Estimates for solutions of nonstationary Navier-Stokes equations

(17) o

seems to be cumbersome and we do not present it.

The similar condition for

The author expresses his thanks to V~ So Buslaev for the discussions of the results. LITERATURE

CITED

J. von Neumann and E. Po Wigner, "Uber merkwiirdige diskrete Eigenwerte," Physik. Z., 50, 465-467 (1929). 2.

T~ Kato, "Growth properties of solutions of the reduced wave equation with a variable coefficient, ~ Commo Pure Appl. Math., 12, No. 3, 403-425 (1959).

3.

A. Poincar~, New Methods of Celestial Mechanics, Selected Works [Russian translation], Vol. 1, Nauka (1971), Chap~ XVIIo

ESTIMATES NAVIER

-

FOR

STOKES

SOLUTIONS

OF NONSTATIONARY

EQUATIONS

V. A. S o l o n n i k o v

1. I N T R O D U C T I O N In the present paper we consider the problem of finding the vector and the function

~<~,b=(o,(~,b), ~,r

p(x,t) , satisfying in the cylindrical domain (~T=1)x [0.T] (zce ~ C ~ ~, ~

~(~,L))

[0,Y] ) the

system of equations $

~-~t-vav+o~(~,tW+~a&~,t,)~. * vp=~-(x,b,

v. v'--o

(i.i)

and the initial and boundary conditions

~It=o =~o(~,

~[~jO,

(1.2)

where S is the boundary of ~Z, 0~(x,t) and 0~,(~,~) are certain given matrices, while V = ( ~ ' ~ ' ~ , ) . In addition, one considers the nonlinear problem

U~

k=~

K

k=l

k

(1.3)

The solvability of the Dirichlet problem for the three-dimensional linear and nonlinear nonstationary Navier-Stokes system has been established by Hopf [I], Kiselev and Ladyzhenskaya [9.-4]. They have Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta ira. V. Ao Steklova AN SSSR, Vol. 38, pp~ 153-231, 1973. 9 19 75 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording o? otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

467

d e t e r m i n e d the g e n e r a l i z e d solutions of these p r o b l e m s in different functional c l a s s e s and have proved that the linear p r o b l e m is always uniquely solvable. However, a unique solvability of the nonlinear p r o b l e m for any ~ and ft. is obtained only for sufficiently s m a l l T , depending on s o m e n o r m s of ~ and ~. and increasing indefinitely when these n o r m s tend to z e r o . Similar t h e o r e m s but in other function c l a s s e s have been proved in [5-8]. The differential p r o p e r t i e s of the solutions of the linear and nonlinear p r o b l e m s are investigated in [9-14]. In p a r t i c u l a r , in [12, 13] one has obtained c o e r c i v e e s t i m a t e s of the solution of the linear p r o b l e m in the n o r m s of /~(Qr) and in the H(ilder n o r m s of ~'~T~

(see definition in Seco 2) and with their aid one has investigated the differential

p r o p e r t i e s of the g e n e r a l i z e d solutions of the nonlinear p r o b l e m obtained by Kiselev and Ladyzhenskaya. It should be mentioned that the c o e r c i v e e s t i m a t e s in L~((~T) have been c o m m u n i c a t e d e a r l i e r by Yndovich [15), however, as f a r as we know, their proofs have not been published. In the p r e s e n t p a p e r we c a r r y out f u r t h e r investigations of p r o b l e m (1.l), (1.2) with the aid of c o e r c i v e e s t i m a t e s for which we give a complete proof, much s h o r t e r than in [12, 13l. It is c a r r i e d out by Schauder's well-known method: f i r s t one c o n s i d e r s p r o b l e m (1.1), (1.2) in the half s p a c e for O.=a~=O

,

whose solution is written out explicitly in t e r m s of the G r e e n m a t r i x , and then, with the aid

of the partition of unity and the local unbending of the boundary, one d e r i v e s e s t i m a t e s for the solution of p r o b l e m (1.2) in an a r b i t r a r y finite or infinite domain ~'1 with a smooth compact boundary S and for an a r b i t r a r y "]". At the r e a l i z a t i o n of this method, well-known f r o m the theory of elliptic and parabolic boundary-value p r o b l e m s , it is n e c e s s a r y , by virtue of the c h a r a c t e r i s t i c s of the nonstationc r y N a v i e r - S t o k e s s y s t e m , to consider not only the o r d e r but also the s t r u c t u r e of the t e r m s which occur at the unbending of the boundary. In o r d e r to prove the solvability of p r o b l e m (1~

(1.2) we

apply the method used in 1161 for the investigation of parabolic b o u n d a r y - v a l u e p r o b l e m s (the c o n s t r u c tion of a regulator).

Finally, f r o m the c o e r c i v e e s t i m a t e s we d e r i v e inequalities connecting different

n o r m s of the solution of p r o b l e m (1~

(1.2) with ~=0

for an a r b i t r a r y

t>0

and for

t=O

~

Some of

these inequalities w e r e established in F17) and used for the proof of the stability of the solutions of the s t a t i o n a r y N a v i e r - S t o k e s s y s t e m in /~(~). Apparently, the r e s u l t s of this p a p e r allow us to c a r r y out s i m i l a r investigations regarding the stability in HSlder n o r m s . In Sees. 2-5 we c a r r y out the a b o v e - d e s c r i b e d investigation of p r o b l e m (1.1), (1.2) in the spaces ~t~(Qr~ ---W~'(0~ , while in Sees. 6-9 in the spaces

~'~((~r) , whose e l e m e n t s have H a d e r continuous

d e r i v a t i v e s occurring in s y s t e m (101). In Sec. 10 we consider nonlinear p r o b l e m (1.3). For this we prove a solvability t h e o r e m of the s a m e c h a r a c t e r as in [3 I, but in the s p a c e s ~ ( Q ~

and ~ " ( ( ~ .

In

addition, we give e s t i m a t e s in different n o r m s for the solutions of p r o b l e m (1.3) when ~=0 in t e r m s of the n o r m s in/,~(~) , ~ 5 2.

and the norm in ~'(s

NOTATIONS

AND

of the vector 1~o~) . AUXILIARY

STATEMENTS

E v e r y w h e r e in the sequel ~ is a bounded or unbounded domain of the t h r e e - d i m e n s i o n a l space ~* with a compact boundary

.~ of c l a s s

whose points a r e denoied by (x,L~; ~

G~ , ( ~ = ~ x [0.T] is a cylindrical domain of the space ~N ,

~ % ~ , re--[0,T], ~T = ~• [0.T] .

We introduce the following Banach s p a c e s . By /..q(~l) we denote the space of functions whose q - t h power ~ t

468

is Lebes~ue integrable and where the norm is defined by

(2 o~) For

~ = ~ we set |t~|.,a:e~ga~h~ I~z)l. By /_~,~(.Q~ for ~ ,

v~

Similarly one defines

Z~(Q~, L~(S) and so on.

we denote the space of functions with the norm T

|~,~,, aT:

,I.

U,(.

,~

(2.2)

9

0

By ~i(fl~ we denote the space of functions having in ~ generalized derivatives up to and includLag the power ~ and belonging to L#(fl) together with all the derivatives. The norm in ~(f~) is given by the f o r m u l a p

where ~:C2,,,3,,jaD, fl~>O Ifll=jlJ,,+3,+N~ ~ f u : By ~ ( { ~

=~r

3'~tt

we denote the space of functions having ~-th power summable generalized

derivatives of the f i r s t and second o r d e r with r e s p e c t to m and of the f i r s t o r d e r with r e s p e c t to 1~;

the norm in ~(,Q~ is

~or

the leadi~ te~ms of ~ e norms in W~Cn~ ~ d

~04~ ~ we introduce special notations

2-

We define now the following spaces of HSlder continuous functions. By

H~)

we mean the

space of bounded and HSlder continuous functions with exponent ~ , defined in ~ (everywhere in the sequel ~ ~ (Off) } . This is a Banach space with the norm

where

<~>a=~p

I ~b

i

I~-.,v,,I It~(~)-uA~c)I.

(2.4)

By ~(QT ) we mean the space of functions defined in QT and H51der continuous with exponent ~, with respect to the variable ~ and with the exponent ~ with respect to ~ . The norm in ~ (

is

(2.5)

t

~,~,t

l~-~'i~

--

~.t.t'

It-t'l "~

(2.6)

469

THEOREM

2.1. Let

tl,<e:.'~.l,(.i~ ~ . If p,~.>q/>t

are such that we have

jo : 1 - 2ul,-z - & p +•c,,,v or

:~--N ~+-c~o or v =~ we assume a s t r i c t inequality in these relations), then

(for p = ~

.P

p-i

(2.7)

or ~~- il__p,~,~ [~] T~o(~ .~ [~1~0,~Z-t ~#,~T), . r e s p e c t i v e l y , with any 5 ~-C0.5.) ; 5. depends on ~ .

LtHQ ~0(.5 and under the condition

(2.8)

Moreover, if 0-~ ~ - ~ < t , then

~.~llq,Q~t~

Ilullr

(2.n)

O
_pr__ooof_- As is known, e v e r y function

ti ct il4,n.~ .~c il ~ fIr

IDc~r

can be extended to the entire space ~ such that

I[_u.i14.~, ~C 11u, II ~,tQ.;)

T h e r e f o r e , if one obtains (2o7)-(2.10) for smooth finite functions defined in ~", then the theorem will be proved. Moreover, it is sufficient to establish both inequalities for 5 =1 : replacing ~t~c,b ~,~)

by

we obtain (2.7)-(2.10)for any ~>0. We make use of the r e p r e s e n t a t i o n

where

(~vtf ~~p r J ) for t>o, F(~.b= is the fundamental solution of the heat conduction equation

(2.11) for

t
3F ~ - ~ h [ ~ = o(~.L) . We shall consider h e r e

that v=1 . Let ~(~) be an infinitely differentiable function of the positive variable ~, satisfying the conditions

0~2(~{,

~(~)=0 when ~>~I and ~(~)=~ when ~ .

t

470

We have

~here

:s ar(~.t,a++ ~ao,,),~r +r-c,,t)(~d)qo+,)_~,r ) 3

Let

++-:+-+++ 4

We h a v e t .o:

t-I

and s i n c e

3 _~=_t+?

+ ~+- - ~4 , f r o m h e r e we o b t a i n the e s t i m a t e (207) f o r

~=~ . E s t i m a t e (2.9) f o l l o w s

from

.,( a~ IFC~~t,t+h-+)-r(+-+j,t-+)l+t+),+-~--A+l+,m+ + +,+L,

p

+_.

+"tq,m'(,I i+l It+(+'U't+P+'+)-rCm-~'t-+)la~)--cCl - l +I+

+ +-+

S i m i l a r l y one e s t a b l i s h e s i n e q u a l i t i e s (2.8), (2.10).

The t h e o r e m i s p r o v e d .

T h e a b o v e - i n t r o d u c e d s p a c e s c a n b e d e f i n e d a l s o f o r the v e c t o r s

t~=Ct~,,u,,,t~+) with o b v i o u s

m o d i f i c a t i o n s in t h e d e f i n i t i o n of the n o r m (e.g., D ~ l ~ n = Z . n ~ t ~ a ,, , ). The n o t a t i o n f o r the s p a c e s r e m a i n s the s a m e a s b e f o r e . It is known [18], t h a t e v e r y s m o o t h f i n i t e v e c t o r of two o r t h o g o n a l t e r m s in

~(~) c a n be r e p r e s e n t e d in the f o r m of the s u m

L2C~)

~(~ ~+(~ +~(~), =

where

~(~) is a v e c t o r s a t i s f y i n g the c o n d i t i o n s

(2.12)

?.~ =0 , ~ .~(~)lS:0. T h e n ~ i s d e t e r m i n e d f r o m the

Neumann problem

where

n(~)

is the e x t e r i o r n o r m a l v e c t o r to S ~ In t h e c a s e of an unbounded ~ one m u s t a l s o r e q u i r e

that

~

O.

(2.14) 471

THEOREM 2.2. F o r any smooth finite (for unbounded l] ) v e c t o r ~ C~,) we have the inequality

II v~ II,1,,~ + ~ I,~,~ ~c I1~:~,~,~. In the c a s e of a bounded ~ ,

(2.15)

(2.15) follows f r o m the e s t i m a t e s of the solution of ~)roblem (2.13)

obtained in [19]. H e r e we shall give a different proof which applies also for an unbounded ~ . We introduce the s p a c e W~(5) for 0
o This is the set of functions, defined on ~ , with a

finite n o r m

(2.16)

R is known [20] that

<<~))~.
{2.17)

is an a r b i t r a r y bounded subdomain of fi such that ~ ~Cfix~') =0 . LEMMA 2 ~ ~ F o r a solution vanishing at infinity (in the c a s e of an unbounded ~ ) of the p r o b l e m

we have the e s t i m a t e s

for

(r ' ~l~,~, ~c,{:~'~,~,

'

(2.1o)

J~>o.

Proof. R is well-known that (2.20)

where ~C~)= Cq~l~l)"~ is the fundamental solution of the Laplace equation and

~, is the solution of the

integral equation (2.21)

w h e r e the sign % " c o r r e s p o n d s to the i n t e r i o r and " - " to the e x t e r i o r of the domain S with respect to ~ . The function

s

472

satisfies an equation s i m i l a r to (2o21) (2.21') s

and, expressing it in t e r m s of the resolvent, it is easy to obtain the estimate IIc,i~s_~c I16olI,, s.

By virtue of Stokes's formula, we have (2.22) 5

and therefore

We proceed to the estimation of the f~nction ~ :'-~(~'+~"), where

s

5

~,=~ s Since the norm in L~(~) of the second derivatives of the simple layer potential is estimated in t e r m s of the norm in W~~(5) of the density (this is equivalent to the known coercive estimate for the Neumann problem), we have

Iivq311 ~_~cim-c~ll ,-r,.>~ q,, W~ (s) },~

(2.23)

ll(~llq,,d~C lt&-altr s ~C <<0,~s .

(2.24)

Moreover,

Now we estimate the potential

Since for an unbounded ~ it follows from (2o21) that

r -- ~.0.) we have V~-L~,~.,

and in every case

473

which together with (2.23), (2.24) proves (2.19). We proceed to the proof of Theorem 2.2. We represent the solution of the problem (2.13) in the form q =q, +q~ , where

~ =v-A(x), A(=,)=I$(~,-~)#(t~)d~, and ~. is the solution of the problem

A%=O, ~, s=~.~-v~.~,, subject in the case of an unbounded ~

to the condition (2~

By the Calderon-Zygmund theorem

on the boundedness in / ~ of the singular integrals [21], we have

and since ~-Vq~=-~ ~&~, by virtue of L e m m a 2ol we have

and the theorem is proved. It follows from Theorem 2.1 that the decomposition (2.12) holds for any ~.q(f~), where V~ and satisfy for any V~)~i~,(fi) the integral identities

{v~, vr =(~, v~), (q, ~r

(2.25)

where (rE ~) = ~t~"t~~m = ~.~ lat$~(m)~(m)~m o We denote by C-~(.O.)the set of all vectors vq~.t~(~), and by

~(fl)

the set of all vectors

satisfying identity (2.25). Let ~C(~ and ~,CQr) be the subspaces of ~_~((~r) , consisting of the vectors which for almost all t belong to C~(~) and ~(~) . Formula (2.12) means that

Let ID. and P, be the projectors onto Gr and 3~ : 7 q = ~ ,

~:Pj~.

Similar decompositions take place when ~= ~ or ~ = ~+ - ~ :~>0], and estimate (2.15) can be established in a simpler manner with the aid of the explicit formulas

474

(2.27) where )i~,~) = ~9C~-~) +~ (~-@

is Green's function for the Neumann problem in ~ ,

LEMMA 2.2. Assume that in the semisphere

~=-[~r

while

there is given a vector

~(~)~Lq(~) which satisfies the identity (2.14) for any smooth function vanishing on the surface i351--.~ (i~

in the generalized sense

onto the semisphere

V.~ =0, ~1~=~=0). Then, there exists an extension

g+ such that ~ (~)r

q'Cm) of this ~ector

and

(2.28

II

Proof. By a similarity transformation the lemma reduces to its own special case corresponding to ~=~ o Let us prove it for a smooth ~(~) , satisfying the condition 7.~ =0, ~slz,=o0o We extend the components ~,, ~, in an even manner and ~ in an odd manner with respect to z~ onto the suhere %=~z:~r

~ and we solve the Neumann problem

in the domain ~ \ 0 , .

We verify that the vector ~(zr ~;C~),~v : ~ \ ~ ,

gives the required extension.

Representing ~ in the form

where ~/

V' is the solution of the Neumann problem in

--N(~)Iq~-9~V~(~'~)'q(~?~q]l

estimate (2~

~ \ ~ , with the boundary condition

=-~(~5)'~ ~x~(~-tJ)qC~)~ql

, and making use of L e m m a 2.1, we obtain

Then, from the identity

there follows that ~ ~3~(.~) ~ Finally, since I~ is an even function of m~, it follows that are also even while C]~ is an odd function. One can conclude from here that

t-

~

~ cJr

5

q[ and ~ and the

!emma is proved. For the investigation of problem (I.I), (1.2) we shall make use of different estimates of the solution of the Dirichlet problem for the stationary Stokes system

In the case of an infinite ~ we shall consider ~ and ~ finite and we subject the solution to the condition t~ ~

0.

(2.30)

It is known that when the n e c e s s a r y condition cl~]~=0 is satisfied (inthecase of a bounded ~ ), the problem under consideration has a unique solution which can be written in the form 475

~=~,t4, p=~+s,

(2.31)

where

~

s

(2032)

a r e the solutions of the s y s t e m -W~ +w =t , Y w=J~ while u~and S a r e the solutions of the p r o b l e m -?ht~§

vW=0,

~zls=-~15, tg~.~o>0,

(2.33)

whose existence is established in the theory of the hydrodynamic potentials of Odqvist [4, 22]. In (2.32) by ]J" we have denoted the fundamental m a t r i x of the Stokes s y s t e m with the e l e m e n t s

=

+

satisfying the conditions ~U ~ > _ 0

THEOREM 2.3. F o r finite ~ and j~ , the solution of p r o b l e m (2.29), (2.30) s a t i s f i e s the inequality (2.35)

where the constant does not depend on Proof. F o r t~ and ~, e s t i m a t e (2.35) is obtained at once f r o m r e p r e s e n t a t i o n s (2.32). M o r e o v e r , making use of the known e s t i m a t e s f o r integrals with weak p o l a r k e r n e l s , one can e s t a b l i s h that

If ~<~ , then the constant in this inequality does not depend on ~ ,

~p~j~ since for the

e s t i m a t i o n of the volume i n t e g r a l s in (2.32) one can make use of Sobolev's inequality [23]; e.g., for the I

f i r s t t e r m ~t of the right-hand side of (2.32) we have

As e s t a b l i s h e d in [4], the solution of the p r o b l e m (2.33) is subjected to the inequality

The t h e o r e m is proved.

476

THEOREM 2.4. Assume that in (2.29) we have

~=0 and

pc~)=v~(z)%c~), and also ~ O = ~ j ~ o c ~

(2.36)

, Then (2.37)

where )>0

, 0~>1 in the case of a bounded

Proof. Let

V

I~ and ~>5 in the case of an unbounded

P be the solution of the problem (2.29) with j3=O and with a r b i t r a r y finite ~ .

We have

f tr'l&~'lt~'vP&=-I ~Pg:~"-I ~'p&+ I ~ vP&-I R.,'.,PJS, fl

fl

~1

fl

~I

8

and, consequently,

where

~ =~4_ ~

If we take

Cng]~,,~ by virtue of Theorem

< 9 4 4 , then all the norms 2.3 and we obtain (2~176

of P in the right-hand side are bounded by

Since the vector ~ must be arbitrary, in the

case of an unbounded domain 1~ one has to require that ~'4~ , i.e., ~>5 9 Theorems 2.3 and 2.4 hold also for ~=~§ and in this case they can be obtained with the aid of the explicit representation of the solution of problem (2.29) in t e r m s of Green's matrix [24], which for ~=0 has the f o r m

g+

. GREEN'S

R,

MATRIX

FOR

BOUNDARY-VALUE

A

(2.38)

SEMISPATIAL

PROBLEM

We consider in the domain ~ ( ~ > 0 , t > 0 ) of the space ~ the problem

4

, V.g=O, ~/~=o=0,

~/1~=o=0

(3.1)

with a smooth finite (or well decreasing for ~ I - * ~ ) right-hand side ~<m,D , satisfying the conditions V'~ = 0 , (in the notations of See. 2, ~ d ~ (~) with any

~3 %'~ =0

(3.21

~>~ ). Our aim is to construct Green's matrix for

problem (3.1). F i r s t we find the solution of the homogeneous system with nonzero condition on the boundary ~3 = 0

477

-•-VA• + Yp :0,

ut:o:O '

tt.

Y t~:0,

:acz,,x,,t,),

After applying a F o u r i e r t r a n s f o r m with r e s p e c t to

(3.3)

~,~T~O

~c,,x~ and a Laplace t r a n s f o r m with r e s p e c t to

with the aid of the f o r m u l a ao

(3.4) we convert (3,3) into a boundary-value problem for the s y s t e m of o r d i n a r y differential equations

+ Lp :o,

(3.5)

1~0~:0

~'~+~

We shall seek %t, in the f o r m of the sum w=~+9~, where q

(3.6)

is a harmonic function in ~§ and U~ is a solenoidal vector satisfying the heat-conduction

equation. Since t r a n s f o r m (3.4) of e v e r y function satisfying the heat-conduction equation and vanishing for ~ ---~r

is equal to 0,(~,S)e" ~

, while that of a harmonic function is equal to

0~C~,s)dm~ , it

follows that (3.6) is equivalent to

where

The functions

~ , ~, ~ are easily determined f r o m the boundary conditions. We have

Solving this s y s t e m , we obtain the f o r m u l a established in [12]

478

(3.7) o

~

We make use of (3~

:-

,

,

=-v

(J~I~l)~

'~'~'.

{3.~}

and (3,8) to solve problem (3.1), We shall seek the solution in the form O' = 'u,

+

~,

(3.9)

where

If

''

FC~,t) is the function (2.11)o Obviously,

while

where

t

~,~=0. Vector

~(~,~) m u s t be the solution of p r o b l e m (3.3) in which 0~ is d e t e r m i n e d by f o r m u l a (3.10).

Now we compute ~ , we i n s e r t ~ into (3.7), (3.8), andwe p e r f o r m the i n v e r s e F o u r i e r - L a u l a c e transform.

In this connection we shall make use s y s t e m a t i c a l l y of the convolution t h e o r e m and of the

f a c t that t r a n s f o r m (3.4) e s t a b l i s h e s a c o r r e s p o n d e n c e between the following functions:

~(t).~i~cP =~

e

'I 8~c3

Therefore,

t o

~§ 479

~

t

":

u,c'j

aJ.

8

O~C

"

r

o

O

a J

J

]

Y'- , /

(3.11)

t

p:,

F o r the solution of problem (3.1) we obtain the e x p r e s s i o n s t

%

t

(3.12)

I v &c=-~)~,~Ii,Ircy.', t-'~)I,c~,'~)i,-Ul, p(~;,t) = ~

These formulas hold under conditions (3.2). In the general case we make use of the d e c o m position ~ = ~ ~p . Since ~

+~

and we leave in the right-hand side the vector ~

while adjoining

~t:~

to

satisfies the conditions (3o2), we have

(3.13} The solution of the Cauchy problem

-vA~-vp=~., ,7.~':o, c~,b eFIT=P,,x [O,T] (3.14)

b0=0, is expressed by similarf o r m u l a s : for ~_JpC~T)

t

p=o; if, however,

(3.15)

~eLpCi]T), then r

(3.15,)

THEOREM 3.1. For any ~L$~,~(~>4, ~C<*[0,'t]), problem (3.1) has a uniq.e solution ~E. ~ C ~ ) , vp~Z~(~,~ for which we have the inequ~ity

(3.16)

460

T

the constant

being independent of 6 ~

Proof. If ~ is a smooth v e c t o r f r o m functions (3~

(3.12).

~r

, then the solution of p r o b l e m (3.1) c o n s i s t s of

Let us p r o v e f o r them inequality (3.16)o F o r the proof we shall make use

of the C a l d e r o n - Z y g m u n d t h e o r e m on the boundedness of singular i n t e g r a l s in

/-p (in general, with

discontinuous c h a r a c t e r i s t i c s ) and of the well-known e s t i m a t e

f o r the t h e r m a l volume potential

It is a consequence of the M a r c i n k i e w i c z - M i k h l i n t h e o r e m on the m u l t i p l i e r s in F o u r i e r i n t e g r a l s

[25]. With the aid of (3o17), we obtain for the f i r s t t e r m

in (3.12)

W e consider the second term ,

t

o~

-

~"

It is e a s y to v e r i f y that any d e r i v a t i v e ~

-

-

-

can be e x p r e s s e d in the f o r m of a linear combination of

the singular i n t e g r a l s

where

i

, ~,

p<3 , and of the function

3:~ ~e n,(~,b)

T h e r e f o r e , as a consequence of the C a l d e r o n -

Zygmund t h e o r e m , we have

t~,0~Jr S i m i l a r l y one e s t i m a t e s VP : ~ - ~ + v ~ .

From

~=~

'

~. T~

9 T

; however, it is e a s i e r to e s t i m a t e the vector Vp f r o m the equation

Thus,

here, from (3.13),mldfrom the boundedness in

obtain inequality (3.16) for an arbitrary

~ Lr

L,(~

of the operators

~ and Pj we

.

The uniqueness of the solution of problem (3.1) is established in the following manner: t ~ ~,t~r

Vp~-L~(~ is a solution of the p r o b l e m with ~ =0, then for any solenoidal

if

~)~ ~,(gT') ,

481

satisfying the conditions

q) t~0. q)l =0, we have the relation T

from which it follows that if=0, since the problem

,,.,---o, %.,---o, %--o is solvable for any ffcLq:(g~ . The theorem is proved. In the following theorem we establish estimates for the lowest derivatives of the solution. THEOREM 3.2. If ~_J~(~_~, then d ~/ F

nt~ ~,,~ ~.eT ~l g~,~, (3.19)

Proof o For potential (3o18) we have 3

and since

and p in (3.11), (3.12) are expressed in t e r m s of the thermal volume potential and dif-

ferent singular integral operators of it, from (3.20) there follow estimates (3.19). Estimates, similar to (3.16) and (3.19), hold also for the solution of Cauchy problem (3.14). 4. T H E

PROBLEM

(1.1),

(1.2)

W e start with the investigation of the problem

~ -vAo'+vp=~, v.~--O,

(~L)cQ, (4.1)

v.~:o=O ' vs=O THEOREM 4.1. Let $ ~ 6 : . Ve~((]l) ,

vp6/s

For any ~_Lr

, ? > t , problem (4.1) has a unique solution

, for which we have the inequality [gl#,a +llVplI%QT~lllI%Q+C,llfl}l,fiT,

(4.2)

the constants C, and O2 being independent of 7-. Proof. As we have mentioned,without loss of generality we may assume that ~Jr

9 In

this case problem (4.1) can be considered as the problem of the inversion of the bounded operator

which is defined on the space ~ t t ~ 0 , ~1~0, and acts from

482

~((~

of all solenoidal vectors from

~CQ,~ into 3~r

~ r (Q~, satisfying the conditions

- Let us prove that i ~ r a ~ e

~r

fUls out all of

](Q~ wheneverT is sufficiently small. To this end we construct a bounded

, acting from

into I~((]~') and such that (4.4) where

II~,II<{ for sufficiently small T . Then ~(~)_~(~)=Jr

.

Let ~ , k=0,~ . . . . J~-,bea covering of ~ , where ~ for k>~ is adomain, ~ is diffeomorphic f to the semisphere [~L=[ze-Z+:lzl=~)~],0
form a covering of ~ and we assume that in the local Cartesian system of coordinates

[~] with center at the point to ~ at

The varieties

~'v. ~ , and whose ~ axis is directed along the interior normal

~(~)

~' , the variety ~ is given by the equation

(4.5) where the functions F have norms in is the circle

~ +~2

g~(~.~'),boundedin their totality for all k and ~,.<~, ; Z__L

. Obviously, k k

k=

k~

(4.6)

We also assume that the inverse transformation, effecting the diffeomorphism of ~kand

~k

, has

the form

(4.7) so that if

k.~]~ (m.~) ( Vk is an orthogonal matrix), then

Finally we can assume that the multiplicity of the covering of ~ by the domains ~k does not depend on ~, and that there exists a partition of unity subordinate to this covering (4.8) =

such that ~ # p S ~ f i k , ~ C C ~ ) ,

-

~

,-tyl

l~=~l~C~

, lJ~i~<~,.

We proceed to the construction of the operator 2~. For each function given in ~kand for each Lr(~) given in ~ we set

§ On the set of the vectors defined in ~ ,

we define the operator


(4.9)

~. ~]~=0 , then

~'~lz,.0=0 . O n the set of these vectors 483

we define the operator ~ effecting their continuation on ~ , for each

according to L e m m a 2~

In addition,

~J~(l~) we set

We introduce now on J~CQ~ the operators

where OJ is the restriction of ~o. ~ = ~ < 0 , ~ ,

~ on

~.~0,~=0. ,wh.e

r

, Q, andS.are

given by formulas (3012), (3.15). Finally, we set

~--~.p~j, ~=~.~+~,f, where ~ = t a

and ~ = s

is the solution of the problem

-w~,~s:o, ~,.~---v-&J~. %:o. Obviously, ~

1~CQt).

Let us prove (4.4). We form the expression

Since the vector or =~k5QJ~Vk%5~

and the function tt =7~, V, ~ V ,

I~-VatI,+VO~for ~ : ~ , ~=~.

%.6,~ satisfy

in Oz the

system

where Yk: V+~ V~V Ir ~-~

-

, as a result of the substitution we obtain (4.10)

(4.11)

F r o m (4.10) there follows

i.e., (4~

v=~5+P.~,.

We estimate the norms of the operators S and ~,~, . The f i r s t two t e r m s in (4.11) are estimated with the aid of Theorem 3.2 for Cauehy problem (3.14) by

The remaining t e r m s have the form

484

~r, r

where

5~p~0kr k . It is easy to show that

Now taking into a c c o u n t inequalities (4,6), we have ~~

~

k

k "~"~T ~

~

~t:2

k "h~ T

~

{:~

~ "}'~T /

> "f

and estimating the norms of the derivatives of ~ 2~

with the aid of Theorems 3.1, 3.2 and Lemma

we obtain

#S~.tlq,,~cf(); T +l T)ll~4tq,,Q++.m~mma,mtvF1(~7".11~,,~+, '~,:' u I/,~d. ).,'t ";-

++ (z.z+,,+:,,+. z+,C;+j :+++(+,++,+o! We c o n s i d e r the o p e r a t o r

We v e r i f y that the finite function Taking into account that

~)t~+ o The v e c t o r

++'+)++,+.o+

uJ':+t~,+ is the solution of the p r o b l e m

] - V~t+2Vo~ s a t i s f i e s condition (2.36). L e t ~ =~J]+2{o~ , Vo=V .

V~. u%0 (k~0) , we obtain

+2Wv)+.~.+Z+~,~/Z'j:'c+:_t)vy~j+L ~+o

We trmsform

~+, (.v- -vy-t;~ ++. +--[-

~+,

the r i g h t - h a n d side by making u s e of the identity

(4.12) which is valid f o r any v e c t o r functions

k

4 ~

4

k

k

a(~s) and ~Cm) . We have

-i

-+

k

Since (?k-~)~=%~-~+(V~vF~'~)=-v'n(+)~(V~vF~'9) , from here we obtain condition (2~

where p,

r e p r e s e n t s the l a s t t h r e e t e r m s and

485

+cJ, I

~r

~r

' ~' Ur

(5,=S~,I:o.1]),

,,o,1~o ~: (~k % ll~%~To,) ,c ~,~,~,~,--c~ ~:h '/') I%.~ ~,t~7 ,o

'

F r o m these estimates, by T h e o r e m 2.4, there follows that for any ~>~ in the case of a bounded domain ~ , and for any ~ >5 in the case of an unbounded domain l~, we have

I~'~,11,~.~,.~.f +cd') q-k~ , . Thus, for the indicated ~ we have

.,~.~,,~,C,~s~%~,It%r p~ --.(J,~,~-r '~+a:'T) ' ~l.k~, l and t h e r e f o r e one can fix ~--~o and for O-
T:I

such that

IIz~ll-~ll~llo

The solvability of problem (4.1)

is established.

For

Assume now that T > ~

t-'l

the solution exists. We extended it in an e~en manner

with r e s p e c t to the plane t =T. into the domain Q~, and we denote the extended functions by tl~ We define now g' and p' for T. ~ ~

as the solution of the problem

fT, -t) v.~=o,

f g' and for %~.To we set lem (4.1) in

Q~.

~'=0 ,

p'=O.

t~'lS=0,

=0 b--To

'

It is easy to verify that

O=~+g', p =p§

is the solution of p r o b -

. Repeating this argument, one can construct the solution of problem (4.1) in (~

Now for the case of a bounded domain ~ one can prove completely the t h e o r e m . F i r s t of all, f r o m the solvability of the conjugate problem (see T h e o r e m 3.1) there follows the uniqueness of the constructed solution. Estimate (4.2) is established in the following manner~ Since for %~
(this inequality is proved in the same way as the previous one, by making use of the T h e o r e m s 2.3, 2.4, 3.1, 3.2), for

Let

T~T. we have

T>T~ and let

I

=%~,(.b) be a smooth partition of unity on

[0,T] such that ~ p

[z,,j~,] , j~k=~+T . The functions t~k=~ , zk=p~ f o r m the solution of the problem k

uJk

486

=0,

u~kI$=0

~,~

on the interval [z~ ,fi~] o Applying to t0k the estimate we have just obtained and summing with respect to all k, we prove (4.2). For an unbounded ~ the solvability of problem (4.1) in the class I~CQ~, ~>~, is proved~ If we would have:established a priori estimate (4,1) with any ~>t , then,.approximating 1r Lr vectors, we would have established the solvability of the problem under consideration for any

by finite ~>~ ,

and the theorem would have been completely proved. A deficient a priori estimate will be obtained later, at the end of Sec. 8o We proceed to problem (1.1), (1.2). We assume that

do(X) belongs to ~'~(~) , the closure of

the set of smooth finite solenoidal vectors equal to zero on 5 , in the norm of ~'~r (/xl,b)} ~1§ [l'l,b'~ fl

for

#

where p(~) is the distance from ~ to 5 and

I

for ~4J~, for ~>~;

if, however, 0v=~ , then the seminorm \,,w,,,,,~,~=(<1~))~,~, defined in See. 2~ It is known (see, e.g., [26]) that every vector t~)v--J;~(~l) can be extended in Q~ and ur

and ~5=0 ; in order to

construct u,C~,b) it is sufficient to solve the boundary-value problem

~{=,t,~ va~.x,'b=O, v,,C:r,,t,'s> =0, at,

u,c~,o)=t~(~:).

Therefore, an equivalent norm in J:~(~) is

where the in~ is taken over all the extensions gC~,5) of the vector v,(~) in (~( from the indicated class. THEOREM 4.2. Let

5r

and

~OAs,~,~.t.ZT j ]O.,j~s,,~,.o.,~' where

Qa.ls,~,a , =~,~ IIC~ ',~li,,~,Q , w h i l e the n u m b e r s

s,d, ~, ,d,

s a t i s f y the c o n d i t i o n s

3 t~-< 4 I , !~s, ,,. ~. ~ . ~-~

Then, for any ~ ~.~CO.~ and 1/.r

vp

(4.13)

~q) problem (1.1), (1.2) has a unique solution t~ r

(4.14) ,

and (4.15)

487

where the constants ~ and C~ do not depend on Proof. F i r s t we consider the case

T~

~ =0 and we write the problem in the form

(4.16) where

is operator (4.3) and ~g = ~ 0, g=~+0~ . We have

Here

and, consequently, 3. ~ 5" 5

t . E #

~,~ _ 5

I

By virtue of Theorem 2.1, 3

[g

.Z.~,II@lse~. +e ll~llssa) ~,=i

w

tl

Y

,

'

*

(C ~

(4o17)

Ua,llsf~,0T+C~llccllsea~t/t~ll~Q--@ ":

9

,

'

"

,

l

where

is a quantity which can be made a r b i t r a r i l y small by selecting 5 and T small. We fix these numbers in such a manner that

[~'~#,~

~/~[r

; then Eq. (4.16) will have a unique solution q~r

~

Reasoning now in the same manner as in Theorem 4.1, one can prove the solvability of the problem under consideration in the cylinder (~T of a r b i t r a r y height T o Estimate (4.15) for

~0=0 follows

from (4.2)and (4.17). Let tI0(~0 and let ff'(Z,~) be a vector from

~(QT~ such that tI~,0)=tI0(:~), il5 =0, IIt~71t~((~

g III~oII1%,n ~ We write ~J= ~J'+~ +t6, p=v*S, where -vhu.?~,=O, 7.u=-7.g ', ~JS=0,

(4.18)

(4.19)

~t,:o =0, u;is=O. According to Theorem 2.4, for any 0vH in the case of a bounded domain and for any ~>5 the case of an unbounded domain, we have

488

in

,

,

%,QT+ ll~'~

E~35,Q~, II~ p lira ~-~c, 01~ li~,Q+ ill t/o Ill~,a + II~II$,QO + O~ll~ II~,Qr. F r o m the l a s t i n e q u a l i t y and f r o m (4.17) we obtain (4.15). Thus, the t h e o r e m is proved f o r the case of a bounded ~_ w h i l e f o r r unbounded ~

for all ~>~

also f o r the case of an

~ At the end of See. 8, the estimate (4.15) w i l l be obtained as an a p r i o r i estimate, but

Approximating 'Jo~ ~~(Gh by smooth finite solenoidal vectors, equal to zero on 5

one

can easily prove the solvability of problem (4.1) for any ~/>t . COROLLARY 1. For any real T r 0

-~t -~t z6=~ , v =pe .

The estimate follows from (4.15) for

COROLLARY 2. For any ~ >C~> 0 we have the inequality

r

Q

~

T

II v" U~$(&~t tlv II where C, and r Indeed, for

"

(4.21 ')

do not depend on T . ~>C2 we have t

0r

-!

~ ~($ IItql$,Qt + 65~r~ (~).

&IIl/,llq,,a+q,llIIIlo,,O.t(6,~,(t')+r Since I!VI*{,o --- ~ IItr ~'q,,~t , we have t

and, consequently, for

~'~(C,,~) we have T

T

0

0

t 0

$

,

T 0

C~ $ _. ~T

IIO q,,Qr,~W 'ra, (T)e .

489

F r o m these e s t i m a t e s and f r o m (4.15) and (4.20) we obtain inequalities (4.21) and (4.21'). These inequalities may hold also for s m a l l e r , sometimes even negative, values of ~ (see Sec.

5). In conclusion, we mention that the boundary conditions in problem (1.1), (1.2) can be taken to be nonhomogeneous:

where ~ ~_W,

( 9 (for the definition o~ this space, see, e.g., [26]) and ~.~=0

the right-hand side of (4.15) the norm

]]~llW[m,-'&Cs ) occurs.

. In this ca.,e, ~n

In the proof of the theorem the v e c t o r

g'(~,~) has to be chosen so that

II{II~/,c~,~r

IIr ~ ~' ,-~(s,)")

The remaining arguments are left unchanged. One can show that estimate (4.15) does not hold always if q . ~ r

.

Condition (4.13) in the case of an unbounded domain ~ c a n be relaxed, requiring instead the boundedness of the local n o r m s

' ~ ~ ~t 5'~ %,(T)=Z_~CI,a.,

t)~

where oJ is the intersection of ~ and the unit sphere with the center at an a r b i t r a r y point ~E.~ . by a countable number of such domains c0k , we obtain an inequality s i m i l a r to

Indeed, covering (4.17)

r,-T

~,

a=4 Jo

k 0

"b

K

K

~

J P"mk

r, k

'~' ~

% c0 II~ U~,a,,

which, just as (4.17), is used for the derivation of the estimate (4.15). 5.

ESTIMATES

FOR

THE

RESOLVING

OPERATOR

In this section we shall consider problem (1.1), (1.2) as a Cauchy problem in the space j~(l]) ,

(5.1) where

Act)v = A~ + Bcbtr,

490

(5.2)

are o p e r a t o r s defined on the set, dense in J~(~), of the solenoidal vectors f r o m ~r on

, which vanish

~ o

We denote by 1)vtt,%) the resolving o p e r a t o r of Eq. (5.1), which a s s o c i a t e s to the solenoidal vector the vector t~(t)=~Ct,%)9, solution of the problem

&~ &t + Act)~~0, ~ t,Cq By virtue of T h e o r e m 4.2, the o p e r a t o r s

(5.3)

~(~,~) a r e defined on the set of the solenoidal vectors f r o m

J : ~ ( ~ ) , dense in j~(O) ~ They p o s s e s s the semigroup p r o p e r t y

~r

=~(t.s~,

t >.v~s, 11 Q,b -- I.

(5.4)

The solution of problem (5.1) is e x p r e s s e d by the f o r m u l a t 0

Let us show that

~(t~5) is bounded, while for t>s it is a smoothing operator in J~(~) .

THEOREM 5. 1. Assume that the conditions of T h e o r e m 4.2. hold. There exists a number such that for all

~>~o , ~,
wehave

II%ct.~)~Pll~a-~o(t-s) ~oct-9 ~,Ct,~)q)n -
~

He, a,

(5.6)

e

wr

(5.7)

IWltr ~

(5.8)

8

(5.9)

e.

~
~

IW]lt~.

F i r s t we shall assume that the operator m does not depend on t , i.e.,

o~--~(~)r

0~,(m)~Ls,(~) , 5 >~/%, 5,>~. M o r e o v e r , we shall prove estimates (5.6)-(5.9) under the conditions

~>~>~ ~ 5

3~

and if

~,=oo, then ~>-~ ,

(5o6')

> ~~> ~ 5 -3-

andif

~ =oo, then ~ > 5 ,

(5.7')

0<~ < ~ 0 ,

0~I

s {),

__5 9'

(5.8')

(5.9')

r e s p e c t i v e l y . If A does not depend on t , then ~(t,9] =~(t-5) and it is sufficient to orove (5.6)-(5.9) Integrating (5.3) with r e s o e c t to t , we obtain for s=0. Let 0-- ZC%)q, W(t)= ~ V ( z ) s 491

(5.3') and, according to C o r o l l a r y 2 of T h e o r e m 4.2, t

t

(5.i0)

G>C,

for any

o

Let

~(~,~) =(~-~(~)~ d(%), where ~ is the function defined in T h e o r e m 2.1. We have

Therefore t 0

0

Let us p r o v e now (5.6), for example. Under condition (5.6'), for any v e c t o r for

t=O ,

t~(~,t)~p((~,) , vanishing

we have the multiplicative e s t i m a t e

(5o11) where the constant ~ does not depend on Oo This e s t i m a t e is equivalent to inequality (2.7) with v = ~ . Setting U : t(C "~: V(~ - ~ (~))e "~ we obtain

and for

~ >d

IIt~ II~,a ~
WIIta.

S i m i l a r l y one p r o v e s e s t i m a t e s (5o7)-(5.9), but instead of (5.11) one has to make use of the inequalities, which also follow f r o m T h e o r e m 2~

<>~c,~,Qjl~,~ q + , , II~ll~,~), ?T:~T_ ~_r s 3 4';, (5.n,)

%

valid under conditions (5~176

Let us eliminate now the restrictions (5.6')-(5~176 It is easy to

verify that for any ~ and ~, connected only by the relation ~ >0r>] , one constructs the numbers ~ o , . . . , ~ , possessing the following properties: 0Vo=#, #~=~,~.> ~>~-~k.~3- 9 By virtue of (5.4),

492

and, according to what has been proved,

-~t ; ' )

i

Consequently,

~C$~)

~

lleg(t)qli~.a.
Wlla~m.

E s t i m a t e s (5.7)-(5.9) a r e obtained f r o m the already established inequalities, e.g.,

/..~(tq)~l-
v~

])~(.g)q)ll~,n-
II~llto ,

satisfies condition (5~176

Thus, the t h e o r e m is proved for a constant operator A , in particular, for A:Ao, where ~o=C,

,

We p r o c e e d to the general case~ We denote by ~0 the resolving operator for A = #,o and we make use of the relation

Assume f i r s t that

a,(z,t'):O

~a,a,:~t~=P: .~_~o

and

Then

11~ cb~1$,n~ ~cb~ll = % , np,,n, where

dcb:~xllc~Cx,t)ll~neL~CO,~),

Ildll~.<J~, ~~= g~ - ~~.

Let

%ct, s)~ =~ct, s) <>~#a=~ ct,@), We have t (5.13) and, according to what has been proved,

~(t,s3~ee ({-s)U~,o+cI ~

m'I<~,) ~,

S

where

~ >Q

and -, - ~,k,'~- PU" ~ - ~

~

i.

~(~-3)

Consequently,

(5.t4)

and for the functions

493

$

(t-tO'

t+.,-tJ" 'ok-s)"

C

"',J

b

we have the e s t i m a t e

mF! ~

f'+'

~( {k-,-tD+L~+;

K,ct,s++ML!ct-t,+ w h e r e ~I

- % : ~1-

~+ -

,•

+

~r

s i n c e f o r any

~, ~0

~

+k

M+

Ct+-s).+,++I ~ .r .

[+?+;CC~++X+~,~h)

F r o m h e r e it follows that I~ 4 -d,

Ll§ CM+cts)~~

<++eI{{-S)r+

i~,+

+' ',r

~ , )+)0 we have

if we take 0, >0 )~. Thus, the n u m b e r ~ depends on M, C,, +~,,+~0 L e t u s p r o v e that e s t i m a t e s (5.6)(5.9) hold f o r ~(>~(o+_P o If 8>~,>+o+J~ , then t $

F r o m h e r e t h e r e follows the e s t i m a t e , s i m i l a r to (5.10), t

.~

$

-oZc~-s),

.

o~

(5.t5)

5

f r o m which, as we have s e e n above, t h e r e follow e s t i m a t e s (5.6)-(5.9). A s s u m e , finally, that

0,c~,t) #0

. ~(~])=~'(t)+ ~'CD, ~"(t~=P:~u, ~]--~o +~' , U' is the c o r r e s p o n d -

ing r e s o l v i n g o p e r a t o r f o r which e s t i m a t e s (5.6)-(5.9) have just been p r o v e d .

It is connected with U

by the r e l a t i o n t + In o r d e r to conclude the proof of the t h e o r e m , we only have to r e p e a t the a b o v e - g i v e n a r g u m e n t s , r e l y i n g on the fact that //

where

~ # {, T = -T

#

~'#v)=tla(z,},)!lsa~L~(oT)

R e m a r k 1~ With the aid of f o r m u l a s (5.4) one can show that the c o n s t a n t s in inequalities (5.6)(5.9) and the n u m b e r p do not depend on the n o r m s of but only on

494

~ p ! 11%C:c,~),l ~d]~ and

~p v

~i "+

ak(z,t) and

c+cz,t) in the e n t i r e i n t e r v a l (0,T) ,

R e m a r k 2. We have deduced all inequalities (5.6)-(5.9) f r o m e s t i m a t e (5.6) with 0v=P o Let us show that under the s a m e condition

7 >~o we have e s t i m a t e s (4.21), (4.21'). L e t L<~, 4~. F r o m (5.5)

and (5.6) it follows that t

t

o

o

and, consequently, T

T

~,

T

t'

which together with (4.20)yields (4o21), (4.21'). A s s u m e that o p e r a t o r ~ does not depend on ~. In this c a s e , as proved in [17], ~ o ~ ~ Re~ , where d(-h3 is the s p e c t r u m of the operator-A o The proof of this fact is b a s e d on the e s t i m a t e s of the r e solvent o p e r a t o r - ~ , which we shall d e r i v e , following an idea of Sobolevskii [6], f r o m (4.15)o THEOREM5.2. e x i s t ~ >0

Let

~

, 0 , . ( ~ ) ~ L s ( ~ ) , acg0)~-L~C~), where

~>~

, ~5

o Then t h e r e

~ >0, depending on the constants f r o m inequality (4.15), such that the domain

~-2,~ =

~l<~+q] belongs to the r e s o l v e n t set of the o p e r a t o r - A and (5.16) where 6 does not depend on J~ ; E{~,)=(A+~,I~' , Proof,

Let

tY= E(~)J, ~(f.) be the same function as in T h e o r e m 2olo The vector

~, =v'e~tC4-~(~[~))

is the solution of the p r o b l e m

(5.17)

~I~:o=0 and according to (4.15) we have

i.e.,

where T

T

=l c4 0

;t 0

495

Obviously, e s t i m a t e (5.16) will be proved if

and 7 such that

? >C~ and if we s e l e c t 4

We take ~=~ +1s

(5.18)

. If ~c~>0, then

T

so that (5.18) can be satisfied by the selection of a sufficiently l a r g e

T.

The s a m e holds for [~c~ =0.

If, however, ~.~<0 , then ,~ir. mc~r,l'/,'l r~'~r~~ ~,"~"~

-illE " ~ d ,;~ ~

and (5,19) w i l l be satisfied only f o r sufficiently large on ~ and ]

; f r o m here we obtain a r e s t r i c t i o n

.

E s t i m a t e (5.16) is proved; the solvability of the equation A~+i~=~ or, which is the s a m e , of the problem 3

)~u-vA~+o.,s+L
r

(5.19)

can be proved by the method of extension with r e s p e c t to the p a r a m e t e r .~ , since for

sufficiently l a r g e positive )~ this p r o b l e m is solvable. As is known [27], T h e o r e m 5.3 has as a consequence the f o r m u l a ~t

{5.20) F where F, e.g., is the boundary of ~--A~" Assume that the s p e c t r u m of the o p e r a t o r A is situated in the half-plane ~eL ~,~o . Then the contour r" can be d e f o r m e d into a finite p a r t of the complex plane so that it is situated in the half-plane ~e)~<~ , Y>~o. Then, f r o m (5.16) and (5.20) it is e a s y to derive the e s t i m a t e IIp,~,

(5. 2 1 )

f r o m which, as proved above, t h e r e follow inequalities (4.21), (5.6)-(5.9) for ~ >~o .

6. THE SPACES ~

(@,"cQ~, J""cQ~

In this section we s t a r t with the investigation of p r o b l e m (1.1), (1.2) in the s p a c e s of functions which a r e HSlder continuous together with their d e r i v a t i v e s . We give their definitions. 496

By ~"~'C~(k>0 is an integer,

0 ~ , ~ ) we denote the Bananh space of functions f r o m

~(,~)

whose derivatives satisfy the Hblder condition with the exponent ~ , In this space one can introduce the norm

where

is the norm in the space

C"(~) , while
f o r m u l a (2~ By ~ " ~ ( ( ~ we denote the space of functions, defined on (~v and having the finite norm

(6.~) ~.,I.:,.

%

O,t

where ~g];) is defined by f o r m u l a (2.6). Basically, we shall have to do with the spaces rives of the functions ~ e ~

Finally, for any

~ (~) and ~"((~T) . It is known that the d e r i v a -

[Q~ have a finite s e m i n o r m

~,,g~ (0, 1)we define the s e m i n o r m I

"~

. {.~,,D

~10. : ~p It-%l I~-~i~I~(~,t)-~c~,t?-wc~,%)+ u(~,~. T

(6.2)

:=,~,t,~

It is easy to see that

Ittl~ ~ ~ P It-~l In particular, if

St =V~, E ~

l~(m,t)-~(m,~)l * ~ , ~ t ( ~ , t ) > n.

(Q~, then

w let ~ C Ivlo.T For

~:~

or 1~ = ~ one can replace h e r e

l~J(~ by

(6.3)

eT "

All these definitions apply in an obvious manner to vector functions t ~ , t ) =(tq,u~,u,~') . In addition, we introduce the following spaces of vector functions: the v e c t o r s f r o m

~-I~(;l) and ~ z ( ( ~ having the f o r m

~(/1~, ~(QT)

are the subsuaces of

w = ?q, where q is a function continuously dif-

ferentiable with r e s p e c t to ~> and having HSlder continuous derivatives ~ ; J ( l ~ , j((~T~ are the subspaces of the v e c t o r s f r o m

H~(I~) and

~(QT) satisfying the conditions

t~.nls=O and

V. t~ -=0 (the l a t t e r

is considered in the generalized sense (2.25) for any smooth finite (~). 497

In o r d e r to i n v e s t i g a t e the o p e r a t o r s

~ and P~ in t h e s p a c e

H'~(/).]

, we obtain first some

e s t i m a t e s f o r the s o l u t i o n of p r o b l e m (2.13). T h i s , a s it is known, is e x p r e s s e d in t e r m s of the Newton p o t e n t i a l and the s i m p l e - l a y e r p o t e n t i a l

(6.4)

(6.5) Let 9e

H~(D.), ~_~)9 ,"

m o r e o v e r , in the c a s e of an unbounded ~ we s h a l l a s s u m e t h a t j3 d e c r e a s e s

s u f f i c i e n t l y f a s t a t i n f i n i t y s o t h a t i n t e g r a l (6.4) c o n v e r g e s .

Wi n Now we e s t i m a t e

5~p

L E M M A 6.1.

I ~)'~V!, ~14,

It is w e l l known t h a t

.

(6.6)

~eH (.0_'), jr/,<].

(6.7)

-
.

Let

.~Cx,)=~7.gC~C)§

~/~(.(1),

T h e n f o r any v >0 we h a v e

(6.8) where

I~1--~, and

the c o n s t a n t 5 d o e s not d e p e n d on ~ . M o r e o v e r , if m a ~ C j ~ , ~ - $ ) ~ $ <~ , then

(6.9) Proof.

We m a k e u s e of t h e i d e n t i t y

The l a s t i n t e g r a l c a n be t r a n s f o r m e d b y i n t e g r a t i o n by p a r t s in the f o l l o w i n g m a n n e r :

(6.11)

I n e q u a l i t y (6.8) c a n b e d e r i v e d f r o m t h e s e two r e l a t i o n s w i t h the a i d of e l e m e n t a r y e s t i m a t e s : in c o n n e c t i o n with t h i s one h a s to t a k e into a c c o u n t that the s i n g u l a r i n t e g r a l

is b o u n d e d f o r a l l

498

~

, v ~ b y s o m e c o n s t a n t w h i c h d o e s not d e p e n d on ~ .

L e t us p r o v e (6.9)~

It is e a s y to show that

(6.12) We have

By v i r t u e of (6o12) the f i r s t two i n t e g r a l s axe bounded by

while the l a s t one c a n be w r i t t e n in the f o r m

F r o m t h i s f o r m u l a i t is c l e a r t h a t i t does not exceed

Ol~-a~{~'~<~,~ ~ o The e s t i m a t e (6.9) is proved~

L E M M A 6.2. If in (6.5) ~ ( ~ ) : ~,c~").~ta,c~)Is ,

(6.13)

t h e y f o r any ~ > 0 we have (6o14) moreover, r

c.L)

twl, .-eta> s

(6.15)

P r o o f . By v i r t u e of S t o k e s ' s f o r m u l a we have

$

and t h e r e f o r e (6.15) follows f r o m (6.6).

Then,

(6.16)

499

where

~(:~) is the intersection of ~ and the sphere ~,t~:l~-t~l.,.-~,],while ~ is the point of ~ n e a r e s t

to ~ o The f i r s t two t e r m s in (6.16) are bounded by

~(v'Z~/>~'~+,~a~lWI) while the third one is t r a n s -

f o r m e d with the aid of the Stokes formula in the following manner:

(6.17)

Obviously, this t e r m does not exceed c

,b4+$ _

~s

.(~

" The l e m m a is proved.

LEMMA 6.3~ For the solution of problem (2.18) which d e c r e a s e s at infinity (in the case of an unbounded ~ ) we have the estimate C~'~,)

I~)ln

C.O

~
(6.18)

while under condition (6o13) we also have (6.19)

Proof~ We make use of r e p r e s e n t a t i o n (2.20) of the solution of problem (2.18) and of the following e s t i m a t e s for the functions which occur in the integral equations (2.21) and (2.21'): nw~ [~vc.~:)l< o ~,z.~ 1~1,

5

l~vls ..zcl~ls '

5

(6.21)

$

[The last inequality, valid under condition (6.13), follows f r o m (2.22). ] Estimate (6.18) follows f r o m (6.6) and (6.21). Inequalities (6.19) and (6.20) have been already proved for the potential

~) = 6{~>-~)~C~

9 for the potential

with the aid of (6.22):

=

~(~-~36{~)~ they are established 5

(6.23) ~c~a~ 161 -~ c s. The l e m m a is proved~ THEOREM 6.1. E v e r y vector ~ C~) ~ H~(~) satisfying the condition o

(6,24)

500

o

where ~ - L~(fl), while ~@~) is a function with finite norm <~)n f o r m of the sum (2.6), where the vectors

V~P=PGI~~ ( f ~ ) ,

, ~
~ r P~ ~ J ' ( ~ )

are subjected to the in-

equal~ies (6.25)

and

o a

fl

u

o

},u

kk=I ,~',K .U

(6.26)

with an a r b i t r a r y positive 7~ o Proof. In the case of an unbounded l'/ we assume that I is a finite vector. We define q f r o m the problem (2.13). We have q0 = q~ + q,, where

Ii

Estimates (6.25) and (6.26) for V~ and, consequently, for ~ - V q ~

can be easily derived f r o m

L e m m a s 6.1-6.3, provided we apply to V~ the inequality (6o19) where X satisfies the condition of L e m m a 6.1o Assume now that ~ is an infinite domain and ~ is an arbitrary vector satisfying the conditions of the theorem. Let ~

It is e a s y to v e r i f y that for

and

V ~ P 6 ~ ~ we have estimates which differ f r o m (6.25), (6.26) by the o r e s -

ence in the right-hand side of t e r m s of the f o r m ~ ~ o

and

~(~,k) a . Making ~ tend to ~ , we

conclude the proof of the t h e o r e m . Thus, o p e r a t o r s ~ and ~ are defined on the set of vectors f r o m H~(n) , satisfying the condition of T h e o r e m 6.1. F o r a bounded domain ~ , we have fi~(fi)N L$(~) =H~(~) and t h e r e f o r e , H~C~) = G*(~) +J~(~). F o r a~n unbounded ~ , the conditions of T h e o r e m 6.1. impose additional r e s t r i c t i o n s at infinity on the vector ~. One can show that the o p e r a t o r s ~ and ~j are bounded in the space of H51der continuous ,vectors, increasing at infinity not f a s t e r than CIzr , in which the norm is defined, e.g., as r ~ l l l '<~>~' THEOREM 6.2. E v e r y v e c t o r

.o

~C~5 e ~ (Q,) with the components

501

~

o

(6.27) w h e r e ~ i a r e functions with a finite n o r m

, -~-~

9

.o

~,,~0

while ~.(~,t) have finite n o r m

~ [ c t ~ , can be r e p r e s e n t e d in the f o r m

Moreover, r

~)

r

~

~,~

IPGIIQ, +i~li^~c~tli~ T'~" II~k1~.=, 4, +~, t t It~ Indeed, the norms

,4%~

~

.

l~(m,t)-~(m,t)a0,ml

(6.28)

and QT (6.26) for ~=I , while the seminorm ~5~,It-~[~IU~(~,t)-7~(~,~)} , with the aid of inequality (6.26) for ~:~

5~O~) and S~ IVql are estimated with the aid of inequalities (6~

, applied to qC~,t) -~(~,t')o Similar r e s u l t s hold for the space and t h e h a l f - s p a e e . F o r example, for ( ~ g T e s t i m a t e (6.28)

has the f o r m

[PGl)P,, § [ Pfl ]~,~ c[[l]~, §~-~-, ~,,IIJ

+~Pt.~lt-tl("~')ll ~(m't)-~(~'t)ll~"} .

(6.29)

o

In particular, if s~i>P ] c K),(0), then

~[[~]~ *Y_I~,,I~ ~s~plt-$1

l~(=,t) g(mt)l}

e,0

(6.30)

Both inequalities are easily proved with the aid of formulas (2.27). 7.

THE

PROBLEM

(3ol)

In this section we investigate p r o b l e m (3.1) in the classes ~ ' ~

; however, f i r s t we shall give

the e s t i m a t e s for s o m e potentials and for the solutions of the b o u n d a r y : v a l u e p r o b l e m s in the spaee and the half-spaceo We s t a r t with the investigation of the integral

where the kernel is subjected to the inequalities .

~

~,

(7.2)

We shall a s s u m e that integral (7.1) exists as a singular one if ~Cz) d e c r e a s e s at infinity as a power function (in the following, for K( 0~,~ ) we shall have absolutely integrable or singular k e r n e l s

a~,~ ~ 502

'~

~

}-

LEMMA 7.1. Assume that K satisfies inequalities (7.2) and

(7o~)

(7.4) where K~(.~)=t~r

Then for any

I-<4 we have

(7.5) Moreover, if --

where

s~

p.cK~C0~ and < ~ ? ~ ~,~ ,

CD

~0 and ~ ( o , ~ ) we have (7.6)

Proof. Let

~,~e~,

~:s

o Taking into account (7.4), it is easy to prove that

F r o m here we obtain (7.5). Inequality (?~

f r o m the estimate

+~ . LEMMA 7:2. If K(~,~) satisfies the conditions of Theorem 7.1, while

where I ~ I ~ < ~ and if for any fixed D

~T

~ J ~,

sccpQf.c~0),then

~t.~

.... it_t'i@

....

Similar results hold for a potential with a singular or absolutely integrable kernel in the space or in the plane [~' ( ~ : 0 ) , e.g., for

R'

ill,:~ 503

We consider now the thermal volume potential t

2,

l+

t

LEMMA 7.3. If

j and l(=,,X,,O,O}=O, then

~r

(.=*,0

~)

,

(=.~')

(~)

(7.8)

Proof. The function ~C~,t):~x,b-~'(x,t) is the solution of the problem

~t'v~=~, t~=o=O,

~1

(7.9)

=0,

and the function t#(0~,[)=~(=,[)-~'(x,t) can be written in the form

~(=,t)= ~ rcm-~,t-$)J(~,~)&~J+=LI(=.b for =,,0. It is a solution of the Cauchyproblem , =

t.0

=0,

(z,[)E[1,=gx [0,T]

(7.10)

Inequalities (7.8) follow f r o m the known estimates of the solutions of problems (7.9) and (7.10) (see, e.g., [16, 26])

ruin, ,.etJ]~,,

l:Valn,.
(7.11)

We proceed to the investigation of problem (3.1)o The fundamental r e s u l t of this section is the following t h e o r e m . THEOREM 7.1o F o r any

~(r

satisfying the condition C~,,x=, 0,0) = 0

problem (3.1) has unique solution ~ " ( ~ ) ,

vpr

for which we have the inequality

[~]R, + l:vp],~, + Ipl,~, -
(7.12)

(7.13)

and where the constant T does not depend on G.

Proof. Let us prove the existence of the solution of problem (3.1) and estimate (7.13) for it. We estimate the vector 1I and the function I) , computed with formulas (3.11), (3.12) in which ~ ~ ' ( ~ ) satisfies only condition (7.12). It is easy to verify that

504

%~176 %~176 t

t

(7.14)

+-

(7.15) The derivatives ~

are linear combinations of the potentials

I

'l

where ~.~,~ , and t

.

r,

T h e r e f o r e , by virtue of L e m m a s 7.2, 7.3, we have

(7.!6) (,klO

F o r the estimate of

P ~r

we f o r m the finite difference with r e s p e c t to ~ of both sides of

(3.12) and we make use of estimate (7.5) for potential (7.7). We obtain .

,,,

and, consequently, r

~

~)

r

Finally, considering ff(~,t) as the solution of boundary-va.tue problem (7.14) for the system of three heat conduction equations with the right-hand side ~-vp and making use of (7.11) and (7.16), we conclude the proof of estimate (7.13). We make use of this estimate for the functions 1J~(~r

. p~(~,t), computed with the aid of

form~as (3.11) and (3.12) in terms of the vector ~'--J~(~), where ~ satisfies the conditions of the theorem. Since

[0 ]~, + vpJ~+

ac(.[I]e +,~ ,up1~l),

there exist sequences ~' , ~' , converging uniformly together with their derivatives in any bounded subdomain of [~, to I/(~c,b and p(m,t) , for which estimate (7.13) holds. Replacingin the relations (7.14) and (7.15) ~(m,t) by ~"(~,b and taking the limit when k--~ , we see that I/(m,t) and p(x,t) are solutions of problem (3.1). It remains to prove the uniqueness of the solution. Let u , a ~ (~T~ ~ V~C ~'(B~ be the solution of problem (8.1) with ~--0. Thenthe vector w(~,b= t~(~,t)(4-~(l~l)) and the function 5(~,b--~(~,b(~-~(l~l)) 505

a r e solutions of the p r o b l e m

%/_0.

,.~.,~.~,. ,%~ We r e p r e s e n t ur and

~':vJ'+w"

in the f o r m

, $ = s'*5", where

-v,~'§ while ~"and

v-=':=-w~, ~r'I

s* is the solution of p r o b l e m (3oi) with

~-n - ~ - ~~ ' . The vector u~ is e x p r e s s e d in t e r m s

of ~=~-v~ by f o r m u l a (2.37), f r o m where one can see that - ata-"~'~LP(~,~ ' r that ~ = 0

:o,

~---0

), and, consequently,

~,~:~e(~T)

,

~'~(~)

Wr

Le(~T)

with .

p>5 and then

F r o m T h e o r e m 3.1 it follows

, ~=0, and this concludes the proof~

P r o b l e m (3.1) is solvable in the c l a s s

~'"(s

Then in the right-hand side of (7.13) one adds

also in the c a s e when

~ -"" J (~,] but

%~:rc~o.

[Pal]~) and the consistency condition b e c o m e s

~It=o,~=o:0 8.

THE

SOLVABILITY (4.1)

IN THE

OF

THE

CLASSES

Ps R O B L E M ~

((~T]

This section is devoted to p r o b l e m (4ol)o F i r s t of all we p r o v e an a p r i o r i e s t i m a t e for its solution in the n o r m s of ~ " ( Q r ) 9 2*~

THEOREM 8010 We a s s u m e that p r o b l e m (4.1), in which the v e c t o r

~cH

~cj'(Q~

0 Let lJ~ ~'((~T)

' v P ~ ( Q T ") be the solution of

s a t i s f i e s the consistency condition

(8.1)

~ (~c,O) L~s:O. F o r any ~ r

we have the e s t i m a t e

L ~'' § IV],," + IpJQ,'~"'-'- q (,El], +~ l~C~:.U)' where

/

I

(~T=.O x [0,'[] and

_QI

c, ~,~r

(8.2)

is an arbitrary bounded subdomain of ~ such that

S N(~\fl')--0. The

constants 6, and C, do not depend on T .

Proof~ It is s~ficient to estimate L~:,~' since . t e r that Evp3~% is e s t i m a t e d f r o m the s y s t e m , and since ~p=v~o"

is the solution of the p r o b l e m

IS

15

(8.3)

$

(the l a s t condition is posed for an unbounded ~'l), by virtue of L e m m a 6.2 we have

lal ~c(.l~ol r QT

The n o r m 506

Eg]aT

:~

+s~p.It-tl t.(

_sSol~Lec~,t)-~,gc~,bl)-
can be e s t i m a t e d by Schauder's method.

UT

Let d~{z) be the functions f r o m the partition of unity (4.8). It is easy to show that for any

(8.4)

provided domains ~ ,

=~ ~

VK~. Here the constant C depends on the multiplicity of the covering of ~ by the

but not on )~.

We estimate Lff%~Ja, - -~'~ , ~ ,

assuming (without loss of generality) that the system of coordinates

"

[~r

coincides with the local s y s t e m

[ ~ ] at the point ~'~5 . Let 95~=F(~,,Gbc_H~*'(~X) be Eq. (4.5)

for the surface ~ in the neighborhood of the origin (we omit now the index k ). In the coordinates =~

= (%, z , , % - F(z,,~,))

the N a v i e r - Stokes s y s t e m takes the form

~t vv'o+v'p=~, ~,e=o, where

~'=V-VF-~,.

Consequently, ~t~.

,~

1~=t~ _

and

(8.5)

~=p% is the solution of the semispatia~ problem

,qi.:o, 1,,o0o,

+

where ~(~,L)=-VCv~.v)ff-vffv~*" m pv'E . We r e p r e s e n t the solution in the form

and assume that the r e m a i n i n g t e r m s are solutions of the problems

(~.~)

(8.8) 8~t ~ _ vato + vs-~ _ _ ~a-f u ) , , ~v, v)0r +V (v ,t V)~,

~lt:o = 0 ' Let us estimate the solutions of these problems.

v t0=0,

~1%:0:0

(8.9)

By v i r t u e of (7,11) we have

(8.10) Then, the solution of problem (8.8) is expressed by formula (2038) and, as a consequence of L e m m a 7.!,

(8.1t)

507

F o r the estimates of -~- we write ~

in the f o r m

(8.12)

where

F r o m here and f r o m Eqs. (8.5)-(8.7) it follows that

(8.13)

The terms which contain ~ are cancelled because of the fact that ~e j ' ( Q )

.

After integrating by parts in the last integral, it is easy to write ~,c~,t) in the form of a sum at of terms, satisfying the conditions of the Lemma 7.2, and an easily estimated integral with a weak polar kernel

aL,(~.~) . Namely,

'

#L~(~u) ~ , 9F 'u, ~'

~t

'

(8.15)

~',

-Iz~ ~T, t[~' E~',~]] P~Z} a~,~,-f~L,~',~>~

-f~F~ q~t~~,

Here T~ ~ - - ~ - 7 P ' I is the t h i r d component of the vector -[~, (4,9), We have made use of the equality, which follows f r o m (4.12), i

1

,

/

r

/

With the aid of f o r m u l a (8.14) one can estimate the s e m i n o r m

508

(8.16)

Since ~

is a linear combination of the kernels ~-~a~, I~-~1 ' ~ ~ and of the kernel, abso~' ~ ~ ~ -,~'I ' satisfying condition (7.4), we can make use of lutely integrable with respect to ~ a ; , Z , ~ I~ L e m m a 7.2 for the estimate of ~

+,~)~ ~ , .

Choosing in it ~ =+-~, we obtain

c+[q],t~ivFl(Eu,]o +[++]o +Ir ")+Z..~l~+l+~?tVO~l++ LZ_iu+I~ ttpl~, )], which, together with (8.11), leads to the inequality

where

--[t:~]~ +Z_ s,~,l~td+~pl?q,l+~ ~.E-..1~.1~,+lpl,,, .). T

~i=~ 1

I~T

~

~=~

~

LJ.T

L{ T

/

r'%

For the sequel it will be n e c e s s a r y to have such an estimate also for ~ ~-~. By virtue of (6.30)

+;I =

It is easy to see that for

'

~ : t-6 we have

I~, I~ JZ_T t.s~ott-tl. %#p~.c~,b-fi,(~,bl. ._
The function ~

must be transformed beforehand, by raising the singularity of the kernel in the r ~ h t -

hand side of (8.15) by means of a single integration by parts. After this, one can use inequality (7.5), from which we obtain the estimate

I~I. ~c7_[I~I~ +~IvFII

I I

]+

Thus,

We consider problem (8.9). Obviously, vF~, (v"-~')u~=~ [ ( v F ~ - T F ' y t ~ ] - v ( v F ~ ) ,

~:~-~+(V-V)~tV(~-Y)w~Krk and, since (V-V)0~:yF~-~:

we have

509

M o r e o v e r , f r o m (8.13) it follows that ~'(~,0)=0. ct+~'l

T h e r e f o r e , by T h e o r e m 7.1 ~)

C,t,.l)

I Jm,,..

Summing this with (8.10) and (8.17), we obtain

,c c~ i.e., f o r

*$),

~ ~ ~c we have

{Pl,.1~ < +7

+' " "<"+> [vP'C,]<'>+l <'" ~ +:LmC.<Jm+ P,', p~.
k)#

"+ ~<" +.T

I':70 I<"+"++ , +IpL, ~trl + ~ p Ivpt ','/-..ITS, '" ",., ++' ,<+ <+,+>9 O;+

. S i m i l a r l y , even though in a c o n s i d e r a b l y s i m p l e r m a n n e r ,

one e s t a b l i s h e s such an e s t i m a t e f o r k=0 : C28..+)

(~)

(.t~

(4)

[ts~'.]~ . , +[vpl:'.], . , +lpii'.l~ . , <,c[[lir,].., +Z_.

s

I~l

Ivpi ~

+,p,<">' <+;j

<'++

F r o m t h e s e inequalities and f r o m (8.4) it follows that

<"+

{[+]~

~

[t~]~ .+ct, I Q++ -'I~tHal s~"PI~'1+~.'~" ~s~A' ~'iP.

+"

+

<+.++> <.,.>~

tvpi f Z_its=i,,+,<., ,'.'., + ipi,~,.++..f

It r e m a i n s to r e m o v e f r o m the r i g h t - h a n d side the s u p e r f l u o u s n o r m s of t~C~C,b) and pcm,t) + We e s t i m a t e the n o r m of p with the aid of the inequality

which holds f o r

~(~5,

and a l s o with the aid of the e s t i m a t e which follows f r o m (8.3) and L e m m a

6.2, viz., .

~

r

-.v-s.

Selectiug

(,2++,i')

o,,

~c=,t)

.

~"

s

]^+~,~,+Z_~te~i+Z..i~,i.,, ). ~ "Jr l+, + +,++ ++:, , '+,,,#

C',+)

in the r i g h t - h a n d side a r e bounded by (~I

510

C4*D

~ sufficiently small, we obtain

[~] ,c([ Since all the n o r m s of

~

'

ib_t,l~ where

(4-*~.)

(.i++,O

(,L~.~

(8.18)

for ~'.<~(~), from (8.16) we obtain (8.2). The theorem is proved~ THEOREM 8.2. For any ~GJ~c(~ satisfying the condition (8.1), problem (4.1) has a unique solution Ve ~"C(~r') , 7p a ~ ' ( Q ~ , for which inequality (8.2) holds~ Proof. The uniqueness of the solution follows f r o m (8.2): ff tY , p is the solution of problem (4.1) with ~ =0

, then -Eff/~=0a~,/ -0 and, consequently,

-ff~ so that q=0 for t ~ 0 ~

and so also for ~ T

Let us show the solvability of problem (4.1) in the class of functions considered. F i r s t we

L~.CQ~; then

assume that ~.(~,t)c.~'~CQT')O

~r

proved, problem (4.1) has a solution Cr

for

all v.
Vpc-/~(Qr3. F r o m Theorem 2.1 and L e m m a 6.3 it

that d~, , pe[~iQ~ with any d~'~-J o Thus, in (8.6) ~,G~(~T), Vq,GL~(~, f~r ~(%0)=0 o Let us prove that ~r Vq, r ~ t ~ T ) , f i r s t under the condition FeH"~_.%)

T)

follows

and

this case the vector

. in

~ =Tt~ =(v,, , t ~ , t ~ - ~~Ft ~ - ~3F t ~ l- turns out to be the solution of the problem

(8oi9)

at 8F where &(Z) is the matrix with the elements ~CZ) --~,~CZ)--Ez~(~=flZ) ,

where c

~(t)

are matrices with elements f r o m the class

H~''~ and, in addition, for I~l=~ t~(Z)16

Iv F(z)I. Get J~ ~

. Since s,h~pK, sa~p0r162

the matrices ~ ( ~ , ~ )

and ~(t~ in (8.19)can be replaced

by the matrices

which a r e defined on the entire

~'§ and where

the leading coefficients of ~

and the e l e m e n t s of ~' are

bounded by C~, 9 The problem obtained by this replacement does not have solutions other than G r ~(QT) , Vq~~- L~(Q~

if ~ and T are sufficiently small, since for the difference of any two of its solu-

tions ~ , s we have, by virtue of Theorem 7,1, the estimate

511

5

,

The solution

I.=|

~, ' T

' T

9 T

' T

"

, ~ of problem (8.19i can be constructed by the method of successive approxi-

mations:

~=7__~ ~ , ~/ ~ - ~ , where t~

,

(8.201

are defined recursively from the problems

?

~)

c,,~l ,~ t=o =u'

t~)=O' Y~

~:0

(8.21)

=0.

One can show that for sma/1 )v and T the s e r i e s (8.20) for ff converges both in ~(~v(~T~, ~,>5, and in ~ " ( ~ T~ . Let us verify that t~ e . . ~ ( ~ ) ( ] ~ ( ~ r ~ -V~u' +Vr-f' =0,

d'

-0

It=o-

. We have u~')=~,tt '

~r

where

v~'=~v~ ~;J%:o=O, all%:0=0.[

'

Reasoning as in the proof of the previous theorem, one can conclude from representation (2.38) for 1/, v>5 ,and ~ ~_~z' ~r~([~ , andtherefore,

~L~(g,T)('I~'([~

P~ r

.

Finally,

from the relations

~,(~,o)=g,~,o)-~,I L,(~,~)~,o).~.~%~ o& 3

if follows that ~(~,0)~J (~§

~ oo,~__0oand thus the vector ~ satisfies the consistency condition

~: --.~ +~o, e/-4g~n'g'(@

and br virtue of Theorem 6.2 we have P:~k~'r

F r o m here it follows that ~ % ~ ( ~ i ) ~

; moreover,

t~o=O ~

":'~"~'(,-~VT") and so on. The convergence of the series (8.20) for

small J~ and T follows from the inequalities

I~,.IN §

JN+Iq, I~ -~c~,L~]%+[.Pj~ ]~,)-~cXLL~,J~, LTq, J%+lq, "I~,T)+CZ_s~,pl~ -< -r~./r

512

O~.-O~ C~*'c')

r

O~-O'l ~'I'2

CM.-'O Ca',")

and from the similar estimate for [~ ]~.~+~V~ ~,~,. ~§

Thus, for small ~ and T and under the condition Fe ~ (~__~, it is shown that ~

(~).

The same holds also for F~-~'~(~_j~')which can be easily seen by ~pproxim~ting F~}q C~-~ functions ~

~(~)

with uniformly bounded norms

by

]~I;:' and by making use of the estimate of the

solution of problem (8~ proved in the previous theorem~ In order to conclude the proof it remains to consider problem (4.1) in an unbounded i'~under the condition that ~eJ~((~), but ~gL~(Q# for any ~>~ o W e shall seek the solution in the form ~=~+u~, ~=~,,~, where t~,~, v, and S are the solutions of the problems

u'/t:o=~

(8.22)

~'Is=o,

~ -VhU~+V5 =l(~.0),

q~=O, (8.23)

l~lt=o=O, u~l~=0 The v e c t o r s g ( ~ , ~ ) - ; ( ~ ) ~ r

, ~ ( ~ ) = ~ o t , ~ ( ~ ) , ( = ) , where :~,e-r

and

3 fi

a r e finite and p o s s e s s the following p r o p e r t i e s : for v . - ~

uniformly

in any bounded subdomain of fl, P~ (a,C) and ~

~ --~,

(~) =g~'(~) satisfy condition (8oi) and

I in (8.22) and (8.23) we r e p l a c e ~' by

and ~(~.o) by ~

proved, these problems will have the solutions w~)

,

(:r,)

, then, according to what has been

~'~ and for the sum

t ~ ' ) + -~- ~

= 0~) we have the

estimate

JqT~O,t~lqT+C~,~l~ 1-C,I~.IQ +C,T [o ]Q, ensuring for small T the boundedness of

Q,~ [0{~)]{='

and, consequently, the solvability of problem (4.1).

One can get rid of the assumption on the smallness of T by the aid of the method p r e s e n t e d in the proof of T h e o r e m 4.2. The t h e o r e m is proved. At the conclusion of this section we prove for any solution of problems (4.1) and (I.I), (1.2) e s t i m a t e s (4.2), (4.15) and t h e r e f o r e we conclude the proof of T h e o r e m s 4.1 and 4.2. Let ~ % ( Q T ) ,

vpeLr

be the solution of problem (4.1), while m=~ , ~=p~ - t h a t of problem (8.6). The estimation

of t~ , ~ reduces to the estimation of the solutions of problems (8.7)-(8.9). We have

513

Then, from formula (8. 1 2 ) it follows that the derivative tegrafs ~(%,t):]

K(z,~)

-~,~ where i<5 and K= ,

~

~'

is a linear combination of the inelse

G

Making use m 9

.

the f i r s t case of the C a l d e r o n - Z y g m u n d theorem and in the second case of the estimate

where 6 < ~ , we obtain (8.25)

F o r the estimate of --~-w ~q/ " e make use of (8.13). The f i r s t two t e r m s of the right-hand side, !

which we denote by

~

, satisfy the inequality

II~'11r ~ ~ Cfl~fl~,,~ +-,~.~, IvFlilv V~, - v~ +~ I1~.,~ ~ cCIl~ll~,~, + 11~I1~,,~'§], [ ~,]l~.~, +.Liiv~u~,.~,).

(8.26)

g

We consider the last t e r m q/~. Since the integral over ~

of both sides of (8.16) is equal to zero,

we have

where 6(z,t) = [V'~ [7'~ 0]]- pV'~ and

II ~ II%~,T~C,(Jl~I[q,,%+.~ Ilqll$,~,[0,T]+S IIp V'GH,],,~T) . This, together with (8o25) and (8.26), gives

., 3

-,

-~ 3

%

By virtue of Theorem 3.1, such an estimate holds for the solution of problem (8.9). Combining the estimates of the solutions of problems (8.7)-(8.9) and selecting ~ sufficiently small, we obtain 5

K~C,~%Q~*,llvp<~ll~Q;, -<0(11~<11%<*,~ 11r162

5

+LIW~t,.1 ~ ,,% ~* "I)p}l~,,< t li?il~,s~)

and after raising to the ~.-th power and summing with r e s p e c t to k , we get

514

3

.~

~.q% ~-/Ivflto,._,~-<.(,i!~11,d..,~ .~.-7__.,:,Jt%:./~,,~,~,,o,~.Z__~:,li%~',§1o,, II,pIt~.~,~ II? I1~.s). The n o r m s of ~ in the r i g h t - h a n d side a r e e s t i m a t e d with the aid of (8,3) and (2~ n o r m s of V - with the aid of the e s t i m a t e obtained f r o m (2~

while the

and (2.8):

3

Fixing a s u f f i c i e n t l y s m a l l

~ , we obtain

EO~]spQ, + II XTpllq,,Qr~ q ~1~dlckQ +C~II~Jlick~v

(8,27)

and, c o n s e q u e n t l y , (4.2). In a s i m i l a r m a n n e r one c a n e s t i m a t e the solution of p r o b l e m (1.1), (1.2)o If &=0~, =0 , then the e s t i m a t e is obtained in the s a m e w a y as above, only i n s t e a d of p r o b l e m (8~

one h a s to c o n s i d e r the

problem

whose solution s a t i s f i e s the inequality p r o v e d in [28, 26]

As a r e s u l t , i n s t e a d of (4.2) we obtain (4.15). 0, = a : 0 and one c a n p r o v e

With the aid of e s t i m a t e (4.17) one c a n get r i d of the a s s u m p t i o n e s t i m a t e (4.15) in the g e n e r a l c a s e . 9.

THE

PROBLEM

(1.1),

In this s e c t i o n we p r o v e the s o l v a b i l i t y of p r o b l e m (1~ following a s s u m p t i o n s r e l a t i v e to the m a t r i c e s

(].2) (1.2) in the c l a s s

~Jt"CQT) u n d e r the

a(~,t) and 0,,(:~,t) : $

k:4

~X~k

(901)

'

(9.2) H e r e 0~ois a c o n s t a n t , t is the identity m a t r i x ,

~ , ~k a r e m a t r i c e s with finite n o r m ~-sct

,

'

,

.'

+

(9.3)

w h e r e S>{>~ >~ > O, while ~k is a m a t r i x with finite n o r m

~,t.~' and, as usual, ~ ~=~1~jll

It-~l

(9.4)

,

515

LEMMA 9.1. If the m a t r i c e s O~and 0~satisfy the conditions (9.1)-(9.4), then _

(.a.l

.~

.Ea.)

(.aO

(9.5)

where

I~ cz,{)-t~ c~{')l ~-a t

li

,

9

It-t I~ In addition, for all t we have

Proof.

L e t us prove (9.5).

We have

As a consequence of inequality (6.28) for C~=

>5, we have

~ ~ ~(.1~ ~1~ +~lct~M% ) ~c~gN [~]

(9.8)

F r o m (9.7) and (9.8) we obtain inequality (9.5). One proves s i m i l a r l y (9.6). THEOREM 9.1. If ~ H ~

and ff the conditions of L e m m a 9.1 hold, then for any 1 ~ o ~ ' C ~

~v. ~ ' ( Q ~ , satisfying the conditions ~ l e T ' ( Q p ,

problem (1.1), (1.2) has a unique solution { ~

V'~o=0,

and

g'ls =0,

CO~, vp~UC~CQ~ and

The constants ~ and ~ do not depend on T ~ Proof. We prove (9.10) assuming that the solution of the problem exists. Then the vector w{~,t~=0L~,~)-IroC~) is a solution of problem (1.1), (1.2) with z e r o initial condition and with right-hand side ~'-~ +Vh~o-~ . T r a n s f e r i n g ~t~ into the right-hand side, we can consider tg as the solution of problem (4.1) with right-hand side

~"=~'-~J=~+ ~ o

-~ff(x,t). The consistency condition (8.1) for ur is

equivalent to (9.9). By virtue of L e m m a (9ol) and T h e o r e m 6.2, we have

516

since the right-hand side of (9.5) is estimated in t e r m s of coup0)

+

~/~)gl) o Now appling Theorem

8~I, we obtain (m+~)

r.,t.)

(i,lO

(.i)

dl*4-)

r_~sl%+[vp3%, I pl% -~c(.lIl%+I-" <'~,

fl'

and, consequently, i

+/___~

(9.1~)

and f r o m here, as proved in Sec~ 8, there follows (9o10). We prove the solvability of problem (1.1), (1.2) assuming that 7- is a sufficiently small number. F o r the v e c t o r Z we had the relations

~]t=o=0

ws=O.

,

The existence of the solution of this problem can be established by the method of s u c c e s s i v e approximations, based on estimate (9.11) and reasoning as in the proof of the solvability of problem (8o19)o F r o m this estimate there follows the uniqueness of the solution. The t h e o r e m is proved. A consequence of (9.10) is the estimate

{01QT"O(#'gT)(IjIO +lPjila'{g{a),

(9~

with ~ =G+ o Reasoning as in Sec~ 5 (see R e m a r k 2), one can show that the number

X can be taken

here as in inequality (9.14)proved below~ A similar result holds for the problem with nonhomogeneous boundary condition

~162

r) @~=o.

In this case the consistency conditions become

~q

while in the right-hand side of (9.10)

]ql (i'n Sr

emerges.

We p r o c e e d to the investigation of the resolving operator ~(t,~) in the space j'(l~) . This can be done according to the scheme as in Sec. 5; the difference consists in the fact that the domain of definition ~)(A) of the o p e r a t o r A, ice., the set of solenoidal v e c t o r s f r o m

H2~(l~) which vanish on

S , is not dense in j~(l~) o We denote by ~:C~ the subspace of v e c t o r s f r o m j'(~) which vanish on 5 9 Each v e c t o r e,

--~

~aJ.~(~)

can be approximated by v e c t o r s

in the norm of H~(~) with any .~'4~ , while

t~r

in the following sense;

J,lj ~'{( a")

i~ 'n
9§

THEOREM 9.2. Let 5 ~H

and a s s u m e that the m a t r i c e s 0/, &~ do not depend on t and are 517

expressed with the aid of the formulas (9.1), (9.2), where the problem (5o19) is solvable i n t h e class

6, ~<~H*(i~)N L s ( ~ ) ,

H~"(~) for all ~ r

provided

~H'(i~).

Then

I~,l~p , lan~)~l s~+r

j~, q ~0 . The solution ~Y=~09~ is subjected to the inequality

I~lhJl:)+

<M

. c,>

(9o13)

,

Proof. We r e s t r i c t ourselves to the proof of estimate (9.13). We make use once again of the fact that ~ = ffe*t(~- ~ (~)), ~, = ~§

is the solution of problem (5.17) for which inequality (9.10) holds.

For the right-hand side, ~':~e~t(~-~(~))-i~te~t~,(~;~,'),of Eq. (5.17) we have the inequality

Il I,, .<~ le I(lllpcl.l,I ~111 )~cO+ll~el,I)(ima+ I.l,i ~p i~l). ~T f,~r

Mdition, s~lWl~x~le O.f

;t

L~T

I s~pl~l aM ~T

Therefore, for 7c~ >0 inequality (9.13) can be derived from estimate (9.10) for the function zb by taking a sufficiently large T and I)q ; for ~e~ ~0 imposes a restriction on p and ~ .

one has to require that

I~1 ~ 4 +l~e)4 and this

The theorem is proved.

THEOREM 9.3. Assume that the conditions of Theorem 9.1 hold. Then for any j ~ [ 4 , ~ + ~ ] ,

>~o, q~~ ~C~) we have

(5,9~o ~ccT~-s) e

(9.14)

I@a,

o.i

and for

H~ r

we have

I~ct, s)<~l~_~cct-r e (.ICPln+r.f~-s}isWiqt).

(9.15)

If 0J, 0~k do not depend on t , then ~o--au~6(.~)~e)~, #(-A) being the spectrum of the operator-A . Proof. Let ~(%)--~L(%,55~ ,<(%):~(.~]-), ~ =~(.4,t-s).

ff'~z)=~'C%)trC%)is the solution

The vector

of the problem

and by virtue of (9.10) we have (2+z)

~'

where

,~2+~)

~s,t

,

'

,(4)

us,t

Q,.t, -- 0 x Lt,,t:]. Consequently, for ]~[~, s i .9)

"

,

-I

~)

Q~,t

we have

( (:+~)~-~/, ,c~)~4"~i~

IlS(I,)l --c[l~Sl~ J \iVin) 518

,~

.jS..-.~

-~

c~)

~O(.4+e )iVla~.~,~

(9.16)

1%(% 5)?IQ,]

It r e m a i n s to e s t i m a t e

'

~,~,t

F i r s t we consider the c a s e when ~ does not depend on t ~

The

~

v e c t o r u~cb:J[~~(z)@ ~Z is the solution of p r o b l e m (5.3) and f r o m inequality (9,10)we obtain

and, consequently, Cz)

Czt

Ca)

F r o m h e r e for t- s
-,

and for

~l~

s,t

-~C~e Iql n

t-s >~C we get

t-g,t

t-~,b

The last norm. is estimated with the aM of (5,20) and (9o13) in the follo~ving rammer:

tt-s-G)~la-
(Jgla+(t-s-~ ~pi~i),

(9o17)

and this inequality holds for any ~ ] ( ~ ) i ~>~t~)~o Thus, in all c a s e s CA)

lYlfl

~ce

Iql a ,

t-d,t

~> ~

~e~

~Ve@CA)

This inequality, together with (9o16) gives the estimate (9,14) for any ~r

For

~r

and, consequently, for

, we obtain (9,15) by combining the e s t i m a t e s (9.14) and (9.17), where in the l a t t e r

one has to take 6 = ~

:

~
~:)

~--~

._,

We have a s s u m e d that ~ and 0~ do not depend on ~ . In the general c a s e one m a k e s use of Eq. (5.12). By virtue of (9~

we have

and t h e r e f o r e , for the function

~Ct,s)= I~(~,s)~l~+~) we have the inequality f,

~(b,S)-
519

if qcJ:(iD and

ct-=)[~+(t-v) Itch.s)&,

G>s)-~ce
In both inequalities g >~~162

Reasoning as in the proof of Theorem 5.1, it is easy

to derive from here estimates (9.14) and (9.15) for ~ = {+a. and ~o=~',ja, 0o>0 . Then, from q ~J.~(f]) we have t

Cl,,sWIn-~ce l~Intc,]c

('ItG-'~))e

G-s) ct1:lqln ~c,e

I@ln, x~>g>go.

5 #

/

In addition, if t-~..t
lUCt,s)~-u,(t;,s)wl .dl,to(t:s)~l a + (t'-f)~ ws "

~.(g --~)} (~)~(~,S)~~ + v

"/'}t]Uo(b-~:)~b(H,t,(~:s)~l" t, . ~ . , da;,cll, l,o(t-s-~)~l~) +c-~ff~5[~t(t-r,)]e ~g(:,.s)gf; a c,J[4(l~-~:)]e II$(~:.s)~la&~.c,~ Iqo. s ' ~(,t g s F r o m the last two inequalities if follows that ~,

.

.,~)

~(t-~)

I~(b,S)qlQ

~Ce

~)

Iql~,

u

(9.18)

t-~,t

and from here and (9.16) there follows (9.14). Similarly one proves estimate (9.15). Besides (9.14) and (9.15), one can also prove that for ~<$, ~>J~>~ , and ~ J ,

~Le

it_t,i ~

, ,/

Here q r hand side.

; if, however, q'~JoC.O-) , then the t e r m s

w e have

)

-

(t-S) ~ ~]@] can be removed from the right-

Similarly to (9.14), (9.15), these inequalities can be reduced to estimate (9.19) with is proved in the same way as (9.18). 10. THE NONLINEAR

PROBLEM

In this section we consider the problem

U~

520

l~d

K

v..1

=~ , which

(1Ooi)

I = o, %owhich can also be written in the f o r m

where KIy=~ ~Zr..y,f f . . The o p e r a t o r K p o s s e s s e s the p r o p e r t y

K~ =K(ff,t/) , where

Kcv,t~=Pj~_z.l~,t~ is

a bilineax f o r m r e l a t i v e to d and td. LEMMA 10.1. F o r any ~

and any lf, Igr ~ ( Q O we have the inequality

Kcg,ta)~r

~

t~.0)lr

aT

|~(x,

(10.2)

where 5 is a constant independent of T, t~ , and u~, while

Proof. ,,et

;then II K(~,~311~. Q~T ' ~K(~,t~o3',6, o-~T ~ "u~u. ~,q T 7_ ~~ np~,~,~,0. 3r T vq t r, ~=~

For ~

(lO.S)

one can s e l e c t the numbers p, v ~ so that

and one can estimate the norms of t~and ufwith the aid of T h e o r e m 2ol. In o r d e r that the constant in (10.2) should not depend on T , we extend the v e c t o r s t/ and u/ into the domain Q~=~r-tT1 so that

I1~'11~~,,Q ~r,6nltr(.~.o)~lo,~, hal

- ^ .~CAlu~(~,o)~lla,n

(ff T~ , then this cannot be done). From inequalities (10.3), (2,7), and (2.8) with ~.=~ there follows estimate (10.2) in which for T ~

the t e r m s

F r o m (10.2) it follows that for all

~1I(~,0)~ and

~ltff(~,0)~are superfluous~

15, t~'r ~(QT) we have w

(10.5)

We proceed to the investigation of problem (10.1). We shall prove its solvability by the method of s u c c e s s i v e approximations, for whose norms we obtain inequalities of the f o r m

~,., _~s:+ b~ +c

(10.6)

521

It is easy to prove: LEMMA 10.2. If in (10.6)

c,c>o, and if ~o-
o-g
((-g)~>qo~c

g is the minimal root of the quadratic equation ag%Cg-I)~+C---0), then for

all ~,~0 we have L ~ g

.

THEOREM 10.1. Assume that the conditions of T h e o r e m 4.2 hold and let

q,a~ . If

~Cr

(10.7)

where r and ~( are the constants f r o m inequality (4.21') and r ,r f r o m (10.2), then problem (10.1) is uniquely solvable in ~ ( ( ~

and

{IUI{w,I,(.O..) ~ ,?.c,(4+e)({}~i},~,~~ ~ltrAl~a) I+ V'{- ~ r c,~ Proof. Let

~o)=0

(10.8)

O,T~')O+d:)(ll* II + ~ig {it)

and let tr~ , w>A, bedefined r e c u r s i v e l y as the solutions of the problems

J"*'~r176

db

+ ~'{'~L ),

=~C~)-Kv ,

(10.9)

4'=0= tro

F r o m (4.21') and (10.4) it follows that

and, consequently, for the quantities L=IIu'"II~+ ~lg~lg,a we have inequalities (10.6) with C=C~0+e~)01~llq.a, +~[t~lil~e)~--0,a=c,q(~+e,gT)m~(~,TcO~.

By virtue of L e m m a 10.2, all the ~ ' s

right-hand side of (10o8). F o r the proof of the convergence of difference u~"~=gr

r

are bounded by the

[II '~] in ~ g ( ~

we note that the

is a solution of the problem +/~(D~I =-5,Cq ,uJ )-K(vo ,q

), ~r176

and by virtue of (4.21') we have

Since the boundedness of the n o r m s of 13~ has been already proved, f r o m this we obtain the convergence of the sequence ffr =~_ t0r

in the norm of ~((~2~

with some

~-
the right-hand side of (10.8). In the same manner one proves the convergence of tlC~ in ~ ( ( ~

and

so forth. Similarly one establishes also the uniqueness of the solution. T h e o r e m 10.1 is proved. R e m a r k 10.1. Condition (10.7) guarantees the solvability of problem (10.1) for a r b i t r a r y ~

L~(QT') , ~o~j~~(l~) in the cylinder (~T' whose altitude T depends on the norms of ~ and ft, and

i n c r e a s e s indefinitely when these norms go to z e r o . In addition, for any T > 0 one can find ~ =~(T)>0, so that problem (10.1)is solvable in ~(QT~ s m a l l e r than ~. 522

for a r b i t r a r y

~s

qo~j~{~(~l) whose norms are

If, however, one can take in (4.21) Y-~O , then the number P~, as can be seen f r o m

(10o7), does not depend on T and, consequently, the solution of problem (10.1) will be determined for any t ~ 0

only ff II~-II~,Q+lllt/fl~.n ~R,~

We denote by Vet,5)

the resolving o p e r a t o r of the nonlinear problem

~--T +

,

t:s :~~

F r o m T h e o r e m 10.1 it follows that it is defined in the sphere

(10.10) K :[ll~lll~,~-r of the space ~:~(1~)

for 0~t-s ~T, provided T and ~ a r e connected by the relation

In this c a s e for

~ =1~ we have estimate (10.8) with ~=0 and

ft, =q~ . Since for any fixed ~ we

have ff(~,t)~{~(I1) , f r o m this estimate we obtain the boundedness of the operator 1/ :

In addition, for any ~

~'~. K~ the vector

t~:Vct, )r -Vcf.,9~ - U-~] is a solution of the linear problem

&~'~ § ~,ch ~ + K(.',)',w)+ K(~,r

-&t

wtoo=q'-q/.

(lO.11)

T h e r e f o r e , f r o m inequality (4.21') it follows that

IIIVct, 9 ~ -V(t, 9 ~' III~,~ -~c, (t -~, ~) IIIq - q'lll~m, i.e., the o p e r a t o r Vct,9 is continuous. w e show that the operator 1](t,9 can be extended by continuity to all of ~(1~) , ~/~5 . It is connected with the resolving o p e r a t o r U(t,s) of the linearized problem (5.3) by the equation

(1 O.12) which we shall solve by the method of successive approximations, by setting !](~,S)? --~(t), ~(~)--0 ,

s We estimate

s

~

. We fix the indices

d v>{ so that

From the inequality

and from (5.6) and (5.7) it follows that

=C4 ') ~Ct-s~ S

523

a.~!

i

/__llII (t)ll~n~O.,Ct-$)

e

~%n +C,qICt-~)

e

LlltI~ I~.~n ~'

(10.14')

T h e r e f o r e , for the functions

~=~

we have the r e l a t i o n s

$

~O,~(b~(~)~,~C~) we obtain

f r o m which f o r

~. ct) ,.o, [il ~ ~.n+c, r where ,

~t[

(~-~)

-

",~

~t-~)

-~,,-~

+(t-9

j(~-g

e

(~-s)'"'d~1.

(10.15)

g

F r o m L e m m a (10,2) it follows that if

t- ~,d, r

(10.16)

then " (~ +/Z) 2

and so f o r

~;~-(5,t) we have

T h e r e f o r e , f o r any p satisfying the eondition Og~p.<~,' , we have

r ') p,n C,C -?

e

I1~,,~ §

e

~Rv 6)ll~,nOl;~ (bg~(~.~O

9

since

t

t

-~'~) ~t-m,, ,,~_., kl Gr162

524

5

(lO.17)

~(~

_~_ •

45C,e

Ct-5)

-~ .... Similarly,

r162

(~+M) =

A~

_~ Ct-S) -~'~->-~ liRil~,~

"

('I + ~ ) '

~-

4

if q,-< p -<~ , ~- - } < ~, then

/__,~. r Estimating the difference

I.%,~ i~+(~+g)

~r176 '~'t~r

].

(1o.18)

f r o m the equation

S

one can show with the aid of the obtained e s t i m a t e s that

g~) converges to the solution of Eq. (10.12),

for which inequalities (10.17) and (10.18) hold. THEOREM 10.2. O p e r a t o r

V(Ls)

can be extended by continuity to any sphere K ~ : [Iq IIr ~.

of the space ~" ( i1 J . In this case it is defined for o~t-s~T , and T and ~ are connected by relation (10.16), i.e.,

14c,r q~(T)g
~(T) is defined by f o r m u l a (10.15). F o r any cp, q ' e K~, 0-~t-s -
Ilk/ct.s)~ 11rn ~ccE-s, g) H~m,

(lO.19)

IIVcL s) c~_VcLs)~'llcm ~ q (fl- s, g) II ~- cp'IIr

(10o20)

The solvability of Eq. (10.12) is proved while estimate (10.19) follows f r o m (10.17). Inequality (10.20) follows f r o m (10.17) and (10.18) and f r o m the equation

t~cb-orb =Uct,sx~-~')-IUr

v) + {
(10.21)

5

where

v(t)=Vct,5)e, {(t)-Aj(t,s)~'. Moreover, reasoning in the same way as for the proof of estimates (5.6)-(5.9), one can establish

that for t - s > 0

o p e r a t o r V(t,s) is smoothing and acts f r o m J~/(fi) into V~'~(fl) , p~0v; and ~,.ZC~) ,

~<~ . Finally, inequality (10o18) shows that for the operator t

~C'~,s)~ :~]4,~)~-UcL,s?~=-olUcLz)KVc%s)~ & we have the estimate 2

and t h e r e f o r e UCt,s) is the F r 6 c h e t differential for Vet,s) at the point q=0 We consider now problem (10~

in the spaces

.

~

(QT] . We define the space :~'~(~QT) as the

set of v e c t o r s in (~T having the finite norm

525

lol ~,~) ~

r

.r

lgcm,b-gtm,t')l

For any LTC--'~/'~CQT") we have the estimates (10o22)

5

{h,O ~'~/

/,

t*~,

(10.23) =

and for

QT

tr~-~"'(0 T) we have (10.24) T

(~T

~T

We give without detailed proofs the a~a~ogues of Lemma 10.1 and Theorems 10.1, 10.2.

LEMMA 10.3. If O',W~:~ C(~,), c~ZlY=O, then (~) 0..0 IKc~,ml^ ~cC~pIOI'I~I +~mlWllVl(~)) ~'T ~, C~ -5-,r

Proof. Since

(lO.25)

d~ ~ : 0 , we have

~ . ~ = ~ : ~ ~=-~c(~,.) and by virtue of (9.8)

I KCu,w) o

.~c(,}C(m~)la +~__~:15, wlQ,~"").

But by virtue of (10.22) and (10.23)

~c{~,,,,% +0%~) C~,*t) C%' ) C~,~,) I ~c(s~plt~l.lwl(~ ,,.o +~ I~II.I~~/. (~T r aT aT QT T 9 (.#.,~l

Similarly one estimates I l~ to"[ ~T "

and

THEOREM 10.3o Assume that the conditions of Theorem 9.1 hold and assume that ~ J ~ tOT) ~o(~) ~_ H ' L ~ ) N J~ (~) satisfy the consistency condition 3

If ~ where C,, and

2., i~.l~T

[C~C2C3(t,e )] ~ s ~ p Itrol] J< ~

(10.26)

are the constants from inequality (9.12), Ct from (10.25), and C~ from (10.24), then

problem (10.1) is unkluely solvable in ~ 526

~

2+d,

~,

(Q~)NJoCQT) and

C,O

(=,z)

O0

(~ (~+,0

l~l c~*~)~cCT, IJ:Io.+,1~.1n , ~pltr01)(l~lo. +1 oin ).

(10.27)

(a)

We prove that under condition (10.26) the norms of the v e c t o r s

v (~,~), defined by (10o9), are bounded.

By virtue of (9.12) and (10.25),

1'5 IQt ~kl+c ;Ll~r]Qt+looln.+zb=ltT IQT s,~ and since

~;*~.~0~+.~(~,+>1~"~1~; ~, for

u+: ~

~Q+ we obtain the estimate

Uo.r~c,(~++ +~1+1~++~.;/+mgguy'c~p ~,+h?++!_~ .
c,%(r+~O,T)) U , ]

~

~,+

+

,,

The boundedness of U~ under condition (10.26) follows f r o m L e m m a 10.2. The remaining arguments are the same as those in the proof of T h e o r e m 10,1~ Regarding condition (10.26), R e m a r k 10.1 holds. F r o m T h e o r e m 10.3 it follows that operator qr H~(D) ~ J,*(~l)

~(~,s) is defined for 0~b-s ~T for ~ny

whose norms satisfy the condition (10026).

THEOREM 10.4. Operator V([,S) can be extended to any sphere case it is defined for

K , ~ J : ( ~ ) : IqlC~[~ ; in this

0--L-s~T ff T az~d [~ are connected by the relation

where r and 6~ are the constants from inequalities (10.25) and (9.14), (9.15), while t

Indeed, ff for the estimate of the v e c t o r

4

~.~

~t

.~

.~.~

ffc++~ f r o m (10.13) one makes use of the inequality

which follows f r o m (10.25), then instead of (lOA4), (10,14') we obtain

t $

527

S

The remaining arguments are the same as those in Theorem 10.2. LITERATURE CITED

1.

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11o

H. Fujita and T. Kato, "Onthe Navier-Stokes initial-value problem,~ Tech.Repo Stanford Univ., No. 121 (1963).

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14. 15.

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V. A~ Solonnikov, "On boundary-value problems for linear parabolic systems of differential equations of general form," Tr. MIAN, 83, 3-162 (1965).

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