Existence of Periodic Solutions of the Navier–Stokes Equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

208, 141]157 Ž1997.

AY975307

Existence of Periodic Solutions of the Navier]Stokes Equations Hisako Kato Graduate School of Mathematics, Kyushu Uni¨ ersity, Ropponmatsu, Fukuoka 810, Japan Submitted by John La¨ ery Received October 9, 1995

1. INTRODUCTION AND SUMMARY The Navier]Stokes equations describe the motion of viscous incompressible fluids. We assume that the incompressible fluids are homogeneous and that the density r and kinematic viscosity n of the fluids are constants independent of x and t. Then the Navier]Stokes equations are

r Ž u t q u ? =u . y n D u s f y =p, div u s 0. The present paper is concerned with the unique existence of strong periodic solutions of the Navier]Stokes equations with r s 1 and n s 1, u t y D u q u ? =u s f y =p, div u s 0,

x g V , t g R1 ,

Ž 1.1.

x g V , t g R1 ,

Ž 1.2.

tgR ,

Ž 1.3.

u N ­ V s 0,

1

where V is a bounded domain in the N-dimensional Euclidean space R N with smooth boundary ­ V, the vector-valued function u s Ž u1 Ž x, t ., . . . , u N Ž x, t .. is the unknown velocity field, the scalar function 141 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

142

HISAKO KATO

p s pŽ x, t . is the unknown pressure, the vector-valued function f s Ž f 1 Ž x, t ., . . . , f N Ž x, t .. is the given external force, and the notation N

u ? =¨ s

Ý is1

u

i

­¨ ­ xi

N

,

div u s

Ý is1

­ ui ­ xi

is used for the vector-valued functions u and ¨ . Equation Ž1.1. can be considered as the nondimensionalized form of the Navier]Stokes equations with Re s 1 Ž Re: Reynolds number.. The unique existence of strong periodic solutions of the Navier]Stokes equations with Re ) 1 can be studied similarly. The scaling in the smallness assumption on f introduced later will be affected by the choice of Re. The problem we consider is as follows. Let the given external force f be periodic in t with some period v . Then we try to prove the existence and uniqueness of periodic strong solutions u of the Navier]Stokes equations Ž1.1. ] Ž1.3. with the same period v : u Ž x, t q v . s u Ž x, t . ,

x g V , t g R1 .

Ž 1.4.

As is well known, there are a number of papers concerning the initial value problem for the Navier]Stokes equations, that is, Ž1.1. ] Ž1.3. together with u Ž x, 0 . s a Ž x . ,

xgV

Ž 1.5.

Žin particular, concerning the existence and the uniqueness, see Fujita and T. Kato w2x, Giga and Miyakawa w4x, Hopf w5x, Ito w6x, Ladyzhenskaia w9x, Lions w10x, Lions and Prodi w11x, Masuda w12x, Serrin w13x, Temam w15x, Wahl w16x, etc... On the other hand, for the periodicity problem Ž1.1. ] Ž1.4., Kaniel and Shinbrot w7x have shown a reproductive property of the Navier]Stokes equations for N s 3 when the external force f is sufficiently small, i.e., sup 0 F t -` 5 f Ž t .5 L 2 Ž V . is small. For 2-dimensions Takeshita w14x also obtained the reproductive property under no smallness assumptions on f. Recently, Kozono and Nakao w8x Žpreprint. have proved the existence of a periodic strong solution for any dimension N G 3 under the assumption that sup 0 F t F v 5 f Ž t .5 L r Ž V . Ž r ) Nr2. is sufficiently small by considering the integral equation in a Banach space. They have also studied the problem in some unbounded domains. In this paper we shall show the unique existence of strong solutions of the periodicity problem Ž1.1. ] Ž1.4. for N s 3 and N s 4. We would like to

NAVIER ] STOKES EQUATIONS

143

emphasize that we shall prove our uniqueness and existence theorems under the critical smallness assumption, i.e., sup 5 f Ž t . 5 L N r2 Ž V . is sufficiently small.

0FtF v

For this purpose, fairly precise energy estimates will be required, and our result is on the grounds that we are able to apply L2-theory together with the fractional power of the Stokes operator. Since the Stokes operator Ždenoted by A. is strictly positive self-adjoint in the Hilbert space, the fractional power of A is defined by its spectral resolution. However, when we consider the problem in the Hilbert space, we need to place the restriction on the space dimension N because of the nonlinear term appearing in the equations. We shall give the definition of the Stokes operator later. To describe our theorems accurately, we introduce some basic function spaces and notion. We define ` ` C0, s '  w g C 0 Ž V . ; div w s 0 4 . ` 1 Ž . In addition, we define Hs as the closure of C0, s in L 2 V , and H 0, s as ` 1Ž . the closure of C0, s in H V . Throughout this paper, L2 Ž V . represents the Hilbert space equipped with the inner product

N

Ž u, ¨ . s

Ý is1

HV u ¨ i

i

dx.

We denote the L2 Ž V .-norm by 5 ? 5. H m Ž V . is the Sobolev space of vector-valued functions which are in L2 Ž V . together with their derivatives up to order m. H0m Ž V . is the completion of the set C0`Ž V . in H m Ž V .. Let P be the orthogonal projection from L2 Ž V . onto Hs . By the Stokes operator A we denote the Friedrichs extension of the symmetric operator yPD in Hs with domain DŽ A. s H 2 Ž V . l H01 Ž V . l Hs . It is well known that DŽ A1r2 . s H0,1 s . Then, the periodicity problem Ž1.1. ] Ž1.4. is formulated as an ordinary differential equation in the Hilbert space Hs Žsee Fujita and T. Kato w2x., du dt

q Au q Pu ? =u s Pf Ž t . , uŽ t q v . s uŽ t . ,

t g R1 ,

t g R1 ,

by the orthogonal decomposition L2 Ž V . s Hs [  =p; p g H 1 Ž V . 4 Žsee Fujiwara and Morimoto w3x, Ladyzhenskaia w9x..

Ž 1.6. Ž 1.7.

144

HISAKO KATO

Next, let us introduce some function spaces consisting of v-periodic functions. Let X be a Banach space. We denote by C k Ž v ; X . Ž k: nonnegative integers. the set of X-valued v-periodic functions on R1 with continuous derivatives up to order k. Then let us define the norm

½

5 f 5 C k Ž v ; X . s sup

0FtF v

k

Ý

5

5 Dti f Ž t . 5 X .

is0

We denote by L p Ž v ; X . Ž1 F p F `. the set of v-periodic X-valued measurable functions f on R1 such that 5 f 5 L pŽ v ; X . s

žH

v

1rp

5 f Ž t . 5 Xp dt

0

/

- q`

Ž 1 F p - `. ,

5 f 5 L`Ž v ; X . s sup 5 f Ž t . 5 X - q`. 0FtF v

We denote by W k, p Ž v ; X . the set of functions f which belong to L p Ž v ; X . together with their derivatives up to order k, and in particular we write H k Ž v ; X . s W k, 2 Ž v ; X . when X is a Hilbert space. To prove our theorems, we shall use the remarkable following proposition on estimates of the nonlinear term. ŽWe state the proposition in the Hilbert space.. PROPOSITION 1.1 ŽGiga and Miyakawa w3, p. 270x.. If 0 F d - 1r2 q Nr4, the following estimate is ¨ alid with a constant C1 s C1Ž d , u , r ., 5 Ayd Pu ? =¨ 5 F C1 5 Au u 5 5 A r ¨ 5

for any u g D Ž Au . and ¨ g D Ž A r . ,

Ž 1.8. with d q u q r G Nr4 q 1r2, r q d ) 1r2, and u , r ) 0. Now, our existence and uniqueness theorems are as follows: V is a bounded domain in R N Ž N s 3, 4. with smooth boundary. THEOREM 1.1 ŽExistence .. Let f g H 1 Ž v ; Hs . Ž v ) 0.. Then there exists a constant K 0 s K 0 Ž N . ) 0 such that if M ' sup 5 f 5 L N r2 Ž V . F K 0 ,

Ž 1.9.

0FtF v

the problem Ž1.6. ] Ž1.7. has an v-periodic strong solution uŽ t . satisfying u g H 2 Ž v ; Hs . l H 1 Ž v ; D Ž A . . l L`Ž v ; D Ž A . . l W 1, ` Ž v ; H0,1 s . .

NAVIER ] STOKES EQUATIONS

145

THEOREM 1.2 ŽUniqueness.. The solution of Ž1.6. ] Ž1.7. gi¨ en in Theorem 1.1 is unique. In Section 2, we consider the Sobolev inequality Žsee w1x., 5 u 5 L r Ž V . F C2 5 u 5 H b ,

1

1

Ž 1.10.

) 0.

Ž 1.11.

with g s Nr4 y 1r2.

Ž 1.12.

r

G

2

y

b

) 0,

if

N

and the inequality due to Giga and Miyakawa w4x, 5 u 5 L r Ž V . F C3 5 Ag u 5 ,

if

1 r

G

1 2

y

2g N

Here, we note that if r s N in Ž1.11. it follows 5 u 5 L N Ž V . F C3 5 Ag u 5

Further, we shall establish the uniform boundedness of the norm 5 Ag u nŽ t .5 for approximate solutions u n of the problem Ž1.6. ] Ž1.7.. In Section 3, we shall derive estimates of derivatives of higher order. For this purpose the boundedness obtained in Section 2 plays an important role. In Section 4, by standard compactness arguments we shall show the convergence of the approximate solutions, and hence we shall obtain the proofs of Theorems 1.1]1.2.

2. APPROXIMATE SOLUTIONS Firstly, we shall show the existence of approximate solutions of Ž1.6. ] Ž1.7. under the assumptions in Theorem 1.1. Let wi Ž i s 1, 2, . . . . be the completely orthonormal system in Hs consisting of the eigenfunctions of the Stokes operator A. We consider the system of ordinary differential equations,

Ž u nt q Au n q Pu n ? =u n , wi . s Ž f , wi .

Ž i s 1, 2, . . . , n . , Ž 2.1. n

unŽ t q v . s unŽ t . ,

where u n Ž t . s

Ý

c i n Ž t . wi .

Ž 2.2.

is1

Let Wn be the subspace of Hs spanned by w 1 , w 2 , . . . , wn . It is well known that for any ¨ nŽ t . s Ý nis1 d i nŽ t . wi g C 1 Ž v ; Wn . there exists a unique vperiodic solution u nŽ t . s Ý nis1 c i nŽ t . wi g C 1 Ž v ; Wn . of the linear equation

Ž u n t q Au n , wi . s Ž f y P¨ n ? =¨ n , wi . ,

i s 1, 2, . . . , n.

Ž 2.3.

146

HISAKO KATO

Moreover, we can see that the mapping F : ¨ n ª u n is continuous and compact in C 1 Ž v ; Wn .. Thereby, we shall prove the existence of the solution of Ž2.1. ] Ž2.2. by applying the Leray]Schauder fixed point theorem. To apply the fixed point theorem it is sufficient to show the boundedness sup 5 u n Ž t . 5 F C

Ž 2.4.

0FtF v

for all possible solutions of Ž2.1. ] Ž2.2. replaced by l Pu n ? =u n Ž0 F l F 1. instead of nonlinear term Pu n ? =u n , where C is a constant independent of l. In fact, multiplying Ž2.1. by c i nŽ t . and summing up over i we see

Ž u n t q Au n , u n . s Ž f y l Pu n ? =u n , u n . . Hence, from div u n s 0 and Ž1.12., we get Ž u n ? =u n , u n . s 0, and 1 d 2 dt

5 u n 5 2 q 5 =u n 5 2 s Ž f , u n . F 5 f 5 L 2 N rŽ Nq 2.Ž V . 5 u n 5 L 2 N rŽ Ny 2.Ž V . F C3 C Ž N . 5 f 5 L N r2 Ž V . 5 A1r2 u n 5 ,

Ž 2.5.

where C Ž N . ' < V < Ž Ny2.r2 N and < V < ' the volume of V. Therefore, we get d dt

5 u n 5 2 q 5 =u n 5 2 F C32 C Ž N . 2 M 2 ,

Ž 2.6.

where M is defined as Ž1.9. in Theorem 1.1. Furthermore, considering the periodicity of u n and integrating Ž2.6. over w0, v x we get v

H0

5 =u n 5 2 dt F C32 C Ž N . 2 M 2v .

Ž 2.7.

By using the spectral representation A s Hm`l dEl Ž m ) 0: the smallest eigenvalue of A., we obtain the inequality 5 A a u n 5 F m ay b 5 A b u n 5

Ž0 F a F b . .

Ž 2.8.

Hence, from Ž2.7. there exists t* g w0, v x such that 5 u n Ž t* . 5 2 F my1 5 =u n Ž t* . 5 2 F C32 C Ž N . 2 my1 M 2 .

Ž 2.9.

NAVIER ] STOKES EQUATIONS

147

Integrating Ž2.6. again from t* to t q v Ž t g w0, v x. we can see 2 2 sup 5 u n Ž t . 5 2 F 2C32 C Ž N . M 2v q C32 C Ž N . my1 M 2 ,

Ž 2.10.

0FtF v

where the right-hand side of Ž2.10. is independent of l and n. Thus, we have proved the existence of the solution u n g C 1 Ž v ; Wn . of Ž2.1. ] Ž2.2.. Before starting to prove our lemma on the uniform boundedness of 5 Ag u nŽ t .5 Žg s Nr4 y 1r2., we note that we can choose the basis  wi ; i s 1, 2, . . . 4 such that the eigenfunctions wi of A are also eigenfunctions of Ag and that we can write Awi s m i wi ,

Ag wi s m ig wi

with the eigenvalue m i of A. Ž 2.11.

Next we shall show the following lemma with an idea. Let u nŽ t . be the solution of Ž2.1. ] Ž2.2. gi¨ en abo¨ e. Suppose

LEMMA 2.1. that

with K 0 s min  Ky2 , 1 4 ,

M - K0

Ž 2.12.

and K ' CŽ N .

y1

my g q C1C3 C Ž N . m gy1r2 .

Then we ha¨ e 5 Ag u n Ž t . 5 F C3 C Ž N . m gy1r2 M 1r2

for any t g Ž y`,q ` . . Ž 2.13.

Proof. Considering Ž2.1. ] Ž2.2. and Ž2.11. we see

Ž u nt q Au n , A2g u n . s Ž f y Pu n ? =u n , A2g u n . , and further 1 d 2 dt

5 Ag u n 5 2 q 5 AŽ1q2g .r2 u n 5 2 F 5 f 5 L N r2 Ž V . 5 A2g u n 5 L N rŽ Ny 2.Ž V . q Ž AŽ2gy1.r2 Pu n ? =u n , AŽ1q2g .r2 u n . F C3 M 5 AŽ1q2g .r2 u n 5 q C1 5 Ag u n 5 5 AŽ1q2g .r2 u n 5 2 ,

Ž 2.14.

where we applied Ž1.8. in Proposition 1.1 and Ž1.11.. By Ž2.8. and Ž2.9. we see 5 Ag u n Ž t* . 5 F m gy1r2 5 =u n Ž t* . 5 F C3 C Ž N . m gy1r2 M,

148

HISAKO KATO

and hence we get 5 Ag u n Ž t . 5 - C3 C Ž N . m gy1r2 M 1r2

at t s t*

by assuming that M - 1. Thus, we set T * s sup  T N 5 Ag u n Ž t . 5 F C3 C Ž N . m gy1r2 M 1r2 for any t g t*, T . 4 . Here, we take notice of the order of M. Then, we get T * s `. In fact, if T * Ž t* - T *. is finite it should follow that 5 Ag u n Ž t . 5 F C3 C Ž N . m gy1r2 M 1r2

for any t g t*, T * . ,

and 5 Ag u n Ž T * . 5 s C3 C Ž N . m gy1r2 M 1r2 .

Ž 2.15.

Therefore, for such a value t s T *, the estimates of the right-hand side of Ž2.14. are C3 M 5 AŽ1q2g .r2 u n Ž t . 5 F CŽ N .

y1

m1r2y g 5 Ag u n Ž t . 5 M 1r2 5 AŽ1q2g .r2 u n Ž t . 5 ,

and C1 5 Ag u n Ž t . 5 5 AŽ1q2g .r2 u n Ž t . 5 2 F C1C3 C Ž N . m gy1r2 M 1r2 5 AŽ1q2g .r2 u n Ž t . 5 2 , where Ž2.15. was used. Hence, the estimate Ž2.14. implies 1 d 2 dt

5 Ag u n Ž t . 5 2 q 5 AŽ1q2g .r2 u n Ž t . 5 2 F KM 1r2 5 AŽ1q2g .r2 u n Ž t . 5 2

Ž 2.16.

with K defined as Ž2.12., since the inequality 5 Ag u n 5 F my1 r2 5 AŽ1q2g .r2 u n 5 is satisfied. Accordingly, from the assumption of this lemma it follows that d dt

5 Ag u n Ž t . 5 2 - 0

at t s T *.

Ž 2.17.

NAVIER ] STOKES EQUATIONS

149

Thus, in a neighborhood of t s T * it follows 5 Ag u n Ž t . 5 F C3 C Ž N . m gy1r2 M 1r2

for any t g T *, T * q d . ,

which implies T * s `. Moreover, this gives 5 Ag u n Ž t . 5 F C3 C Ž N . m gy1r2 M 1r2

for any t g Ž y`, q` . ,

because of the periodicity of u nŽ t .. Consequently, the proof of Lemma 2.1 is complete.

3. ESTIMATES OF DERIVATIVES OF HIGHER ORDER To show the convergence of the approximate solutions we shall derive estimates of derivatives of higher order. Firstly, we recall Lemma 2.1, namely, that if M is sufficiently small the approximate solutions satisfy sup 5 Ag u n Ž t . 5 F C Ž M .

with g s Nr4 y 1r2,

Ž 3.1.

t

where C Ž M . denotes a constant depending on M and independent of n. Moreover, we see easily that if M is sufficiently small then C Ž M . - C for any positive constant C. Hereafter, we shall use the fact. LEMMA 3.1.

Let u nŽ t . be the solutions of Ž2.1. ] Ž2.2. gi¨ en abo¨ e. Set

M0 '

ž

1r2

v

H0 5 f 5

2

dt

/

,

M1 '

ž

1r2

v

H0 5 f 5 t

2

dt

/

.

Then, we ha¨ e sup 5 =u n Ž t . 5 F C Ž M0 , M . , t

sup 5 u nt Ž t . 5 F C Ž M0 , M1 , M . , t

where C Ž M0 , M . and C Ž M0 , M1 , M . denote constants depending on Mi and M Ž i s 0, 1. and independent of n. Proof. From Ž2.1. ] Ž2.2., similarly, we see

Ž u nt q Au n , Au n . s Ž f y Pu n ? =u n , Au n . ,

150

HISAKO KATO

and 1 d 2 dt

5 =u n 5 2 q 5 Au n 5 2 F 5 f 5 5 Au n 5 q C1 5 Ag u n 5 5 Au n 5 2 F 5 f 5 5 Au n 5 q C1C Ž M . 5 Au n 5 2 ,

Ž 3.2.

where we used Ž3.1. and Ž1.8. in Proposition 1.1. By integrating Ž3.2. over w0, v x we get v

H0

5 Au n 5 2 dt F M0

ž

v

H0

1r2

5 Au n 5 2 dt

/

q C1 C Ž M .

v

H0

5 Au n 5 2 dt,

because of the periodicity of =u nŽ t .. Seeing that C1C Ž M . - 1 we obtain v

H0

5 Au n 5 2 dt F C Ž M0 , M . .

Ž 3.3.

By Ž3.3. there exists t* g w0, v x such that 5 =u n Ž t* . 5 2 F my1 5 Au n Ž t* . 5 2 F my1

C Ž M0 , M .

v

,

and by integrating Ž3.2. from t* to t q v Ž t g w0, v x. we have easily sup 5 =u n Ž t . 5 F C Ž M0 , M . ,

Ž 3.4.

t

where C Ž M0 , M . is independent of n. From Eqs. Ž2.1. ] Ž2.2. again we see

Ž u n t q Au n , u nt . s Ž f y Pu n ? =u n , u nt . , and 5 unt 5 2 q

1 d 2 dt

5 =u n 5 2

F 12 5 f 5 2 q 12 5 u nt 5 2 q 5 u n 5 L N Ž V . 5 =u n 5 5 u nt 5 L 2 N rŽ Ny 2.Ž V . , F 12 5 f 5 2 q 12 5 u n t 5 2 q C2 C3 5 Ag u n 5 5 =u n 5 5 =u nt 5 ,

Ž 3.5.

NAVIER ] STOKES EQUATIONS

151

where we applied the Sobolev inequality Ž1.10. and Ž1.12.. In the same way, by integrating Ž3.5. over w0, v x we see v

H0 5 u

nt

5 2 dt F M02 q C Ž M0 , M .

v

H0 5 =u

nt

5 dt.

Ž 3.6.

Moreover, differentiating Eq. Ž2.1. in t, multiplying by the derivative cXi nŽ t . defined as Ž2.2., and summing up over i, we get

Ž u nt t q Au nt , u n t . s Ž f t y Pu n t ? =u n y Pu n ? =u nt , u nt . .

Ž 3.7.

Noticing that div u n s 0, and the interpolation inequality 5 As ¨ 5 F C4 5 Aa ¨ 5 l 5 A b ¨ 5 1y l

Ž¨ g DŽ A b . . ,

for s s al q b Ž 1 y l . , 0 - a - s - b Ž l G 0 . ,

Ž 3.8.

we get 1 d 2 dt

5 u nt 5 2 q 5 =u nt 5 2 F my1 r2 5 f t 5 5 =u nt 5 q C2 C3 5 Ag u n t 5 5 =u n 5 5 =u nt 5 F

1 2m

5 ft 5 2 q

1 2

5 =u nt 5 2 q C2 C3 C4 5 =u n 5 5 u nt 5 1y2g 5 =u nt 5 1q2g . Ž 3.9.

In a similar way, we see v

H0 5 =u

nt

5 2 dt

F my1 M12 q 2C2 C3 C4

v

H0 5 =u

n

5 5 u nt 5 1y 2g 5 =u nt 5 1q2g dt. Ž 3.10.

Now, since g s 1r4 for N s 3, the inequality Ž3.10. with Ž3.6. implies the inequality v

H0 5 =u

nt

5 2 dt v

F C Ž M1 . q C Ž M 0 , M .

H0

F C Ž M1 . q C Ž M 0 , M .

ž

5 u n t 5 1r2 5 =u nt 5 3r2 dt v

H0

1r4

5 =u n t 5 dt

/ ž

v

H0

3r4

5 =u nt 5 2 dt

/

.

Ž 3.11.

152

HISAKO KATO

As for the second term of the right-hand side in the above inequality, v

H0

5 =u n t 5 dt F v 1r2

ž

1r2

v

H0 5 =u

5 2 dt nt

/

is satisfied. Thereby, the estimate Ž3.11. gives v

H0

5 =u n t 5 2 dt F C Ž M0 , M1 , M .

½H

v

0

7r8

5 =u nt 5 2 dt

5

,

which implies the boundedness v

H0

5 =u nt 5 2 dt F C Ž M0 , M1 , M .

Ž N s 3. .

Ž 3.12.

On the other hand, for the reason that g s 1r2 for N s 4 and 5 Ag u n 5 s 5 =u n 5, the inequality Ž3.10. gives v

H0

5 =u n t 5 2 dt F C Ž M1 . q 2C2 C3 C4 C Ž M .

v

H0

5 =u nt 5 2 dt.

Ž 3.13.

In the above inequality, seeing that 2C2 C3 C4 C Ž M . - 1, we get the estimate Ž3.12. for N s 4 also. Hence, similarly, there exists t* g w0, v x such that 5 u n t Ž t* . 5 2 F my1 5 =u n t Ž t* . 5 2 F

1

mv

C Ž M 0 , M1 , M . .

Consequently, integrating Ž3.9. from t* to t q v Ž t g w0, v x. we find sup 5 u nt Ž t . 5 F C Ž M0 , M1 , M . .

Ž 3.14.

t

This completes the proof of Lemma 3.1. Furthermore, we will show the following lemma. LEMMA 3.2. ha¨ e

Let u nŽ t . be the approximate solutions gi¨ en abo¨ e. Then, we sup 5 Au n Ž t . 5 F C Ž M0 , M1 , M . ,

Ž 3.15.

t

sup 5 =u n t Ž t . 5 F C Ž M0 , M1 , M . ,

Ž 3.16.

t

v

H0

5 Au nt 5 2 dt F C Ž M0 , M1 , M . ,

v

H0

5 u nt t 5 2 dt F C Ž M0 , M1 , M . .

Ž 3.17. Ž 3.18.

NAVIER ] STOKES EQUATIONS

153

Proof. Similarly, from Ž2.1. ] Ž2.2. we get

Ž u nt q Au n , Au n . s Ž f y Pu n ? =u n , Au n . , and 5 Au n 5 2 F 5 u nt 5 5 Au n 5 q 5 f 5 5 Au n 5 q 5 Pu n ? =u n 5 5 Au n 5 . Applying Ž1.8. in Proposition 1.1 and Ž3.1. we see 5 Pu n ? =u n 5 F C1 5 Ag u n 5 5 Au n 5 F C1C Ž M . 5 Au n 5 . From this, we find also 5 Au n Ž t . 5 F C Ž M0 , M1 , M . . Moreover, by differentiating Eq. Ž2.1. and making the scalar product in Hs with Au nt we see 1 d 2 dt

5 =u n t 5 2 q 5 Au n t 5 2 F 5 f t 5 5 Au nt 5 q 5 u n t 5 L 2 N rŽ Ny 2.Ž V . 5 =u n 5 L N Ž V . 5 Au nt 5 q 5 u n 5 L 2 N rŽ Ny 2.Ž V . 5 =u n t 5 L N Ž V . 5 Au nt 5 .

Therefore, by using Ž1.10. and Ž1.12. we see 1 d

5 =u nt 5 2 q 5 Au nt 5 2 F

2 dt

1 2

5 ft 5 2 q

1 2

5 Au nt 5 2

q C2 C3 5 =u n t 5 5 Agq1r2 u n 5 5 Au nt 5 q C2 C3 5 =u n 5 5 Agq1r2 u nt 5 5 Au nt 5 . Ž 3.19. As for the estimate Ž3.19., moreover, we can see that, in the case N s 3, d dt

5 =u n t 5 2 q 5 Au n t 5 2 F C Ž M1 . q C Ž M0 , M1 , M . 5 =u nt 5 5 Au nt 5 q C Ž M0 , M . 5 =u nt 5 1r2 5 Au nt 5 3r2 , Ž 3.20.

where we used the interpolation inequality Ž3.8.; and in the case N s 4, d dt

5 =u n t 5 2 q 5 Au n t 5 2 F C Ž M1 . q C Ž M0 , M1 , M . 5 =u nt 5 5 Au nt 5 q 2C2 C3 C Ž M . 5 Au nt 5 2 .

Ž 3.21.

154

HISAKO KATO

By applying the Young inequality ab F a prp q b qrq Ž1rp q 1rq s 1. to the right-hand side of Ž3.20. and noticing 2C2 C3 C Ž M . - 1 in Ž3.21., we obtain the desired estimate Ž3.17.. Thus, by a similar discussion we also get the boundedness Ž3.16.. Lastly, we get the following equation from Eqs. Ž2.1. ] Ž2.2.:

Ž u nt t q Au n t , u nt t . s Ž f t y Pu nt ? =u n y Pu n ? =u nt , u nt t . . As for the nonlinear term of this equation we obtain < Ž Pu n t ? =u n , u n t t . < F C Ž M0 , M1 , M . 5 =u nt 5 5 u nt t 5 , < Ž Pu n ? =u nt , u nt t . < F C Ž M0 , M1 , M . 5 Au nt 5 5 u nt t 5 , where we used the boundedness of 5 Au nŽ t .5 Ž n s 1, 2, . . . .. Thus, we can see Ž3.18.. Consequently, the proof of Lemma 3.2 is complete.

4. PROOF OF THEOREMS 1.1]1.2 Firstly, we shall show the convergence of the approximate solutions u nŽ t . obtained above. Since the estimates in Lemma 2.1, Lemma 3.1, and Lemma 3.2 are valid, standard compactness arguments imply that there exists a subsequence u nŽ t . tending to a function uŽ t . in such a way u n ª u weakly* in L`Ž v ; D Ž A . .

Ž 4.1.

u n ª u strongly in L`Ž v ; D Ž A1r2 . .

Ž 4.2.

u n t ª u t weakly* in L`Ž v ; D Ž A

Ž 4.3.

1r2

..

u n t ª u t strongly in L`Ž v ; Hs . ,

Ž 4.4.

and the function uŽ t . satisfies u g H 2 Ž v ; Hs . l H 1 Ž v ; D Ž A . . l L`Ž v ; D Ž A . . l W 1, ` Ž v ; H0,1 s . . Here, Ž4.1. ] Ž4.3. are evident, and hence it is sufficient to show the convergence Ž4.4.. In fact, the sequence < Ž u nt Ž t ., wi . < Ž n s i, i q 1, i q 2, . . . . is uniformly bounded and equicontinuous,

Ž u nt Ž t q h . y u nt Ž t . , wi .

F C Ž M0 , M1 , M . < h < 1r2 5 wi 5 ,

where the wi Ž i s 1, 2, 3, . . . . is the completely orthonormal system in Hs consisting of the eigenfunctions of A as mentioned above. Therefore, using the diagonal process we can finally select a subsequence u nt Ž t . such

NAVIER ] STOKES EQUATIONS

155

that u nt Ž t . converges weakly and uniformly in t g w0, v x to an element in Hs . Furthermore, considering the boundedness Ž3.16. in Lemma 3.2 we obtain the convergence Ž4.4.. Next, considering the above lemmas we see that Pu ? =u is well defined, and 5 Pu n ? =u n y Pu ? =u 5 F 5 P Ž u n y u . ? =u n 5 q 5 Pu ? = Ž u n y u . 5 F C1 5 Agq1r4 Ž u n y u . 5 5 A3r4 u n 5 q 5 Agq1r4 u 5 5 A3r4 Ž u n y u . 5 4 F C Ž M0 , M .  5 u n y u 5 3r4y g q 5 u n y u 5 1r4 4 ª 0,

as n ª `, uniformly in t ,

where we used Ž1.8. and Ž3.8.. Consequently, we see that

Ž u t q Au q Pu ? =u, wi . s Ž f , wi . ,

i s 1, 2, 3, . . . ,

t g Ž y`, q` . .

Ž 4.5. We also find that this relation is valid for any w g Hs , owing to the estimates obtained in the previous sections. Since Pf s f for any f g Hs , we get Ž1.6. ] Ž1.7., u t q Au q Pu ? =u s f

for any t g R1 ,

uŽ t q v . s uŽ t . . Thus, the proof of Theorem 1.1 is complete. Lastly, we shall show Theorem 1.2 under the same assumption of Theorem 1.1. Let u and ¨ be the solutions of the problem Ž1.6. ] Ž1.7.. We put w s u y ¨ . Then, it follows dw dt

q Aw q P Ž u y ¨ . ? =u q P¨ ? = Ž u y ¨ . s 0,

and dw

Ž

dt

q Aw, w. s y Ž Pw ? =u, w . ,

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because of Ž P¨ ? =w, w . s 0. Hence, it follows 1 d 2 dt

5 w 5 2 q 5 =w 5 2 s Ž Pw ? =w, u . s Ž Ayg Pw ? =w, Ag u . F C1 5 =w 5 2 5 Ag u 5 F C1C Ž M . 5 =w 5 2

Ž 4.6.

by using Ž1.8. and Ž3.1.. Since C1C Ž M . - 1, it follows d dt

5 w 5 2 F 2 Ž C1C Ž M . y 1 . m 5 w 5 2 s yL 5 w 5 2 ,

where L ' 2Ž1 y C1C Ž M .. m ) 0. Hence, it follows that 5 w Ž t . 5 2 F 5 w Ž 0 . 5 2 exp Ž yLt .

for any t g Ž 0, ` . .

Since w Ž t . is periodic in t, for any t g Žy`, q`. there exists a positive integer n 0 such that t q n 0 v ) 0 and 5 w Ž t .5 2 s 5 w Ž t q n 0 v .5 2 . Hence, it follows 5 w Ž t . 5 2 F 5 w Ž 0 . 5 2 exp Ž yLn v .

Ž n G n0 . ,

which implies 5 w Ž t .5 ' 0. The proof of Theorem 1.2 is complete.

REFERENCES 1. R. A. Adams, ‘‘Sobolev Spaces,’’ Academic Press, New York, 1975. 2. H. Fujita and T. Kato, On the Navier-Stokes initial value problem 1, Arch. Rational Mech. Anal. 16 Ž1964., 269]315. 3. D. Fujiwara and H. Morimoto, An L r-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Uni¨ . Tokyo 24 Ž1977., 685]700. 4. Y. Giga and T. Miyakawa, Solutions in L r of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 Ž1985., 267]281. ¨ 5. E. Hopf, Uber die Anfangswertaufgabe fur ¨ die hydrodynamischen Grundgleichungen, Math. Nachr. 4 Ž1951., 213]231. 6. S. Ito, The existence and the uniqueness of regular solution of non-stationary NavierStokes equations, J. Fac. Sci. Uni¨ . Tokyo 9 Ž1961., 103]140. 7. S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes equations, Arch. Rational Mech. Anal. 24 Ž1967., 302]369. 8. H. Kozono and M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, preprint. 9. O. A. Ladyzhenskaia, ‘‘The Mathematical Theory of Viscous Incompressible Flow,’’ English translation, Gordon & Breach, New York, 1969.

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10. J. L. Lions, ‘‘Quelque Methodes de Resolution des Problemes aux Limites Non Lineaires,’’ ´ ` ´ Dunod, Paris, 1969. 11. J. L. Lions and G. Prodi, Un theoreme ´ ` d’existence et d’unicite´ dans les ´equations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris 248 Ž1959., 3519]3521. 12. K. Masuda, Weak solutions of Navier-Stokes equations, Tohoku Math. J. 36 Ž1984., ˆ 623]646. 13. J. Serrin, The initial value problem for the Navier-Stokes equations, in ‘‘Nonlinear Problem’’ ŽR. E. Langer, Ed.., pp. 69]98, Univ. of Wisconsin Press, Madison, 1963. 14. A. Takeshita, On the reproductive property of the 2-dimensional Navier-Stokes equations, J. Fac. Sci. Uni¨ . Tokyo 16 Ž1970., 297]311. 15. R. Temam, ‘‘Navier-Stokes Equations,’’ North-Holland, Amsterdam, 1977. 16. W. von Wahl, ‘‘The Equations of Navier-Stokes and Abstract Parabolic Equations,’’ Vieweg, Braunschweig, 1985.