flow. Phys. Lett. A, 106:165-168, 1984. 6. F. Harlow and A. Amsden. Fluid dynamics. LANL Monograph LA-4700, National Technical Information Service, Springfield, VA, 1971. 7. D. L. Youngs. Numerical simulation of turbulent mixing by RayleighTaylor instability. Physica D, 12:32-44, 1984. 8. N. Freed, D. Ofer, D. Shvarts, and S. Orszag. Two-phase flow analysis of self-similar turbulent mixing by Rayleigh-Taylor instability. Phys. Fluids A, 3(5):912-918, 1991. 9. V. H. Ransom and D. L. Hicks. Hyperbolic two-pressure models for two-phase flow. J. Comput. Phys., 53:124, 1984. 10. J. H. Stuhmiller. The influence of interfacial pressure forces on the character of two-phase flow model equations. Int. J. Multiphase Flow, 3:551-560, 1977. 11. A. Prosperetti and J. V. Satrape. Stability of two-phase flow models. In D. D. Joseph and D. G. Shaeffer, editors, Two Phase Flows and Waves. Springer-Verlag, 1990. 12. K. I. Read. Experimental investigation of turbulent mixing by RayleighTaylor instability. Physica D, 12:45, 1984. 13. D. L. Youngs. Modeling turbulent mixing by Rayleigh-Taylor instability. Physica D, 37:270-287, 1989. 14. D. M. Snider and M. J. Andrews. Rayleigh-Taylor and shear driven mix­ ing with an unstable thermal stratification. Phys. Fluids, 6(10):33243334, 1994. 15. M. B. Schneider, G. Dimonte, and B. Remington. Large and small scale structure in Rayleigh-Taylor mixing. Submitted to Phys. Rev. Lett. 16. J. Glimm, D. Saltz, and D. H. Sharp. The statistical evolution of chaotic

148

fluid mixing. To appear in Phys. Rev. Lett. 17. D. A. Drew. Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech., 15:261-291, 1983. 18. R. Menikoff and B. Plohr. The Riemann problem for fluid flow of real materials. Rev. Mod. Phys., 61:75-130, 1989. 19. Y. Chen. Two phase flow analysis of turbulent mixing in the RayleighTaylor instability. Ph.D. Dissertation, University at Stony Brook, 1995. 20. J. Glimm, D. Saltz, and D. H. Sharp. A general closure relation for incompressible mixing layers induced by interface instabilites. To appear in Proceedings of the Sixth International Workshop on the Physics of Compressible Turbulent Mixing. 21. U. Alon and D. Shvarts. Two-phase flow model for Rayleigh-Taylor and Richtmyer-Meshkov mixing. In R. Young, J. Glimm, and B. Boston, editors, Proceedings of the Fifth International Workshop on Compressible Turbulent Mixing. World Scientific, 1996.

149

PAINLEVE ANALYSIS A N D ITS APPLICATIONS

BENYU GUO Department

of Mathematics,

Shanghai

University,

Shanghai

201800,

China

ZHIXIONG CHEN Department

of Mathematics,

University

of Pittsburgh,

Pittsburgh,

PA 15260,

USA

A b s t r a c t . Painleve analysis plays an important role in the study of dynamical systems. In this paper, we review its history, some developments and applications to several fields, such as the integrability and the construction of analytic solutions of integrable and non-integrable systems. Finally, we construct analytic solutions of a nonlinear evolutionary equation governing the wave transmission in nerve system.

1

Introduction

As we know, many phenomena in nature can be described by various mathe­ matical models, which are often governed by differential equations. Thus the study of these phenomena is connected with the study of corresponding equa­ tions. Since most of them can not be solved explicitly, we have to study their qualitative properties. For instance, people look for the conditions under which certain systems are integrable. Painleve analysis plays an important role in judging the integrability. In the next section, we review the history of Painleve analysis and six Painleve equations. Then we present ARS conjecture and ARS test in Sec­ tion 3. ARS conjecture establishes the relation between infinitely dimensional dynamical systems and the equations of Painleve type. It renewed the re­ search of Painleve property. In Section 4, we discuss two different definitions of Painleve property for partial differential equation and their applications to the integrablility of nonlinear systems. In Section 5, we illustrate the technique of constructing analytic solutions via Painleve analysis. In particular, we take a nonlinear evolutionary equation governing the wave transmission in nerve system to show how to generalize this approach to non-integrable systems. Finally, we talk about other topics in this field. 2 Painleve Property for ODE and Painleve Equations

150

In 1887, Picard [1] considered the following equation d2w

dw.

where z is a complex variable, F is analytic in z, algebraic in w and rational in ^ j . The question is whether or not all branch points and essentially singular points of its solutions are fixed. This problem led to the beginning of singularity analysis. Painleve [2-4], Gambier [5-6] and Fuchs [7-8] and many other authors have made a lot of contribution to this problem and so the following concept occurred. A singular point is movable, if its position depends on integral constant. Definition 2.1. (Painleve Property for Ordinary Differential Equations). An ordinary differential equation (or a system of equations) in the complex domain is said to be of Painleve type or to have the Painleve property if the only movable singular points that its solutions can exhibit, are poles. Clearly, the Picard problem is connected with the question whether or not a second-order ordinary differential equation has the Painleve property. Example 2.1. Consider the equation dw dz

o

— +w2 = 0. Its general solution is 1 w = z-A where A is an integral constant. Obviously z — A is a movable pole and thus the above equation has the Painleve property. Example 2.2. Consider the equation d2w _ 2w — 1 dw 2 ~dz^ = w2 + r!z~> ' It has a general solution w = tan[ln(Az — B)] where A and B are integral constants. Clearly, the movable point z = B/A is not only a branch point, but also an essentially singular point. Thus the above equation has no the Painleve property. We now list some of theoretical results as follows.

151

1 The only first-order ordinary differential equation with the Painleve prop­ erty is the generalized Ricatti equation, namely, ^=P0(z) where Po(z),Pi(z)

+ P1(z)w + P2(z)w2.

and P2(z) are analytic in z.

2 There are fifty canonical forms for all second-order ordinary differential equations with the Painleve property. Among them, six equations are distinguished, called Painleve equations. The others either can be inte­ grated in terms of known functions (such as Airy functions or Elliptic functions), or can be reduced to one of these Painleve equations. The six equations are d2w „

2

d2w

o

lz^W

n

+Z

>

d2w_l dw 2 1 dw aw2 + b 3 d 2 dz w dz z dz z w' d2w 1 .dw.9 3 o3 b A 9 rt/92 N —y=— (— + -w + W + 2(z - a)w + - , dzz 2w dz 2 w d2w_ Sw — 1 dw 2 1 dw (w — l)2(aw2 + b) cw dw(w + 1) dz2 2w(w — 1) dz z dz z2w z w—1 2 d w 1,1 1 1 ,,dw,2 ,1 1 1 ,dw z dz 2 w w — 1 w — z dz z z — 1 w — z dz w(w — l)(w — z) ( bz c(z — 1) dz{z — 1) H z 2^(z —7T7i l)2 \0 + -^w2 + 7 (w —7T^l ) 2 + (-w; — z)2 They are named as Painleve equation I to Painleve equation VI, respec­ tively. 3 For the third or higher order equations, Chazy, Garnier, Bureau and the others derived fruitful results (see [9-12]). But the answer is not yet complete. 4 Kowalevskaya treated the problem of the motion of a rigid body about a fixed point. It is assumed that the solution has no movable essential singularities and movable branch points. She discussed three motions with this assumption. One of them was discovered newly, and is called Kowalevskaya motion. Her idea and technique led to deep research in Painleve test (see [13-15]).

152

We can find the more detail of past work in [16-20]. Moreover, the discovery of the relation between the Painleve property and the integrability, renewed the study of Painleve equations and the Painleve property for ordinary differ­ ential equations. For instance, people looked for rational solutions, families of one-parameter solutions and their Backhand transformations (see [21-36]). Also, the asymptotic behaviors of solutions, the connection formulas between equations and the monodromy preserving deformations were discussed (see [37-47]). 3 A R S Conjecture and A R S Test By the end of the seventies, Ablowitz, Ramani, Segur and many other authors studied nonlinear evolution equations which can be solved by the inverse scat­ tering transformation (1ST). It was found that the resulting equations reduced by similarity transformation are either Painleve equations or of Painleve type (see [22-25, 48-52]). Therefore they guessed that it is valid for all equations solvable by 1ST. This is referred to as ARS conjecture. Furthermore Ablowitz, Ramani, Segur, McLeod, Olver and the others used linear integral equation of Gelfand-Levitan-Marchenco to verify this conclusion provided that certain symmetry holds. Example 3.1. Consider the KdV equation ut + 6uux + uxxx - 0.

(1)

Let

-P U =

o".

(3t)i Then Equation (1) is reduced to Painleve equation II with the unknown f and a = 0. Since Equation (1) possesses Galilean invariant, we put u = f(x — ct) and so it reads fxx = - 3 / 2 4- cf + A where A is an integral constant. This equation is also of Painleve type. For many well-known equations in theory of solitons such as modified KdV equation, nonlinear Schrodinger equation and so on, the ARS conjecture is true (see [22-23]). The great advantage of ARS conjecture is to provide a simple and readily applicable test for integrability of nonlinear evolutionary equations. As we know, it is very difficult to check whether or not a partial differential equation (or a system of equations) is integrable. There are two parts in ARS method. The first is the group reduction (see [53-55]). The other is the judgement on the

153

Painleve property of the resulting equation. To do this, we can use a-method by Painleve [17], or ARS test given by Ablowitz, Ramani and Segur [49-50]. Notice that both of the above tests are only necessary condition for judging the Painleve property. The reason is that the solutions may still possess movable essential singularities, even they pass the test. We now focus on ARS test. It includes three steps: 1. Check the dominant behaviors, 2. Find the resonance points, 3. Determine the integral constants. Consider the equation

(

dw

dnw\

^^•••^)=°

,x

(2)

where F is analytic in z, and rational in ^Ly.^ 0 < j < n. Let the solution be of the form of Laurent series OO

3=0

where zo is a pole depending on integral constants, and p is an undetermined negative integer. By substituting it into Equation (2) and comparing the coefficients of terms (z — ZQ)3+P, we get a series of recursive equations for dj. The values of p and a 0 can be evaluated by letting j = 0. But p may have several possible values as well as CLQ. lip is not an integer, then the solution of Equation (2) has algebraic branch point and thus Equation (2) has no the Painleve property. If p is an integer but not negative, then the solution either has essential singularities or has no singularity. Finally if p is really a negative integer, then we turn to step (ii). In this case, we have the recursive formulas as follows QoQ(j)aj =Rj(z0,ao,' • ;aj-i),j > 1, where Qo is a constant, Q(j) is a polynomial, and the coefficient of its leading term equals 1. Rj is rational in all variables. The roots of Q(j) are so called resonance points. If there exists a non-negative integer j 0 such that Q(jo) = 0 and RjQ / 0, then it is contradictory and so Equation (2) has no the Painleve property. If Q(jo) = 0 and RjQ = 0, then we obtain an identity and so a,j is arbitrary, which is an integral constant. Especially j = —1 is always one of the

154

roots of (5(i) = 0, which corresponds to the arbitrariness of ZQ. By CauchyKowalevskaya Theorem, if the number of integral constants is less than n, then the above series is not the general solution of Equation (2). Therefore it is not of Painleve type. Example 3.2. Consider Painleve equation I. Substituting the Laurent series into it, we have p = —2, ao = 1 and i-i

U + 1)0" ~ 6 K = 6 J2 aJ-k*k + z06(j - 4) + 6(j - 5), j > 1,

(3)

k=i

where S(x) — 1 if x — 0 and 8{x) — 0 otherwise. Obviously both j — — 1 and j = 6 are resonance points. By Equation (3), z n ° di — a2 = a 3 = U, a 4 = - —,

1U

1

a 5 = - —. 15

For j = 6, we get an identity. It means that the resonance point j = 6 is consistent and so OQ is an integral constant. The other coefficients dj,j > 7 can be obtained by Equation (3). Hence Painleve equation I passes ARS test. Indeed by analytic transformation of Steeb and Euler [24], we know that Painleve equation I is really of Painleve type. Example 3.3. Consider the same equation as in Example 2.2. It has no the Painleve property, but passes ARS test (see [50]). This illustrates that ARS condition is not sufficient. In the previous paragraphs, we described ARS test briefly. As to some special cases, see [50]. The related theory and applications can be found in [22-25]. 4 Painleve Property for P D E and Its Test The relation between the integrability and the Painleve property is very fas­ cinating. It is believed that there is a deep and intimate connection between them. Certainly it would be better, if one could extend ARS method to treat partial differential equation (PDE) directly, since the group reductions are quite strenuous (see [22]). At present, there are two extensions in this field. The first is due to Ward [61]. The other is given by Weiss, Tabor and Carnevale [62]. _ _ Definition 4.1 (see [61]). Let S be an analytic and non-characteristic complex hypersuface in the n-dimensional complex space Cn. If all solutions of a PDE (or a system of PDE's) are analytic on Cn/S and meromorphic on Cn, then we say that it has the Painleve property.

155 Example 4.1. Consider the system Ut = C\UX + uv, vt = c2vx + uv where c\ and c2 are constants. It has a general solution _ (c2 - ci)F[(x + Cit) ~ Fl{x + c1t) + F2{x + c2t)'

U

V

__ (c2 - c i ) F 2 ( x + c 2 t) ~ F±{x + cit) + F 2 (x + c 2 t)

w/iere Fj is arbitrary smooth function and F- = j-Fj(y). Thus this system has the Painleve property. The above definition is a direct generalization of Definition 2.1 to PDE. But it is hard to be used. We now focus on the other definition. Definition 4.2 (see [62]). A PDE (or a system of PDE's) defined on Cn is said to have the Painleve property, if the solutions are single-valued about the movable noncharacteristic singularity manifold. We next explain the above definition in detail. Let <\> — 0 be the movable, noncharacteristic (i.e., (j)x(j>t / 0) singularity manifold of solutions. Assume that ^

oo

where (p and Uj are analytic functions in a neighborhood of the manifold <j> — 0. Putting the above expansion into a PDE and analyzing the leading part, we get the value of p and a series of recursive relations for uj. We say that this equation (or a system of PDE's) has the the Painleve property, if the following three conditions are satisfied: 1. p is a positive integer; 2. The recursive relations are consistent for all Uj] 3. There are enough free functions in the sense of Cauchy-Kowalevskaya Theorem. Clearly Definition 4.2 is a natural generalization of ARS method. It also gives a practical test. There are the concepts of resonance points, consis­ tency and so on, in the way similar to those in ARS method. Weiss, Tabor and Carnevale guessed that every equation with complete integrability has the

156

Painleve property in the sense of this definition. It is refered to as WTC con­ jecture. It is verified that many of them have really the Painleve property (see [62-86]). But there exist some counter examples. Indeed Clarkson [52] found that WTC conjecture is neither sufficient nor necessary. Therefore some scientists hope that the modification of Painleve expansion may remove this drawback (see [23]). Nevertheless WTC method is still powerful as a detec­ tor for the integrability. Now it has been widely applied to many nonlinear problems, especially the equations with parameters, to find the integrability conditions and other properties (see [64,79]). On the other hand, the proce­ dure of this method brings many equations which often lead to the Backlund transformation, Lax pairs, symmetry operators, bilinear forms and rational solutions (see [62-91]). Finally it is easy to extend this method to construct analytic solutions of nonintegrable systems (see [93-97]). Example 4.2. Consider the KdV equation Equation (1). Let x and t be the variables in the complex plane. Then its singular points are combined in a manifold, say <j) = 0. Insert the ansatz in Equation (1), we obtain p = 2, tio = - 1 2 ^ and Uj-3,t

+ 0 ~ 4)(f>tUj-2

+

"}Tuj-.k(Uk-l,x

+ ( * - 2)Ukx) +

Uj-3,xxx

k=o +30'

~ ^)(/>xUj-2,xx

+ 3 0 - 4)<j>xxUj-2lx

+ 0 -

+30 - 4)0 - 3 ) f e - ! , x + 30 - 4)0 +0-4)0-3)0-2)^^=0,

4)xxxUj-2

WxfaxUj-!

or equivalently after simplification, QU)luj = Rj(,uo,- -,Uj-i)

(4)

with

QU) - 0 + 1)0-4)0-6). Thus j — - 1 , 4 , 6 are resonance points. Evidently, j — -I corresponds to the arbitrariness of (j). Furthermore, it follows from Equation (4) that Ui = 12(j)xx, „

_o&

t

Axxx

f.

157

u3 = — — + —- + — U2. x


(6)

(Px

For j = 4 ; we have -~-((f>xt + (j>xxU2 - 0 x ^ 3 + 0xxxx) = 0.

(7)

It is easy to know that it holds automatically. Thus Equation (4) is consis­ tent for j — 4, and u\ is arbitrary. Similarly we can get u§ and prove the consistency for j = 6. Therefore UQ is also an arbitrary function. All these arguments show that there are three arbitrary functions as expected by CauchyKowalevskaya Theorem. Consequently, Equation (1) has the Painleve property. It is pointed out that since (j)xt ^ 0, we can put (j) = x 4- ip(t) and then simpli­ fies the judgement on the Painleve property. This technique was proposed by Kruskal [62]. If our purpose is only to check the Painleve property, then the procedure stops here. But the success of WTC method is also due to that the deriva­ tion brings other results. For instance, we can apply truncation technique to Equation (1). Since p = 2, we put 02

(j)

Then for j — 5, U2,t + ^ 2 ^ 2 , x + U2,xxx = 0.

(8)

This is exactly the same equation as Equation (1). Thus it leads to autoBacklund transformation. Moreover, we have by substituting uo and u\ into the truncated series that d2 u = 12—~^{ln4>) + u 2 where 0 and u2 satisfy Equation (5), Equation (6) with 1x3 = 0 and Equa­ tion (8). This form is connected with Hirota bilinear transformation (see [72]). Furthermore, by letting (j)x = ip2, we get the Lax pair of Equation (1). We can also use Schwartz derivative and the second-order scattering equation to obtain the Lax pair. On the other hand, Strampp [85] declared that u\ is a symmetry of Equation (1), and derived the strong symmetry operator. Finally some ra­ tional solutions can be obtained from Equation (5), Equation (6) with 1x3 = 0 and Equation (8) which is the subject of the next section. It is clear that WTC method not only gives a test of the Painleve property, but also brings fruitful results related to the integrability. This method is

158

also applied to many systems of PDE' s and some hierarchies of nonlinear evolutionary equations (see [63,65,81]). Newell and some other authors adopted this idea to deal with nonlinear ordinary differential equations (ODE's) and get a new Lax pair of Henon-Helis system (see [76]). 5 Construction of Analytic Solutions In this section, we show how to construct analytic solutions from the resulting equations obtained by WTC method. This technique is due to Conte (see [88-91]). Let 5

= {0:x} = ^

x

-

^

2 cj>l

(9)

and

C = - £ .

(10)

S is called the Schwartzian derivative while C has the dimension of velocity. In addition, let Q _ 1 XX 2 (px which is suitable for computation. We also need the following lemmas. Assume that P(z) is analytic in z. Lemma 5.1. Letv\ andv2 be two linearly independent solutions of the equation g + P ( z ) t , = 0,

(11)

which are defined and holomorphic on some simply connected domain D in complex plane. Then v2(z) satisfies the equation {w:x}

= 2P(z)

(12)

at all points of D where v2{z) ^ 0. Conversely if w(z) is a solution of Equa­ tion (12), holomorphic in some neighborhood of ZQ € D, then one can find two linearly independent solutions V\(z) and V2(z) of Equation (11) such that U>{z) = —TTv2{z)

159 Lemma 5.2. The Schwartzian derivative is invariant under fractional linear transformation acting on the first argument, namely,

faw + P \ . x 1 — : — \ z ) = {w'zs { jw + a J where a,/3,7 and a are constants, and ao~ / /3j. The above lemmas can be found in [24]. Example 5.1. Consider the KdV equation Equation (1). By using Equa­ tion (5), Equation (6) with u% = 0 and Equation (8), we have C-4S-12Q2,

u2 =

Cx — Sx — o, Ct + CCX — 6SCX — 5St — 5Co x — 2SSX — ^Sxxx = 0. Since both S and C depend on (j), the consistency leads to St + Cx^x + 2SCX + CSX = 0.

(13)

5 - C = A,

(14)

S = ^ ( f i + A),

(15)

Thus we have

2 = C-4S-12Q2, (16) u where A is an arbitrary constant, and u satisfies Equation (1). If we take 3 {2 = 0, A = --A; 2 , A;>0,

£/ien

A-2

According to Lemma 5.1, we consider the equation k2 VrXX.r. -^V 4 y

=0

with two linearly independent solutions e*x and e~^x. plies ae%x + (3e~~2x 7e2 x + ae

*x

Thus Lemma 5.2 im­

160

where a,/?,7 and a are arbitrary functions of t, and aa ^ J3j. By Equa­ tion (10), = —T}

ZkT

je 2 * -|- ae 2 s

where £ = x — k2t, and a, /?, a, 7 are arbitrary constants with aa ^ /3j. Finally Equation (16) gives

which is a one-soliton solution. We can put u — u 0 and 0 < e < ^ . The asymptotic waveform was discussed. But it has no waveform solutions. We consider the following generalized equation, u>tt = utxx - (1 - u + bu2)ut - Xu - fiu2

(17)

where 6, A and /i are constants and b > 0, Xfi ^ 0. Now we use Painleve analysis to construct analytic solutions. We first replace u by its Lorentz series in Equation (17) and find the power p — — 1 and its recursion relations. It is easy to verify that the resonance points —1,3,4 are consistent. The truncated series gives u — ea—(ln(j)) + u\,

where e = ± l , a = J | . In terms of 5 and C, satisfies that

CCX - ^C3 + (^-a2 - l \ c + CS-ena

= 0,

161

CCxt - \c2Ct 3

- efiaCx + \efiaC2 * 3

z

+ CSt - (x + ^aA

C =0

and CCtt - 4CCxxt +(l+ Q / i a 2 + \\C2-

^a2\

+ 3e/iaC xx - ^ / i a C 3

CCt + \c*Ct

(^eXa + - ^ a

3

) C - 3CS X , - 3C 2 S t

- 3 C S C t + Se^iaCS = 0. We now focus on the simplest case, in which 5 and C are constants. Then A

5 =

2/i2a2'

/3A

a

\/j,a

2

and 12A3 4 4

// a

5A2

/i^a

2

6A

//^a

A 2

1 J

2/x

rt

rt

\x

The last equation is the condition that A and [i have to satisfy. Finally, we get \e(ae2*** U\ = —

2

M

3^

-fie

2

£

^c)

^

2fj,(ae*^t + / ? e _ 5 ^ )

=

P

—

_i

\

I3T7 6 = 1 1 + fe ^«ft

where a and /? are arbitrary constants, and £ = x — £(^£ + §)£• Recently the authors used this approach to obtain periodic solutions, trav­ elling wave solutions of Fisher equation, Nagumo equation and other related problems. In particular, we found some analytic solutions describing the coa­ lescence of two waves. This analysis was also used successfully for the system of carrier flow equations and the Belousov-Zhabotinskii reaction (see [94-97]). 7 Discussion In the previous sections, we review systematically the Painleve analysis and its applications. But there are several important topics which are not mentioned. For instance, the Ziglin Theorem for singularity analysis of Hamilton system, the applications of Painleve analysis to bifurcation, chaos phenomena and the selfsimilarity constructure (see [98-102]). How to realize the Painleve approach

162

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21. M. Tabor, Nature, 310(1984),277-282. 22. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Trans­ form, SIAM, Philadelphia, 1981. 23. A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985. 24. W-H. Steeb and N. Euler, Nonlinear Evolution Equations and Painleve Test, World Scientific, Singapore, 1988. 25. A. Ramani, B. Grammaticos, and T. Bountis, Phys. Rep., 180(1989), 159-245. 26. M. J. Ablowitz, M. Kruskal, and H. Segur, J. Math. Phys., 20(1979), 991-1003. 27. G. Z. Tu, J. Eng. Math., 2(1985), 1-12. 28. H. Airault, Stud. Appl. Math., 61(1979), 31-53. 29. H. Airault, C.R. Acad. Sc. Paris, 295(1982), 619-622. 30. K. Nishioka, Nagoya Math. J., 113(1989), 1-6, 173-179. 31. K. Nishioks, Manuscripta Math., 66(1990), 341-349 32. H. Umemura, Nagoya Math. J. 117(1990), 125-171. 33. J. D. Gibbon, A. C. Newell, M. Tabor and Y. B. Zheng, Nonlinearity, 1(1988), 31-53. 34. M. Boiti and F. Pempinelli, Nuovo Cim., 51B(1979), 70-78; 56B(1980), 148-156; 59B(1980), 40-58. 35. A. S. Fokas, J. Math. Phys., 21(1980), 1318-1325. 36. A. S. Fokas and M. J. Ablowitz, J. Math. Phys., 23(1982), 2033-2042. 37. H. Segur and M. J. Ablowitz, Physica, 3D(1981), 165-184. 38. P. Holmes, Appl. Math., 50(1982), 53-62. 39. N. Joshi and M. D. Kruskal, Phys. Lett, 103A(1988), 129-137. 40. S. P. Hastings and W. C.Troy, SIAM J. Math. Anal., 20(1989), 634-642. 41. J. G. Byatt-Smith, Stud. Appl. Math., 79(1988), 143-157; 80(1989), 109-135. 42. A. R. Its and V. Yu. Novokshnov, The Isomonodromic Deformation Method, New York, 1986. 43. M. Jimbo, M. Kashiwara, and T. Miwa, J. Math. Phys., 22(1981), 25812587. 44. H. Flaschka and A. C. Newell, Physica, 3D(1981), 203-221. 45. A. S. Fokas and M. J. Ablowitz, Commun. Math. Phys., 91(1983), 381-403. 46. A. S. Fokas, U. Mugan and M. J. Ablowitz, Physica, 30D(1988), 247-283. 47. A. R. Chowdhury and P. Guha, Phys. Lett, 134A(1988), 115-120. 48. M. J. Ablowitz, Physica, 3D(1981), 129-141.

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166 STABILITY OF TRAVELING WAVE SOLUTIONS FOR A RATE-TYPE VISCOELASTIC SYSTEM

L. HSIAO* and T. LUO** Academia Sinica, Institute of Mathematics, Beijing 100080, China

Dedicated to Professor Xiaqi Ding on the Occasion of His 70th Birthday

1

Introduction

The system of conservation laws in mixed type can be used for dynamic elastic bar theory where the stress-deformation relation is not monotone (see [10]) or for the dynamics of a material exhibiting change of phase such as in a Van der Waals fluid (see [21]). There have been a lot of efforts to investigate this kind of system, we refer to [20,21,13,17,8,7,6], and the references therein. A different kind of regularization, made on the mixed type system, has been proposed in [23] to introduce a visco-elastic regularization which allows a pressure relaxation over time (see [5] and [23]). Namely, consider the following rate-type viscoelastic system

i

vt - ux - 0, ut+px=0,

(p + Ev)t =

(1.1)

-
where v and (— p) denote strain and stress, u is related to the particle velocity, E is a positive constant, called the dynamic Young's modulus, r > 0 is a relaxation time. This system is a kind of regularitation of the following system [vt-ux=Q, \ut+pR(v)x

=0,

(1 y . )J2 '

which is a mixed-type system when PR(V) is nonmonotone, because the semi-linear system (1.1) is always hyperbolic and standard existence and uniqueness results apply for any fixed r > 0, and because the system (1.1) approaches (1.2) in the limit r —>• 0 formally. Since the system (1.2) can be obtained from (1.1) by an expansion procedure as the first order, it is natural to expect that the solution of (1.1) converges to that of (1.2) as r —> 0. However, the theoretical proof of this convergence is not known so far, although some numerical experiments on (1.1) have been made (see [19]) and certain effort on the L 2 -estimates for the difference \p — PR(V)\ of (1.1) have been done (see [5]).

* Supported partially by National Natural Science Foundation of China ** The present address is: Department of Mathematics, City University of Hong Kong.

167 As the first step of the program to prove the expectation, we investigate the existence and asymptotic stability of traveling wave solution of (1.1) in the present paper. We will prove that for any given shock wave (a;v~, u~; v+, w + ), satisfying the Generalized Entropy Condition introduced in Section 2 with a ^ 0, for the reduced system (1.2), the system (1.1) admits a traveling wave solution, smoothing out the shock wave no matter whether PR(V) is monotone or not, where a is the speed of the shock with (v~,u~) and (v+,u+) as the limit value from the left and right side respectively. This is given in Section 2. Furthermore, when PR(V) is monotone decreasing in a neighborhood of (v~ ,v+) (if v~ < v+) or (v+, v~) (if v+ < v~) and the shock wave is suitably weak, we consider a initial value, which is a small perturbation of the corresponding traveling wave profile, and prove that the solution of this initial value problem for (1.1) exists globally and converges, in the if 1 -norm, to the traveling wave solution as t -> +oo. Namely, the stability of traveling wave solutions is obtained, which will be shown in Section 3. In the case when PR(V) is monotone decreasing in v, (1.2) is a hyperbolic system of conservation laws, which can be usually approximated by viscosity method. A different approximation method is to introduce some relaxation terms, which has been proposed in [11] from the numerical point of view for more general system. There are a lot of investigations on the stability of viscous shock profiles (see [14], [24] and the references there). For relaxation systems, the stability of elementary waves has been proved in [15] when the corresponding equilibrium equation is scalar. In the present paper, the corresponding equilibrium system (1.2) is a 2 x 2 system, more difficulties occur certainly. A suitable entropy pair is chosen carefully in Lemma 3.3 to get an energy estimate. Based on this estimate, together with the ideas introduced in [12] with some modifications, our results can be established. The investigation of relaxation mechanism have been made in the field of kinetic theories and in other branches of continuum physics. We refer to [2,18,9,1,12,15,4,25,22] and the references therein.

2

T h e E x i s t e n c e of Traveling Wave Solutions

Consider the following rate-type viscoelastic system

{

vt ~ ux = 0 ut+px=0

{p + Ev)t =

(2.1)

-te-*«W

where v and (—p) denote strain and stress respectively, u is related to the particle velocity, E is a positive constant, called the dynamic Young's modulus, r is a relaxation time. Moreover, PR(V) is a given smooth function defined on (6, oo), where b is a positive constant, and is not necessary to be monotone, say, p'R(v) < 0 for b < u < va or v > vp for some positive constants va < vp, while p'R(v) may change sign when v G [va,vp].

168 A discontinuity (a;v

, w ;v+,u+)

in the weak solution of ' vt - ux = 0

(2.2)

^+PJR(U)X=0

is called a shock wave satisfying Generalized entropy condition if A. The Rankine-Hugoniot Condition is satisfied and the speed a(v+,v~)

is defined, namely,

cr(v+ — v ) = —(u+ — u ) a{u+

- u~) = pR(v+)

-

PR(V~) +

a(v+,v

) = T\

PR(V )

-

(2.3)

~PR(V

B. The Generalized Entropy Condition holds, namely, for any v between v a2(v,v+)

< a2(v~,v+)

= a2(v+,v~)

< a2(v,v~)

and v+,

for

a = a2

for

a = ai

(2.4) 2

+

cr (v,v ) where a*(v,v*)

= -PR{V)

2

> a (v

+

2

+

2

,v ) = a (v ,v

~P?{V'],V'

= V

)>cr (v,v

)

or «+.

Remark 2.1. Compared with the Generalized shock E condition introduced for mixed type system of conservation laws in [8] and [7], the Generalized Entropy condition used here is much stronger. The latter is possibly satisfied only when both (v~,u~)

and (v+,u+)

are

located in the hyperbolic region of the system (2.2). But, it is possible for the former to be and (v+,u+)

satisfied when one of the state (v~,u~)

is located in the elliptic region.

Remark 2.2. In the case when PR(V) is monotone decreasing, this Generalized Entropy Con­ dition is the same as the shock E condition for a shock, introduced for the hyperbolic system of conservation laws which is not genuinely nonlinear (see [3]). For simplicity, we only consider a shock (cr; v~, u~; v+, u+) for which a > 0 and it holds a2(v, v+) < a2(v~,v+)

< a2(v,v~)

for any v

between

v~ and v+.

(2.4)*

For definiteness, assume v+ > v~. We are looking for smooth traveling wave solution of (2.1), namely, solutions in the form (v,u,p)(x,t)

= (t7,iZ,p)(0,

£= ^

^

(2.5)

satisfying i>(q=oo) = vT,u(^oo)

= u^iPfaoo)

= PR{V^)

= p^.

(2.6)

Obviously, the system (2.1) becomes ' — crv£ — U£ = 0

-(7U£+p€ = 0 ^a(p + Ev)z

(2.7) =P~PR{V)

169 which, combined with (2.6), implies a(E - a2)v£ = -a2(v

- v~) + p~ - PR{V) (2.8)

= -a2(v

- v+)

+p+

—

PR{V).

Assume that it holds 0 < a < y/~E(the case -y/E

< a < 0 can be discussed similarly),

(2.9)

namely, the so called subcharacteristic condition (see [15] ) is satisfied, then d£=9(v)

(2.10)

.,, ,_x -a2(v - v~) + p~ - PR(V) -a2(v - V+)+p+pR(v) TJ_. . L with g(v) = * —4—£j-— y i t K J = ^ —^—£_—^*v >. It is easy to see, due o\hj — a ) a{_b — a ) to (2.4)*, that v~ and v+ are the only roots of g(v) = 0 and dv — > 0 for ve(v~,v+). (2.11) d£ Thus, /

—— is finite and monotone with respect to v for an

JV! 9\T)

(v~

,v+). We claim that for any given v\ G (v~

yvi

+ :v ),

9(r)

JVl

y v G (v~,v+) and v\ G

g(r)

In fact, it can be shown by (2.4)* and (2.10) that g(v) = 0(\v - v+\)

as

g(v) = 0(\v — v~\)

as

v->

v+

v—¥v~.

These imply (2.12). Thus, by integrating the equation (2.10), one obtains ._

rv*<« *> dr

~ JV1

9(T]

This yields a implicit formula for v(f) which is uniquely determined (up to the choice of v0) due to the properties of g, discussed above. u(£) and p(f) can be easily determined then by (2.7) and (2.6). Then it reads that T h e o r e m 2 . 1 . Under the Generalized Entropy condition (2.4)* and the

subcharacteristic

condition (2.9), the system (2.1) has a smooth traveling wave solution which is unique up to a shift in f and satisfies v^ > 0.

170 To be definite, we take

Thus, the traveling wave solution (v,u,p)(£) is determined uniquely. We finish this section with the following estimates which are useful for the next section and are easy to be proved by using (2.7) and (2.10). L e m m a 2 . 1 . If (2.9) and (24)* hold, then N

< Qv+ - v~\, K | < C\v+ - v~\, \pt\ < C\v+ - v~\,

and it is also true for the second and the third derivatives of v, u and p, respectively. 3

Stability A n a l y s i s

Our objective in this section is to show the asymptotic stability of traveling wave solutions constructed in the previous section. For this purpose, we consider the initial value problem for (1.1) with initial data (v,u,p)(a;,0) = (V0,U0,PO)(X/T)

(3.1)

which is a perturbation of the smooth function (U, u,p) ( ^ ) , defined by the smooth traveling wave solution (v, w,p)(f), £ — X ~T with t = 0, which smoothly interpolates the asymptotic values (v±,u±,p±) with the speed a. We assume, without loss of generality, r = 1 in the system (2.1). For showing the stability, we make the following hypothesis Hi) there exists a neighborhood D(v~) for v~ such that PR(V) is differentiate up to the third order in D(v~) and it holds —E\ < p'R{v) < -E2, for v G D(v~) for some positive constants E\ and E2 with Ei < E,i — 1,2, H2) inf p"{v) > 0, veD(v-) H3) p"{v) and p'"(y) are bounded in D(v~). To get the stability result, we reset the problem on the moving coordinate (£,£),£ = x — at. Let (v,~P,X)(€,t)

= (v(x,t)

-v(0Mx,t)

- u ( 0 , p ( x , 0 -p(0)

(3.2)

We rewrite the system (2.1) in the form

{

vt - oV^ - Jl^ = 0 T*t ~ °H + Xs = 0

(3.3)

(X + EU)t - a(x + EV)f: = -x + (PR(V + V) -

pR(v)).

Introducing

v(Z,t)= f J— oo

Hv,t)dy,

it{t,t)= [ J— oo

Hy,t)dy,

x(U) = x(U)-

(3.4)

171 We seek the solution (^,/x, x)(£>0

wrfcn t n e

property 2 X(;t)€H

{v{;t),»{;t))eH\ for the following system

i

Vt - GV^ - [l£ = 0

Ht-VLi(:+x

= 0

(3.5)

(X + Ev()t - a(x + Ei/t)t = -x + Pfi(^ + i/$) - Pii(iJ). Combination of (3.5)2 and (3.5)3 gives Xt ~ oxz

+ ^MCC + X + A(iJ, I/ € )I/ € - 0

(3.6)

where A{v,vt)=

[ -P'R(v + evz)dO. (3.7) Jo We assume v+ G D(v~). Thus, v 6 D(v~), since zJ^ > 0. If |i/^| is suitably small so that v + #z/£ G JD(V~) for 0 < 9 < 1, then the hypothesis Hi guarantees 0 < E2 < A(v, i/€) < Ei < E.

(3.8)

Our main result is: T h e o r e m 3 . 1 . Suppose the conditions in Theorem 2.1 hold, the function PR satisfies the hypothesis Hi-H$, and the initial data (vo,^o,Po) satisfy (v0 -V,U0

-U,PO

G H2,

-p)

X

/

rX

(vo —v)(y)dy

and fio(x) = /

(uo —u)(y)dy

are

furth -co

J — CO

well defined for x G R, with (fo^Mo) G L2. Then, there exist suitably small positive constants 8o and 770, such that if \v+ —v~\ < 80 and

\\(v0-v,u0

-u,po

-P)\\H*

+ IK^o,MO)||L2 < Vo

(namely ||(^o, McOH/f3 + ||xo||f/2 < Vo), the initial value problem (2.1), (3.1) will have a unique globally defined smooth solution (v,u,p) which satisfies (v-v,u-

u,p-p){-,t)

G H2

t > 0,

and asymptotically converges to the traveling wave solution in the i7 1 -norm, namely \\(v,u,p)(x,t)

— (v,u,p)(x

- o-t)\\Hi -> 0

as

t -> +00.

We first investigate the Cauchy problem for (3.5) with the initial data MO

= /

(vo -v)(y)dy,M€)

J—00

Xo(0

=Po~P

= / J— CO

(uo

~u)(y)dy

172 in the Banach space X(0,T)

■ {v,ii) G C°(0,T,H3),X

= {(v,ft,X)

€

C°(0,T,H2)},

the norm for which is defined as sup (||(^)WII 2 tf3 + ||x(*)lli) 1 / 2 -

N(v,»,X,T)=

0
In the following, we assume a priority that (v, fj,,x) is the smooth solution of (3.5) and

(u,fi,x)eX(otT). To prove our main result, we need the following a priori estimates. L e m m a 3 . 1 . Suppose the conditions in Th. 3.1 are satisfied, then there exist suitably small constants rji and Si such that if \v+ —v~\ < Si and iV(i/, /z, x, T) < rji for some T > 0, then it holds iV(i/,/i,x,T)2+ / Jo 2

\\(^,^x)(;t)\\hdt (3.12)

2

d

<# JV (*/,//,X,0)

f

2

2

= K N (0),

for (I/,/J,,X) £ X(0,T), where K > 1 is a positive constant which does not depend on T. To prove this lemma, we establish the following lemmas 3.3-3.6 next. Take Si is small, such that v+ G D{v~) and v G D(v~) when \v+ — v~\ < 5\. By Sobolev embedding theorem, Hk+1 <-+Ck,k>0. Thus, if 7V(i/,/i, x, T) < Sx then |z/| c 2 ,|//| C 2

and

\x\c*

<0(l)Si.

Hereafter we use the symbol 0 ( 1 ) to denote a generic constant which does not depend on t. L e m m a 3.2. Suppose N(v,fi,x,T) < T)I, \V+ - v~\ < Si for suitably small rji and 8\, and the conditions in Theorem 3.1 are satisfied, then we have, rt

r + oo

lh(*)||2 + H^WIP + HxMII2 + / / JO


+ O(l)\v+-v-\

f

J-oo

f

JO

l(x-PRM]2(t,T)dtdr

°°(x + Ev(:)2dZdT

(3 13)

-

J-oo

0
the function PR is defined by PR(V)=PR(V

Proof We will work with jl, V and x Define

next

+

V)-PR(V).

since ^ = /Z, v^ — V and x — X>

$(z, v, y) = pR(v + Z ) - pR(v) +Ez-y,

(3.14)

173 it is obvious that $ ( 0 , v " , 0 ) = 0, — (0,t;",0) = p'R(v~)

+ E>0

(due to the hypothesis!^).

Then, by using the existence theorem of implicit function, it can be shown that there exists a positive constant £o > 0 such that a smooth function z — h(y,v) is determined from $(z,v,y) = 0 uniquely if \z\ < £o, \y\ < eo, \v — v~\ < eo- Namely, for any given (y,v) with \y\ < £o and \v - v~\ < eo, there exists a unique root z with \z\ < £Q for the equation $(2,tJ,y)=0,

(3.15)

expressed by z = h(y, v). That is pR{v + h{y,v)) -pR{v)

+Eh(y,v)

- y = 0,

for

\y\ <e0,\v-v~\

< e0.

(3.16)

Due to V£ > 0, it is known that |U(f) —v~\ < \v+ - v~ \ < Si which implies \v(£) — v~\ < £o,£ € R, if S\ is suitably small. Let y = PR(V) + Ev. Choose 771 and £1 suitably small so that \PR(V)

4- Ev\ < £Q if

M < Th­ in view of the fact that z = V satisfies (3.15) with y — PR{V) + EV, and the fact that \V\ < £0 if r/i is suitably small, it turns that V = h(y,v), namely V = h(PR(V) + EUtv).

(3.17)

For any w with \w\ < eo, we define rW

(p{w,v) = / \PR{V) - pR(v + h(r,v))]dT Jo = / Jo

-PR(Hr,v))dr.

It is not difficult to calculate with (3.3) and to show that j

^

+ ^

+ ^ X + E Z ^ ) } + [x + §%(x +

Ev,v)](x-PR(v)} (3.18)

= { § of

+ § x 2 - EM + M*

+ EV)}

-a^£(x

+

Ev,v)vK

where we have made r\\ suitably small such that it holds |x + Ev\ < e0 when |x| < Vi \v\ < m , and $ £ = -PR(h(x

+

an

d

Ev,v)).

Define A(A) = -PR(h(X(x

- PR(y))

+ PR(V) + Ev),v)).

(3.19)

It is clear that \(1) = -£(X A(0) = -PR(h((PR(v)

+ Ev,v), + Ev),v)).

(3.20) (3.21)

174

In view of (3.21) and (3.17), it holds A(0) = -PR{V)

= pR{V)

- pR(v

+ V).

(3.22)

By the mean value theorem, A(1)-A(0) = A'(0i)

for some 0 < 0X < 1,

(3.23)

where A'(#i) can be written as A'(#i) = -p'R[v + hiO^x ~ PR&) + PR(P) + EU,v)] '$Vi(x

" PR(V)) + PR{V) + EV,v) • (x - PR(V))

(3.24)

d

^m(x-PR(V)).

We estimate SA now. Differentiating (3.16) with respect to y, one obtains dh.

*

1 M

=

(3 25)

* + A(*+fc(y,,lO)

-

where Vi = 0i(X " ftC7)) + ft*(*) + ^ 7 -

(3-26)

By a similar argument as used to obtain (3.17), it is easy to see h(0,v) = 0.

(3.27)

\im h(y,v) = 0.

(3.28)

Thus, due to the continuity of h(y,v),

Therefore, it can be guaranteed that v + h(yi,v) G D(v~), by choosing rji and £i suitably small, and it follows then from the hypothesis Hi that -Ei < p'R(v + h(yuv))

< -E2,0

< EuE2

< E.

(3.29)

This, combined with (3.20), (3.22), (3.23), (3.24)-(3.28), yields f- , du = (x-PR(V))2(l

+ m) (3.30)

= (x-PR(u))2 y-j^ix-PniV)?. We estimate (p(x + Ev,v) next.

£ E+p'R(v + h(yuv))

175 It is known PR{JI{T,V))

= PR(V + h{r,v))

- pR{v)

(3.31) = PR{V + 93h(T,v))h(r,v),

for some

0 < 63 < 1.

Similar to (3.25), it can be shown that

ty^

(3 32

= E + P'J+h{y,V)y

- )

Since

(3 33)

^Ehr^iTy^^Ehr,

-

if \y\ is small, and it holds T,v)

h(

= /i(r,U) - h(0,v) =

dh(9f'v\

for

we arrive at

some0

< 04 < 1,

(3.34)

_

^k

*-

^

+

M ( r , l O ) ^ ^ >^

>0 +

provided | r | is suitably small. This implies that if |x|, \V\ < 771, \v 2,EE*E2)

(3.35)

— v~\ < 8\ then it holds

(X + Euf

(3.36)

where 771 and 8\ are suitably small. On the other hand, it follows from (3.35) and the definition of (p that \
+ EV)\

At last, we turn t o estimate

It is known from the section 2 that N

<0{l)\v+

-v~\.

(3.37)

By the definition of ip, it holds

^

^

= ~ lW\p'R(v + h(T,v)) -p'R{v)]dT - J PR(V + h(r,v))

K

d_

J

dr.

It is easy to show by differentiating (3.16) with respect t o v, that dh{r,v) dv

=

p'R{v) - p ' R ( v + h(r,v)) E + p'R(v + h{r,v))

176 Thus, dip(w,v)

fw

= / Jo

\PR(V) -PR{V

+ Hr, v)]

E

, , , - , ■ , , -, dr. &+PR\V + n[T,v)

Due to (3.33) and (3.34), it holds \h(T,v)\<0(l)\r\

(3.38)

if \T\ is suitably small. This, with the help of # 2 , implies \p'R(v)-p'R(v

+ h(r,v))\

< 0(l)|A(r,tT)| < 0 ( l ) | r | ,

(3.39)

if \T\ is small. It follows from the above estimates then that Ev^vt <0(l)\v+

a-gziX +

-v-\(x

+ EV)

if 771 and Si are choosen suitably small. Integrating (3.18) over (—oo,+oo) x [0,£](0
GV^ - ^

o-xz + Efts

= 0

(3.40)

+ x + A(v, I/$)I/ C = 0

(3.41)

where (3.42) A{v,vi)

= - / p'R(v + 0i/z Jo

Thus, it is known from H\ that 0<E2


<EX

(3.43)

if \v$\ and \v+ — v~\ are suitably small. L e m m a 3 . 3 . Suppose N(v, fj,,x,T) then we have

□

< 771 and \v+ — v~\ < Si, ifrji and Si are suitably small, rt

r+OO

\Ht)\\h + \Mt)Wh + HxWII2 + / / JO rt

(3.44)

r+oo

2


(x2 + »l + n\ + ovifj,2m,T)dt;dT

J-00

J-00

(»& + /&+x?)(£.T)de*-,

o
177 Proof. By a similar approach as used in [12] and [25] we discuss the equation i/Li(i/,/i) - A-lnL2{v,n)

= 0.

(3.45)

The left-hand side is reduced to [\v2 + ^ " V - A-1 fiX + ^ ( A " 1 ) ^ 2 } + + %*(A-1)&2 - A-'x2 +{A-1)MX

+ VLH-^)

EA-1^2

+ (E- e r ' X A - 1 ) ^ • m

(3.46)

+ {•••}*

where {• • •}$ denotes the term which disappears after integrations with respect to £ G R. In order to dominate x» we make a calculation on the following equation, /i£Li(i/,Ai) + A - ^ L a ^ M ) = 0-

(3-47)

The left of which can be reduced to {\EA-^\

+ \A-^x2

- ( 1 + \aE{A-^

+ IH»}t + (A-1 + \a(A-^

+ § {A-i)t)n\

+ EiA-^tHm

-

\{A-')t)X2

(3.48)

+ {■ ■ •}«•

Hence, the combination (3.45)+ (3.47) xA with a positive constant A yields Ft + [EA-1 - A(l + \cE{A-^

+ § (A"1)*]/*?

+ {(A - 1)A"! + ^ ( A - 1 ) ^ - ^ ( A - 1 ) , } x 2 + +(E - c2){A-x)^

■ ^ + (A-1)^

\a{A-*)iV? (3.49)

+ <^« - 2A»)

+A J B(A- 1 )^ ( / i e + {---} 5 =0, where

F=\»2 + | ^ V ~ A"Vx + ^ ( A - 1 ) ^ 2 + ^ABA-V? + ^ A " V + \*H"It can be calculated that

<^>< = * £ • * + <«?■'* (3.50) = A- 2 ' In view of H2 and (3.8), it holds A~2 / P'R{V + 6v()d6 > —^ > 0 for some positive constant E Jo i

a0 > 0.

(3.51)

178 This, combined with (3.50) and the fact of av^ > 0, yields a{A-^>^L*v^

_ ^1( 1 + E/Ei), Take A =

it is known, due to E > Eu '

<

»

<

!

(dA~l\

1

+

v

(3.52)

ttl*

that (3.53)

■

On the other hand, EA~l

>

(3.54)

Ex

Therefore, it holds that E def, - A) > — - xA = &i > 0. E\

A - 1 > 0, (EA'1

(3.55)

In view of (3.50) and (3.51), it is possible to choose 771 so small that if |/x^| < 771 and \va\

Viitnen

<

(A - 1)^4 _1 + ^{A~X)^<J {EA~l

>b2>0

- A(l + ^aEiA-1)^

for some constant

+ ^(A'1)^

> b3 > 0

6 2 , and

for some constant

63.

Thus, we are able to show, by integrating (3.49) over (—00,+00) x (0,t) and using lemma 3.2, that H * ) l l 2 + M*)ll 2 + &2 A x ( r ) | | 2 d r + 63 f\Wm?dr+£-f JO rt


+ O(l)\v+-v-\

rt Jo

*&2

J-00

(X +

E^)2(^r)d^dr

J-00

r+OO

/

Jo

r+oo

/ Jo

+ /

[+0°av^2dtdT

JO

-,

[(a2 - E){A'1)^

■ H - (A-^Mx

- am - -it) -

J-00

XE(A-1)iiHlti]d^dr.

*

(3.56) It is known from integration by parts, I /■' r+°°i

<\\*

(dA-x\

2jtJ

(d2A~1_

* r+°°

S:L

dvfdv

d2Adv\ (3.57)

rt

r + OO

< 0(1)^ /

/

av^d^dr

rt

+ 0(1)^ /

JO J-00 rt r + OO

+0(l)m / / Jo

J-00

(*f + l$)d(dr.

JO

r+OO

/ J-00

(y\ +

v\^dr

179 It can be claimed, due to Young's inequality and lemma 3.3 that I

rt

r + OO

I

Mo J-oo

I

I ff f+oc =

dA~l

| / O / - O O I rt

+ /

( < T 2

^

I )

f)A~l

r + OO

\J0 J-oo rt

M

H I

(a2 - E)^—-

/

^ " ' ^ '

vumdtdT\

OH

(3.58) I

r + OO

rt

~ Ukh I oo ™^+0^V+

r+OO

-V~\J0 J „ ^dT

+0(1)% j J 4 + 0(1)7,, j J fidtdr. With the help of integration by parts and the fact ,._i,

dA-1

dA~\

one obtains

\j JiA-^tHix ~ Ofl^dA < 0(1)7?! J | ( 4 + fa +x2+ itydtdT,

I f f \ (d2A~l_

d2A~l

\

dA"1

1

< 0(1)7?! j j av^2 + 0(1)7,, j j(^

1

I

+ ii\ + V2()d£dT. (3.59)

A t last,

\J J XElA-^iHKdtdrl < j j XEiA-1)^^

- X)lHdidr\

< 0(l)(in + \v+ -v~\) j J(fi2 +

(3.60) 2 X )didr.

The estimates (3.56)-(3.60) imply that rt

rt

\W(t)\\2 + IM*)II2 + / (llx(r)|| 2 + I M r ) | | 2 ) d r + / JO

< O(l)7V(0) + 0(l)\v+ -v~\

r+OO

/ JO

rt

v2{^T)d£dT

J-oo

r + OO

/

avtfdt-dr

JO J—oo rt

+0(1)7,1 /

r+OO

/

{v2u + , 4 + xl)(S, r)d^dr,

Jo J-oo

provided 771 and \v+ — v~ | are suitably small.

(3.61)

180 By virtue of lemma 3.3, it holds ft

/

f+oo

ft

{x-PR(vd?d£dT<0{\)\v+-v-\

/

r + oo

(x + E^)2didT

/

JO J — oo

+ 0{\)N{Q). (3.62)

JO J — oo

Due to Hi, it is easy to obtain E\v\ < (Pfi(^)) 2 < E\v\

(3.63)

if 771 and Si are suitably small. It follows then < (PRW)2

E^\

< 2[(X - PR(n)f

+ x2]-

Therefore, (3.62) and (3.61) imply rt

/

r+oo

rt

/

v\didr < O(l) /

JO J — oo

f+00

x2d£dr,

/

(3.64)

Jo J — oo

if \v+ — v~\ is small. This, combined with (3.61) and lemma 3.3 together, yield ftrt

r+°° f-tOO

\Ht)\\h + Mt)\\h + HxMII2 + / / Jo

ft

< O(l)iV2(0) + 0 ( l ) m / Jo

J-oo

(x2 + "l + ov^ + nlmr)didT (3.65)

f + OO

/

( 4 + fa + Xt)(t,T)dtdr,

0
J -oo

The lemma 3.4 has been proved then.

□

Turn to the estimates on higher order derivatives. Lemma 3.4. If r\i and Si are suitably small, then it holds that II^WII 2 + IK«MII2 + I|X«WH2 + [\\\xdr)\\2 < O(l)7V2(0)

+ \\m(r)\l2 + ll"«(r)|| 2 )A-

Jo

0 < t < T.

Proof. Differentiating Li(v,fi) and L2(v,ii), expressed in (3.40) and (3.41) with respect to £ respectively, we obtain <%Za(z/,/i) =LI(J/£,/X£)

<%L2(i/,/i) =L2(^,fi^+A^. A similar approach as used in lemma 3.3 shows that ^<%Z,i(i/,/x) - A'1 fi^L2{^

+EA-1p\i HA~l)t^(Xi

AO

- A-'Xl + (E- a2)(A'^a + o~fiu - fa) - A^A^fi^

•^ +

faA'1)^

+ { }<: = 0

181 and HitdsLi (i/, fj) + A~ 1 X4^L 2 (i/, 11)

= {\EA~l»h + 2 A_1 ^ + ^«^}t + (V 1 + £(A" V - i ^ ) X? - (l + ^(A- 1 )* + § (A"1),) ^ H-E^"

1

) ^ ^ + A " 1 * ^ - ^ = 0.

Thus, using (3.13), (3.65) and the method used in lemma 3.4 it can be shown that if 771 and Si are suitably small, then it holds

ik«(0H2 + iix«(')ii2 + /"* llxttoii2 + f iiweWii2*Jo

Jo

rt

2


r+oo

/

(3.66)

4({,r)d£*-.

^0 ./-oo

We estimate / / i/|^(^,r)d^dr now. ./o ./-oo Let us consider the equation (X£ + E i / t t ) 0 « L i ( i / , / i ) + i/«^L 2 (i/,Ai) = 0. This can be expressed as

f ( l / « ) t _ f ^ ) + (*«**)*+X^« + A ^ « + ^ 4 + { >€=0' from which we can show, with the help of Young's inequality and the estimates obtained,

ll"«(*)ll3 + f ll"«MII2 < O(l)N2(0), if 771 and Si are small. (3.66) and (3.67) yield the lemma 3.5 then. L e m m a 3.5. //771 and Si are small, then

lk«{(*)ll2 + ll««WII3 + llx«WII2 + / " ( | | « « ( T ) | | 2 + l k « ( r ) | | 2 + | | X « ( T ) | | 2 ) * - < O(l)AT 2 (0). Jo

Proof. Since %Li(z/,/x) = 1 , 1 ( 1 / ^ , ^ ) = 0 and %L2(^,/i) =L2(i/«,/x«) + 2^i/^+i4Ki/4 = 0 ,

(3.67) □

182 it can be verified that A-lxzzditiL2(v,Li)

/ie«0«Li(i/,AO +

+ (V 1 + \{A-^a - &p±} X\ - (l + ^(A" 1 ), + i ^ p i ) ^ - ^ ( A " 1 ) ^ ^ / ^ + 2A-1xaAiva

+ A - ^ A t f i/c + { h = °>

and i/«%Iri(i/,/i) - A - V « ^ 2 ( ^ ^ )

= (He + 2 A _ 1 ^ - ^"V«x« + H^4-1)^?*)* +E(A-1)/i|a - A " ^

+(£ - ^ ( A "

+ ( A " 1 ) ^ « ( X « +
2

1

) , ^ •^

+ ^ ( A "

1

^ ~ V ^ z ^ ~ ^ ~ V ^ ^

) ^ + {

k = °>

where the term {•••}•£ may concern the forth derivatives of /x or i/, but this will not cause trouble since we may assume /z £ H4 and z/ £ H4, x £ # 3 first and use Friedrichs modifier then to deal with the original case. By a similar approach as used in lemma 3.5 we can finish the proof of lemma 3.6 then. D Now, lemma 3.2 can be proved by combining lemma 3.3-3.6. We will prove theorem 3.1 next. By a standard method, the following local existence result can be obtained first. Suppose fjto £ H3, v0 £ H3 a n d xo € H2, then there exists a To > 0 such that the problem of (3.5) with initial data rtt, 0) = /xo, "(£, 0) = i/o, x(£, 0) - xo

(3.68)

possesses a unique smooth solution (/LA, Z/, X) o n [0, To]. Moreover, (v(;t)^(;t))eH3,x(;t)eH2.

(3.69)

In fact, we may first consider the system (3.3) with initial data (I7,P,X)K,0) = (n>*,fio*,Xo)(0

(3.70)

which belong to H2. This is a Cauchy problem for a hyperbolic system for which the approach in showing the well-known local (in time) existence theory ([16]) can be used to prove that these is a positive constant T 0 , depending only on the upper bound of \\V,~P,X(Q)\\H2,

sucn

183 that the above problem has a unique solution (I7,/Z,x) € C°(0,T0,H2) Furthermore, sup

nC

||(l7,7!,xW||jf2 < c\\V,-p,x(0)\\H2

1

^,^^

1

).

(3.71)

0
for some positive const, c depending only on Ei,E2 With the (17, //, x) obtained, we define

and the bounds of p{(v)(i

= 1,2,3).

I HU)= [ v(t,W + f{t) Jo

\l*(t,t) = J ~P(Z,t)dt + g(t) Let 1/(5,0) = MO = f Let

0
* ( { , 0 R , we get / ( 0 ) = /" V((,0)d£

- z/0(0). Similarly,

£

/(*) = ^o(O) + / (
0 < t < To

It is easy to check that this (i/,/i,x), defined above is a smooth solution of (3.5) and (3.68). To see (v,/i,x) € ^ ( 0 , T 0 ) , we use the first two equations in (3.5). The first equation gives "(t, t) = MO + [ WHZ, r) + ?Z(f, r)]dr, Jo

for t € [0, T0]

This representation immediately yields v G C°(0,T, i7 2 ) and sup H ^ r ) ^ < | M | 2 + t sup ( H H l / ^ r J I I ^ + llTZ^r)!!^). 0
0
It follows from i/€ = F € C°(0, T, H ) that i/ G C°(0, T, # 3 ) . Similarly, it can be shown that /x<EC°(0,T,# 3 ) and sup \HT)\\H2

< WnoWw+t

0
sup ( H H I / M I I ^ + H x W | | ^ ) 0
Combination of the above two estimates and (3.69) yields SUP (||Mr)||tf3 + | K T ) | | H 3 + | | x ( T ) | | H 2 ) 0
< C ( l + t X I M / r s + \M\H* + IIXolUO = C(l +

t)N2(0),

where C > 0 is a constant, independent of t.

0 < t < To

(3.72)

184 Since the smooth solution (F,/Z, x) of (3.3), (3.70) is unique, and ( ^ , / ^ , x ) > defined from the smooth solution (v,fi,x) of (3.5), (3.68), satisfy (3.3), (3.70), then the smooth solution of (3.5), (3.68) is unique. The local existence and uniqueness is finished. We now turn to the proof of Th 3.1. In view of (3.72), we can choose To such that N2 (i/, /i, x, *) < 2CN2 (0)

0 < t < T0

(3.73)

We take = N2(0) = r& = min { J ^ , ^ } r,\

N\v,n,x,V

where r)i and K > 1 are the constants cited in Lemma 3.2. Thus, (3.73) implies that N2(iy,^x,t)<\f]2
0
(3.74)

Next, we take #o = | ^ + — v~\ — Si, (Si is the constant cited in lemma 3.2), it follows from lemma 3.2 N\v^,x,t)
= \ril

0
(3.75)

particularly, | K T o ) , M r 0 ) | | „ 3 + ||x(To)||„3 < \

n

l

(3.76)

By using the local existence theory, there exists a constant t > 0, the solution of (3.5), (3.68) exists on [To, T 0 + i\ and sup

(||(^/x)(i)ll2H3 + | | x ( * ) l l ^ ) < » ? ? .

(3.77)

T0
Therefore it follows by using lemma 3.2 that (Mv,fi)(t)\\h+\\x(t)\\2H>)
sup

(3.78) 4

T0
In particular, \\(v,n)(T0 + t)\\m + \\x(To + m*

< \nl

(3.79)

Repeat the above procedure, we obtain the global existence of smooth solutions. We establish the delay result at last. Lemma 3.2 and the above analysis show that r+oo

/ Jo

(\\(vt,K)(t)\\h + \\xmh)d* <+<*>-

Thus, it follows from (3.5) that +oo

/

r+oo I J

dt < +oo. 2

lh(i)ll d* < +oo, jo

\jt\\Mt)\\

185 This implies H^WH 2 - + 0

as

*-»+oo.

II^WII 2 -^ 0

as

*->+°o-

Similarly, it can be shown that

Thus, H^ll/fi -> 0 as t ->> +00. The same result holds for ||/i^||/fi and Hxll//1This finishes the proof. References 1. R. E. Caflisch and G. C. Papanicolaou, The fluid dynamic limit of a nonlinear model Boltzmann equation, Comm. Pure Appl. Math. 32(1979), 589-616. 2. C. Cercignani, The Boltzmann equation and its applications, Springer-Verlag, New York, 1988. 3. T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynam­ ics, Longman Sci. & Tech., 1989. 4. G.-Q. Chen, C D . Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47(1994), 787-830. 5. C. Faciu, Numerical aspects in modeling phase transition by rate-type constitutive equation, Int. J. Engng Sci. 29, 1103-1119 (199). 6. H. Hattori, The Riemann problem for a Van der Waals fluid with entropy rate admissibility criterion-nonisothermal case, J. Diff. Eqs. 65(1986), 158-174. 7. L. Hsiao and P. de Mottoni, Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type, Trans. American Math. Soc. Vol.322, No. 1. (1990) 121-158. 8. L. Hsiao, Uniqueness of admissible solutions of the Riemann problem for a system of conservation laws of mixed type, J. Diff. Eqs., Vol. 86, No.2, (1990), 197-223. 9. K. Inoue and T. Nishida, On the Broad well model of the Boltzmann equation for a simple discrete velocity gas, Appl. Math. Optimization 3(1976), 27-49. 10. R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73(1980), 125-158. 11. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48(1995), 235-277. 12. S. Kawashima and A. Matsumura, Asymptotic stability to traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101(1985), 97-127. 13. B. L. Keyfitz and M. Shearer, Nonlinear evolution equations that change type, Springer-Verlag, 1990. 14. T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Memoirs of Amer. Math. Soc. 328(1986).

186 15. T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108(1987), 153-175. 16. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, New York, 1984. 17. R. Pego, Phase transitions: stability and admissibility in one-dimensional nonlinear viscoelasticity, Inst. Math. Appl. Univ. Minnesota, No. 180, (1985). 18. T. Platkowski and R. Illner, Discrete velocity models of the Boltzmann equation: a survey of the mathematical aspects of the theory, SIAM Review 30(1988), 213-255. 19. E. B. Pitman and Y. G. Ni, Visco-elastic relaxation with a Van der Waals type stress, Int. J. Engng. Sci. Vol. 32, No. 2, (1994), 327-338. 20. M. Shearer, Admissibility criteria for shock wave solution of a system of conservation laws of mixed type, Proc. Soc. Edinburgh Sect. A 93(1983), 233-244. 21. M. Slemrod, Admissibility criteria for propagating phase boundaries in a Van der Waals fluid, Arch. Rational Mech. Anal. 81(1983), 301-315. 22. M. Slemrod and A. E. Tzavaras, Self-Similar fluid dynamic limits for the Broadwell system, Arch. Rational Mech. Anal. 122(1993), 353-392. 23. I. Suliciu,On modeling phase transitions by means of rate-type constitutive equations, shock wave structure, Int. J. Engng Sci. 28(1990), 827-841. 24. A. Szepessy and Z. Xin, Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal. 122(1993), 53-103. 25. Z. Xin, The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks, Comm. Pure Appl. Math. 44(1991), 679-713.

187

Existence and Uniqueness of Discontinuous Solutions for A Class of Nonstrictly Hyperbolic Systems FEIMIN HUANG Institute of Appl. Math., Academia Sinica, Beijing, P. R. China, 100080 E-mail: [email protected]

Dedicated to Professor Xiaqi Ding for His 70th Birthday Abstract. In this paper, a nonstrictly hyperbolic system ut + f(u)x = 0, vt + (f'(u)v)x = 0 is considered. Its main feature is that v in solutions is not a usual function, but a measure. Through choosing a special smooth mollifier of u and using the method of Ding-Wang [2], we prove the global existence and uniqueness of solutions to the Cauchy problem under the Oleinik entropy condition. Besides, we also consider another system ut 4f(u)x = 0, vt — (f'{u)v)x = 0 and find that its solution does not contain the delta function. Moreover, its solution is not unique under the Oleinik entropy condition.

1

Introduction

In this paper, we consider the system ut + f(u)x = 0, (i.i) vt + (f'(u)v)x

= 0,

{u,v)\t=o = (u0(x),vo(x)), 2

(1.2)

where f(u) € C (R) and f"(u) > a > 0. System (1.1) is nonstrictly hyperbolic because it has only one characteristic speed A(u,v) = f'(u). In the case of f(u) = \u2, system (1.1) is a simplified model of fluid mechanics with a constant pressure, which is applied in plasma physics (see [3]). For nonlinear hyperbolic conservation laws, the main feature is that solutions generally are discontinuous. One usually tries to find L°° weak solutions in the sense of distributions. A considerable progress has been made in this field so far (cf. J. Glimm [6], P. Lax [9]). In recent years, however, some authors (see D.

188 Korchinski [8], K. Joseph [7], T. Zhang et al [13]) discovered that solutions do not either exist or unique for certain conservation laws even for some Riemann data in the usual L°° space. By numerical or vanishing viscosity methods, they got new weak solutions which are not usual functions. A new type of nonlinear waves, the delta wave, arises for such systems. For system (1.1), it is easy to show that there is no L°° solution to some Riemann initial data in the sense of distributions. When f(u) = \u2, K. Joseph [7] proved that the limit of viscous solutions (u£,v£) of the Riemann problem for (1.1) may contain delta measures as e —> 0 in 1993. In fact, the function v in the solutions (see [1,11]) contains a delta function when u is a shock wave. It contrasts with the standard theory of nonlinear hyperbolic system. However, it might be well-posed if we extend the concept of weak solutions. In order to overcome the product problem of a generalized function with a discontinuous function generated by delta wave, Ding (see [1]) first applied generalized poten­ tial and Lebesgue-Stieltjes integrals to study this singular wave and introduced a new definition of generalized solutions. Using the idea, X. Ding and Z. Wang [2] proved the existence and uniqueness of solutions of the Cauchy problem (1.1)-(1.2) for f{u) = | u 2 under the Oleinik entropy condition. Moreover, the solution which may involve delta wave is the limit of viscous perturbations as the viscosity vanishes. In addition, F. Huang, C. Li, and Z. Wang [5] also proved the similar results of [2] for the following nonstrictly hyperbolic system |

ut + (\v?)x

= 0, 91.3)

[ vt + (Xuv)x - 0, where the constant A > 0. It has two eigenvalues K\(u,v) = u^K2{u^v) = Aw, which is different from (1.1). And some interesting properties of delta wave were obtained. The main aim of the present paper is to generalize the results of [2] to a more general system (1.1). The main difficulty is the first equation of (1.1) is not the Hopf equation. It is hard to get good properties of the viscous solutions ue like Burgers' equation when we apply the vanishing viscosity method used in [2]. But, we can get an explicit entropy weak solution due to Lax [10]. Through choosing a special smooth mollifier depending on the structure of the weak solution u, we can obtain a sequence of approximate smooth solutions u£ satisfying formula (3.9), which is critical to our proof of the existence. Moreover, the approximate solutions u£ converge to u everywhere in the upper half plane. This is important in the proof related to the uniqueness. Here let us briefly recall the definition of generalized solutions by the LebesgueStieltjes integral due to X. Ding [1]. First, we introduce the generalized potential r(x,t)

u(x,t)

—