Contributions to Current Challenges in Mathematical Fluid Mechanics (Advances in Mathematical Fluid Mechanics) [1 ed.] 3764371048, 9783764371043 [PDF]

Zitiervorschau

Advances in

Mathematical Fluid Mechanics

Contributions to Current Challenges in Mathematical Fluid Mechanics

ZZPA O"M Nup"A"M

Giovanni P. Galdi John G. Heywood

Rolf Rannacher Editors

Birkhauser

m 99

Advances in Mathematical Fluid Mechanics Series Editors Giovanni P. Galdi

John G. Heywood

Rolf Rannacher

School of Engineering

Department of Mechanical

Department of Mathematics University of British Columbia

Engineering

Vancouver BC

Institut fur Angewandte Mathematik Universitiit Heidelberg Im Neuenheimer Feld 293/294

University of Pittsburgh 3700 O'Hara Street Pittsburgh, PA 15261 USA

Canada V6T 1 Y4

69120 Heidelberg

e-mail: heywood*math.ubc.ca

Germany

e-mail: rannacher®iwr.uni-heidelberg.de

e-mail: galdicengmg.pitt.edu

Advances in Mathematical Fluid Mechanics is a forum for the publication of high quality monographs, or collections of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its mathematical aims and scope are similar to those of the Journal of Mathematical Fluid Mechanics. In particular mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on fluid mechanics.

The monographs and collections of works published here may be written in a more expository style than is usual for research journals, with the intention of reaching a wide audience. Collections of review articles will also be sought from time to time.

Contributions to Current Challenges in Mathematical Fluid Mechanics Giovanni P. Galdi John G. Heywood Rolf Rannacher Editors

Birkhauser Verlag Basel Boston Berlin

Editors:

Giovanni P. Galdi

John G. Heywood

School of Engineering Department of Mechanical Engineering University of Pittsburgh

Department of Mathematics University of British Columbia

3700 O'Hara Street

Vancouver BC Canada V6T 1Y4

Pittsburgh, PA 15261

e-mail: [email protected]

USA e-mail: [email protected]

Rolf Rannadter Institut fur Angewandte Mathematik Universitat Heidelberg Im Neuenheimer Feld 293/294 69120 Heidelberg Germany e-mail: rannacher®iwr.uni-heidelberg.de

2000 Mathematical Subject Classification 76D05, 35Q30, 76N10

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ISBN 3-7643-7104-8 Birkhauser Verlag, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. 0 2004 BirkhSuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland, Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF Printed in Germany ISBN 3-7643-7104-8

987654321

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Contents Preface ...................................................................... vii

A. Biryuk

On Multidimensional Burgers Type Equations with Small Viscosity 1. Introduction ................................................................1 2. Upper estimates ............................................................ 4 3. Lower estimates ............................................................. 7 4. Fourier coefficients ......................................................... 17

5. Low bounds for spatial derivatives of solutions of the Navier Stokes system.27 References .................................................................29

D. Chae and J. Lee On the Global Well-posedness and Stability of the Navier-Stokes and the Related Equations 1. Introduction ............................................................... 31

2. Littlewood-Paley decomposition ........................................... 33 3. Proof of Theorems ......................................................... 36

Appendix ..................................................................47 References .................................................................49

A. Dunca, V. John and W.J. Layton The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain

1. Introduction ............................................................... 53 2. The space averaged Navier-Stokes equations in a bounded domain .......... 55

3. The Gaussian filter ........................................................ 58 4. Error estimates in the (LP(Rd))d-norm of the commutation error term ..... 59 5. Error estimates in the (H-I (ft))`r norm of the commutation error term .... 66 6. Error estimates for a weak form of the commutation error term ............ 68 7. The boundedness of the kinetic energy for u in some LES models .......... 72 References .................................................................76

T. Hishida

The Nonstationary Stokes and Navier-Stokes Flows Through an Aperture 1. Introduction ............................................................... 79

2. Results ....................................................................85 3. The Stokes resolvent for the half space ..................................... 90

4. The Stokes resolvent ....................................................... 96

5. LQ-L' estimates of the Stokes semigroup .................................. 103 6. The Navier-Stokes flow ................................................... 112

References ................................................................120

vi

Contents

T. Leonavi&ene and K. Pileckas Asymptotic Behavior at Infinity of Exterior Three-dimensional Steady Compressible Flow 1. Introduction .............................................................. 125 2. Function spaces and auxiliary results ...................................... 129

3. Stokes and modified Stokes problems in weighted spaces .................. 132

4. Transport equation and Poisson-type equation ............................ 140 5. Linearized problem ....................................................... 143

6. Nonlinear problem ........................................................ 147 References ................................................................150

Preface

This volume consists of five research articles, each dedicated to a significant topic in the mathematical theory of the Navier-Stokes equations, for compressible and incompressible fluids, and to related questions. All results given here are new and represent a noticeable contribution to the subject. One of the most famous predictions of the Kolmogorov theory of turbulence is the so-called Kolmogorov Obukhov five-thirds law. As is known, this law is heuristic and, to date, there is no rigorous justification. The article of A. Biryuk

deals with the Cauchy problem for a multi-dimensional Burgers equation with periodic boundary conditions. Estimates in suitable norms for the corresponding solutions are derived for "large" Reynolds numbers, and their relation with the Kolmogorov-Obukhov law are discussed. Similar estimates are also obtained for the Navier-Stokes equation. In the late sixties J. L. Lions introduced a "perturbation" of the Navier-Stokes equations in which he added in the linear momentum equation the hyperdissipative term (-0)`3u, fi > 5/4, where A is the Laplace operator. This term is referred to as an "artificial" viscosity. Even though it is not physically motivated, artificial viscosity has proved a useful device in numerical simulations of the Navier-Stokes equations at high Reynolds numbers. The paper of of D. Chae and J. Lee investigates the global well-posedness of a modification of the Navier Stokes equation similar to that introduced by Lions, but where now the original

dissipative term -Au is replaced by (-0)°u, 0 < a < 5/4. Existence, uniqueness and stability of solutions is proved in appropriate Besov spaces depending on the parameter a. Space averaged Navier--Stokes equations are the basic equations for large eddy simulation of turbulent flows. In deriving these equations it is tacitly understood that differentiation and averaging operations can be interchanged. Actually, this procedure introduces a "commutation error" term that is typically ignored. The main objective in the paper of A. Dunca, V. John and W. L. Layton is to furnish a characterization of this term to be neglected. The authors go on to provide a

justification for neglecting this term if and only if the Cauchy stress vector of the underlying flow is identically zero on the boundary of the domain. In other words, neglecting the commutation error is reasonable only for flows in which the boundary exerts no influence on the flow. Since the appearence of the paper of J. G. Heywood in the mid-seventies, the problem of a flow through an aperture (the "aperture domain" problem) has

attracted the attention of many researchers. But even now, a number of basic questions remain unresolved.The article of T. Hishida provides a further, significant contribution. Specifically, the author proves Lv - L'' estimates for the Stokes semigroup in an aperture domain of ]R n > 3. These estimates are then used to

viii

Preface

show the existence, uniqueness and asymptotic behavior in time of strong solutions of the Navier Stokes equations having "small" initial data in L" and zero flux through the aperture. The mathematical analysis of the well-posedness of the Navier-Stokes equations in the case of a compressible viscous fluid is a relatively new branch of mathematical fluid mechanics, its first contribution dating back to a paper of J. Nash in the early sixties. Many problems remain to be solved in this area, despite the significant contributions of many mathematicians. In particular, there remain very interesting problems concerning steady flow in an exterior domain, especially regarding the asymptotic behavior of solutions. This latter problem is analyzed in the paper of T. Leonaviciene and K. Pileckas, in the case when the velocity of the fluid is zero at large distances and the body force is the sum of an "arbitrary large" potential term and a "small" non-potential term. We would like to express our warm thanks to Professors H. Beiriio da Veiga, A. Fursikov and Y. Giga who recommended the publication of these articles. Giovanni P. Galdi John G. Heywood Rolf Rannacher

Advances in Mathematical Fluid Mechanics, 1-30 © 2004 13irkhi(user Verlag Basel/Switzerland

On Multidimensional Burgers Type Equations with Small Viscosity Andrei Biryuk Abstract. We consider the Cauchy problem for a multidimensional Burgers type equation with periodic boundary conditions. We obtain upper and lower bounds for derivatives of solutions for this equation in terms of powers of the viscosity and discuss how these estimates relate to the Kolmogorov-Obukhov spectral law. Next we use the estimates obtained to get certain bounds for derivatives of solutions of the Navier-Stokes system. Mathematics Subject Classification (2000). 351310, 35A30, 76D05. Keywords. Kolmogorov-Obukhov spectral law, hounds for derivatives, degen-

erate state.

1. Introduction We study the dynamics of m.-dimensional vector field u = u(t, x) on the ndimensional torus T" = x'/(fZ),. described by the equation

8=u + vf(u)u = v0u + h(t, x). Here v is a positive parameter ("the viscosity"), f : R"

(1.1)

R" is a smooth map, h is a smooth forcing term and Of(u) is the derivative along the vector f (u), i.e.,

of(u)u = (f(u) V)u. If in = n and f (u) = u, we have the usual forced Burgers equation. In a potential case (i.e., if the initial state uo(x) = u(0, x) and the field h are gradients of some functions) this equation can be reduced to a linear parabolic equation. As it is shown in [1], [12], appropriate bounds for derivatives imply estimates for averaged spectral characteristics of the flow. The purpose of this work is to obtain such bounds for solutions of the Cauchy problem for the generalised m-n multidimensional Burgers equation (1.1).

We describe notations used in this article. If v is a vector in R'. Z or C', then JvJ denotes its Euclidean (Hermitian) norm

_1

If we have to

A. Biryuk

2

stress the dimension, we denote the norm in R' as IviRa. etc. By B(r) we denote the ball of radius r centered at the origin. If A : IIY'1 -, iIP'2 is a linear map then IIAII denotes the operator-norm of this map associated with the Euclidean norms ' I on 18'1 and II812. If v = v(x), then we write I

1/2

(1.2)

IvI = sup Iv(x)I = sup (E Ivi(x)I2) x

x

Sometimes we will denote this norm by ' IL_. If v = v(t,x), then IvI = sup. Iv(t, x) I is a function of t. For a multi-index a we denote Ial = E jail. I

We set (1.3)

H(t) = f , sup Ih(r,x)IR,., dr. 0 WET"

We also denote Ak

F

sup

(

Im

{uER

F_ Iau

mar}

2

1/2

,

fA

1

and m

IIuIIk =Tr.f U,(-°)ku,dx = i

1

(I r') IDavi1L2(Tf)

i=1 M=k rn I

dku

L f3T.j, ... Tjk

I2 L2(Tn).

,=1 31...,3k=1 Inl ) _ al ar,. 0 is an integer and (hI) Here k a _ nl,...,a" in the generalised binomial expansion (x1 + x2 + ... + x )k u = u(t.x), then IIuIIk = Ilu(t, -)Ilk is a function of t.

coefficients (

)xa

. If

Our main results are stated in the following two theorems, where u(t, x), t 0, is any smooth solution for the equation (1.1):

Theorem 1. For any k

0, t > 0, and v > 0 we have

Ilu(t, )Ilk

(Bk(t) + 1)" IIuIII'

v

and IIuIIk >

IIhIIk-l,

then IIuIIk is decreasing.. (2.10)

We denote the right hand side of (1.6) by Fk(t). It is clear from the definition of the function Fk that 11u(0, )IIk Fk(t) then IIuIIk is decreasing by argument (2.10). Since Fk(t) is a non-decreasing function we obtain that IIuIIk never can be greater than Fk(t). We arrive at (1.6). Theorem 1 is proven.

3. Lower estimates In this section we prove Theorem 2. Throughout this section we use standard facts from linear algebra about linear transformations. For the convenience of the reader we very briefly outline the proofs. See reference [7], for an elegant, coordinate free presentation. We start from brief discussing of the notion of the degeneracy of a vector field.

3.1. Degeneracy condition

Using the fact that an n x n matrix A is nilpotent if A" = 0, we can give a definition of degeneracy that is equivalent to the previous one, but more robust. Definition 2. The vector field uf1 is degenerate if

(()f(d(,(X)) n

- 0.

Let j (x) = f (ull(x)). Consider the characteristic polynomial of the matrix

2: xx(A) = det(81 - Al) =

(-A)" + (-A)"-1Il(x) + ... + In(x)

Expanding the determinant, we obtain 8s,i

.. -

Ax,k

...

f)xk

det

Ik (x) 1')I LW dT > .la r(T, 1

T

l

k/2

f(Iuol +H(T))

/l

dT

k

2

k/2

('1. 82u

(T Jo I

dT)

a-IT

(3.8)

(2nTv)

4-2

k/2

-2

v 3(IuoI + H(T))

v 3(Iuol + H(T))

This concludes the proof of (1.7) for k > 2 with the constants (2nf.T)

rk =

k_2

f(IuoI + H(T)) -

(3.11)

In the next subsection we will specify the values of T and c (see (3.16) and (3.33) respectively). 3.3. Technicalities In this subsection we introduce a more general approach to the estimates of Theorem 2 which applies to the non-periodic case. From the previous subsection we already know that the crucial condition for Theorem 2 is the "negativity" rather than the non-degeneracy of the matrix Of( m ) . Let u : [0, oo) x R' --+ IIYn' be a C'-smooth vector-valued function and f : II8' Rn be a C'-smooth map. In this subsection x = (x1, ... , xn) are coordinates in 1R'. W e define gi : (0, oo) x Il8n -. IR, (i = 1, ... , m) to satisfy: n

Otui+Efj(ul,...,urn)A =gi(t,x1,...,xn). j=1

(3.12)

Multidimensional Burgers Type Equations

13

Let j (x) = f (u(0, x)). If this function is C2-smooth then we consider the norm a2fi

fI2 = SUP

2

C7xjaxk

a i,j,k=1

(3.13)

For any u E Rm the derivative Vu f : IR' - R" is a linear map. For any domain E C 1R' we denote

lVufIE = sUPIIVufII = supmaxl(Vuf)vl, uEE ef" 1.1-1

(3.14)

where {(Vu f )v} j = E a vi. We note that for the Burgers' (=NS') nonlinearity (i.e., m = n and f u) we have IVu f I - 1. Theorem 5. Suppose that u : [0, oo) x IR"

IRand f : IR' - R" are C'-

smooth. Let e be a positive real number and l E { 1, 2. ... , n}. Suppose that there

exists x* E R" such that I, (x*) _ -e < 0 and that for k = 1 + 1, ... , n we have Ik(x*) = 0, where 1; are defined in (3.1). Let

IfI = max Ie (x*)I,

(3.15)

where j (x) = f (u(0, x)) and let T = 2n 11/2!11i-'

.

(3.16)

Then there exists a positive function c2(c,) such that if

f sup Ig(r,x)I dr < c, ,

(3.17)

zER"

then

I (T - r) sup Ig(r,x)I dr i c2(c,).

(3.18)

WER"

U

If the function f is C2-smooth and the norm (3.13) is finite, then one can take c2(cl) =

411 n2"-2(If

IIT)2n-21 IVuf

(3.19) *B(Iu0I+c, )If l2T .

Here B(r) denotes the ball in IR"' of radius r in the. Euclidean norm, centered at the origin.

Proof. First of all we note that f (u(t, x)) and a f (u(t, x)) are a continuous vector function and a continuous matrix function, respectively. It follows from this that 1° 3! solution of the Cauchy problem for the following ODE in R":

de7(t, t:) = f (u(t, V, , with the initial state 'y(0, t) = F E R". 2° This solution ^y E C' ([0, T] x IR"; IR")

(3.20)

A. Biryuk

14

3* This solution satisfies

det ( -1b 11

) = exp fo EL O

f; (u(t, y(t, C))

see [8].

The positiveness of the Jacobian in 3° implies inequality (3.18). The rest of this subsection is devoted to proving this fact.

In the proof of Proposition 1 below we will use the fact that there exists

a continuous second derivative 1This follows from the existence and continuity of the first partial derivative j f (u(t, x)). For the quantities (3.1) we have: IIk]

(3.21)

(A)02IfIi.

Indeed, the right hand side of (3.1) contains (k) terms and each of them is no greater than kk/2 If l i . Here we have used the fact that the volume of a k-dimensional

parallelepiped with sides of length less than or equal to f I f I, is no greater than

(fIfll)k.

In the proofs of Propositions 2 and 3 below we will use the inequality

Till, > 1.

(3.22)

It follows from (3.16) and (3.21) with k = I since e = Ili(x*)I. We fix T at the value given in (3.16). For any t, y(t, x) defines a mapping from 1R" into itself. We take t = T and decompose this mapping as follows: 'Y=(T,x) = pi (x) +qt(x),

i = 1,...,n.

(3.23)

Here p(x) comprises the zeroth and the first terms of the Tailor expansion:

pi(x)=y,(0,a)+ dty,(0,x)T=x+f(x)T.

(3.24)

The remainder of the Tailor expansion can be represented as T

qi(x) =f0 (T

d2

(3.25)

-r)dt2'y.(T,x)dr.

Proposition 1. For the Euclidean norm of the vector q we have:

J (T -T) sup

dT.

(3.26)

We recall that I I denotes the Euclidean norm.

Proof. Combining the chain rule with (3.20) and (3.12) we have

dtu'(t,y(t,x)) =9.(t,y(t,x))

Using this equality and assumption (3.17) we obtain that the function u takes values in the ball B(luol + ci) if t S T. We calculate the second derivative of y;

Multidimensional Burgers Type Equations

15

(for i = 1.... , n): 2

-ff(u(t,'Y(t,x))Lr = E

d

Ofi

jgj(T,'Y(T,x)) =I (VUf)g)j

i=1

From this formula we obtain y

I dtzl'(T,x)I < llvufllmlu"I+c,) sup lg(T,x)l zER'

Multiplying this inequality by T - T and using (3.25) we arrive at (3.26).

Consider the linearization of the map p at the point x* (we recall that x* is the point where the leading non zero IA is negative):

Pi(x) = p,(x*) +

fix` (x*)Axj + c,(nx). j=l where ox = x - x* and d,(nx) = o(nx).

(3.27)

We need to investigate the matrix

A=) -b,+T0

(3.28)

which is the linear part of the right hand side of (3.27).

Proposition 2. The determinant of the matrix A is negative and bounded away from zero:

det A < -2n-111/2(TIfI1)'-'.

(3.29)

Proof. The determinant is expressed by polynomial (3.4): det A = P. (T). Using (3.21) we have det A

()11/2Tlill + ... +

li)'-'


0 and a set TA c (0, 1] such that for any v E TA we have

`

()2P1.

SEEZZ"

Then for any positive real number y and any v E Tk we have IeIL>y

ag(V) < cky-21V-2r'(k)

Proof. For any positive real y we have Lia < y-2k 181>-Y

Using (4.1) we arrive at (4.2).

L 1812k&2. < y-2k 1: I8I2k52e 181>-Y

aEZ"

(4.1)

Multidimensional Burgers Type Equations

19

As a corollary, taking y = a)for any positive real numbers z and Al ,\2 < +oo and any v E Tk we have

F-

&. (V) < A I

2kCUV2kz-2p"(k).

Lemma 5. Suppose that there exist real numbers 0 < k1 < k < k2, p"(k)), p'(k), p"(k2), ck, > 0, ck > 0, ck, > 0 and sets Tk1, Tk, Tk2 c (0, 1] such that Is12k;a2 (Vl

c

()2P

i = 1,2

for any v E T",

(4.3)

QEZ"

and (v)2p(k)

1812ka2(,) > ck

for any v E T.

(4.4)

aEZ"

Then for any p E (0,1) and any real A < A(v) and B > B(v) where n

_

k-k1

\1 )2k-2k, A(V) _ 1(/a l2 fSTI

and f3(v) _ (?

11

fk

I

2k1 2k (1 `

P'1k2 -P fk)

V/

k2-k

and any v E T", n Tk n Tk2 we have 1812kaa > (1 - N) E IsI2kas aEZ"

A 0 we have 1812k,as \ A2k-2k, [: IBI2k,Q2 [ Is12Aae < A2k-2k, IaI 0 and b be any real numbers and Then the coefficients suppose that Sk(v) < and Sk(v) > dk(y)2k,,-26.

dk(V)2k7-1b

ae(v) could be as follows: v2b

a2(L)

0

for s = ([

],0,...,0),

otherwise.

To connect the results obtained with the turbulence theory we give a possible

rigorous definition of K -O type law and then obtain bounds for two the most important parameters of the law. Definition 4. We say that positive quantities aa(v) obey a K-O type spectral law if there exist positive real numbers v11, K1 < K2, x, c, C, C1 positive real functions a1(v) and a2(v) such that log(ai(v)) = o(log(v `)) as v 0 (and sup %(V) < 1 if K1 = K2) such that for any v E (0, vo) we have

Multidimensional Burgers Type Equations

21

62(v) < C(I)' for r > a (v) 2. ir-C, lim inf ' kk') condition 3 of Definition 4 holds if the value k2 in (4.10) is replaced with y. = lim p kk',) To do this we take a subsequence {kj, } such that lim inf Then

kk'') -?} = 0(1) as i

max{0,

oc.

(4.13)

Now applying (4.2) for each k = k, we obtain

a(v)

0 (and the functions and interpolation). It follows that K2 < y. Hence x.2 < lim inf p kk`) . 0 Lemma 9. a) Under the assumptions of Lemma 5 suppose that TA, n T' n T(2 3 0, and that the quantities ae(v) obey a K 0 type spectral law. Then p'(k) - p"(kI)

(4.15)

k - k, b) If, in addition, we have K1 < p (k)-p"(k') then

x

2p4+2k+1 ifp'(k)>0,

min r:z

ifp'(k)=0,

I+ 2k x < -2p'(k)

p'(k) < 0. y'(k)-p

Proof. a) We denote p_ = k-ki (k') and p+ = p kj-k (k). To prove (4.15) we 0 not faster show that for any e > 0 the sum cv_ _,, a2s (v) decays as v than some finite power of v.

A. Biryuk

24

Fix any e > 0. Then for small enough v such that v-(P--E) < A(v) (the quantities A(v) and B(v) are defined by (4.5) with p = 1/2) due to Lemma 5 we have

E ae(v) > IeI>v

(P

E

ae(v)

A(v) then it follows from Lemma 5 and the fact that the sum a;(v) decays faster then any power of v. The assumption icl < p- implies that for small enough v we have vl (v)v-", < A(v) and so, by the condition 1 of Definition 4, we have B, (v)

I8I2ka2 < Const

22k-xd2. A(v)

A(v) 0, we have (v)k-q... JIU11k2 > Knrk-q" T1 0 Here rk, q" and K,, is given by (3.11), (4.16) and (4.18), respectively.

(4.20)

Proof. i) Using (1.6) we arrive at (4.19) with 2k supj,dIhllA-, 2k 2 max{1, Hr o vo Ck = Rk(T) ek(o) vo }. ii) Using inequality (4.17), the Cauchy-Schwartz inequality and Theorem 2 we obtain:

7J IIIuIIkdt>KnTJ,IaL

>Knvk-9^.

Consider the orthonormal basis on L2(T"), formed by the exponents e,(.), 8 E Z", where

e,(x) = e exp The Fourier expansion of u(x) has the form

r u(t,x) _ >. f&,(t)e,(x), where u,(t) = J u(t,x)e,(x)dx. T

eEZ^

For the Sobolev quasinorm II

Ilk (1.5) we have IIuIIk = (-i )2A EaEZ" 181111 u,l2.

Suppose that the initial state uo is a non-degenerate vector field. Consider the corresponding solution of equation (1.1) and define the quantities &,2(v) by the formula

r

a; = 1

Jo

Iu.(t)I2dt,

(4.21)

where T = T(f, uo) is defined by (3.16). Then we have 2k 1

(27r) T, IIu'IIkdt.

eEZ"

Lemma 10 implies that for any k and any v E (0, vo) we have the following inequality Is12kaa(U) sEZ^

< (2:r)2k'k (ti)2k

(4.22)

A. Biryuk

26

Under the assumptions of the second part of Lemma 10 we see that for any v > 0 and any integer k > q,, + 2 there is the inequality: sEZ"

IsI2k&e(V) > Knrk-4 `1

\2k

//

lv)k-9o

(4.23)

We recall that qn = [L] ] + 1 (4.16).

Theorem 6. Let u = u'(t, x) be the solution of (1.1) with a non-degenerate initial state u,). Take T = T(f , uo) by (3.16) and consider the averaged Fourier coefficients &2. (v) given by (4.21). Suppose that the forcing term h in (1.1) satisfies the condition H(T) < , where c is given by the right hand side of (3.33). Then for any k > 2+qn, where qn is given by (4.16), there exists ck such that for any e > 0 and any small enough v we have Vk+2ke+q

(4.24)

&.2 (v) > con Stk

For any k > 0 and vo > 0 there exists Ck = Ck(vo) such that for any positive real numbers z and Al < )2i and any v < vo we have &e(V) < al a,( )'slsl5al( )'

2kCk V2k(z-1).

(4.25)

Besides, for any y > 0 we have (4.26)

E aa(v) < Ck(yV)-2k. lel>y

Proof. The quantities &2(v) satisfy inequalities (4.23) and (4.22). Hence we can

apply Lemmas 4 and 5 with p"(k) = k, p'(k) = k/2 - qn/2 for k > qn + 2, Choose k1 = 0 and k2 > k such that

c'A = (2r)2J Ck c'k = l

k2 - (k/2 - qn/2)

k2-k

where A(v) and B(v) are defined by (4.5) with p = 1/2. Now we apply Lemma 5 to obtain

[-`

(

)

IsI

2k-2

ae

I

((1

lek lv)

k-9n

-9A -

A. Biryuk

28

where R = Iu(t,')IL-

IucIL is the Reynolds number. It is clear that if To < oo, then oo as t

To.

Remark 1. Since the mean value of u is constant, then the spatial derivatives of u t.blow tip with Iu(t, )IL-. Due to this reason we assume that max Iaku Lm azj .1=1.....n

oofor t>To land any k1.

Theorem 8. Suppose that the initial state no is a non-degenerate vector field. Then there exist a v-independent positive real numbers T and x2, x;1, x4,... such that for any v > 0 and for any k > 2 we have: Tr

J sup I fl 0

.ET"

(t, x) dt, max T

sup

7=

0

(t, x) I dt

ET"

Viz V

(5.3)

As the next result shows, the assumption of non-degeneracy cannot be removed. To show this we can restrict ourself to the 2D case since any solution of the 2D Navier-Stokes system generates a solution of a higher-dimensional system by adding zeroth components of the field u and fictitious variables. The nondegeneracy is preserved by this operation. We remark that in the 2D case the non-degeneracy is a necessary and sufficient condition for theorem 8.

Lemma 11. Let u be a solution of 2D Navier-Stokes system with a degenerate initial state. Then for any t > 0 and k > 0 we have maxlu(t,x)ICk(T2) max Iu(0,x)Ick(.t.:) and Vp =_ 0.

Proof. In the 2D case any degenerate initial state has the form:

uc(x)

(_b;)Wo(blxi + b2x2) + (c'2)

(5.4)

and real numbers b1, b2i c1, and c2 (see Theorem 3). for a suitable function In this case, the solution of the Cauchy problem for (5.1), (5.2) remains of the form (5.4): u(t, x) = (-bi) ,(t, b1x1 + b2x2) + CIO where the function

2) such that if "

1

T0

sup IVp(t,x)Idt


, rk

If 1

f T sup IVp(t,x)Idt > i. 0

xET"

then for any k > 2 we have: max I

r

j=1...n T 0

sup I

xET" axe

(t, x) I dt, > zek-

IT,

Inequality (5.3) is proved with the constant xk = min{rk, 2fk

;

n }.

References [1] Biryuk A. Spectral Properties of Solutions of Burgers Equation with Small Dissipation. Funct. Anal. Appl. 35, no. 1 (2001), 1-15. [2] Biryuk A. On Spatial Derivatives of Solutions of the Navier-Stokes Equation with Small Viscosity. Uspekhi Mat. Nauk 57 no 1. (2002), 147--148. [3] Dubrovin B. A.; Novikov S. P.; Fomenko A. T. Modern geometry-methods and applications. Part 1. The geometry of surfaces, transformation groups, and fields. Part II. The geometry and topology of manifolds. Graduate Texts in Mathematics 93, 104. Springer-Verlag 1984, 1985. [4] Friedman A. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [5] Frish U. Turbulence. The legacy of AN. Kolmogorov. Cambridge Univ. Press, 1995.

A. Biryuk

30

[6] Jefferson D. A numerical and analytical approach to turbulence in a special class of complex Ginzburg-Landau equations. Heriot-Watt University Thesis, 2002. (7] Halmos P. R. Finite-dimensional vector spaces. Springer-Verlag, New York - Heidelberg, 1974. 181 Hartman Ph. Ordinary Differential Equations. John Wiley & Sons Inc., New York, 1964.

191 Hormander L. Lectures on nonlinear hyperbolic differential equations. SpringerVerlag, Berlin, 1997. [101 Kolmogorov A. N. On inequalities for supremums of successive derivatives of a function on an infinite interval. Paper 40 in " Selected works of A.N. Kolmogorov, vol.1 " Moscow, Nauka 1985. Engl. translation: Kluwer, 1991. [11) Kukavica I., Grujic Z. Space Analyticity for the Navier-Stokes and Related Equations with Initial Data in L°. Journal of Functional Analysis 152, no. 2 (1998), 447-466.

[12] Kuksin S. Spectral Properties of Solutions for Nonlinear PDE's in the Turbulent Regime. GAFA, 9, no. 1 (1999), 141-184. [131 Pogorelov A. V. Extensions of the theorem of Gauss on spherical representation to the case of surfaces of bounded extrinsic curvature. Dokl. Akad. Nauk. SSSR (N.S.) 111, no.5 (1956), 945-947. (14) Spivak M. A comprehensive introduction to differential geometry. Vol. III, Publish or Perish, Boston, Mass., 1975.

Andrei Biryuk Independent University of Moscow 11, Bol'shoj Vlas'yevskij pyeryeulok Moscow 121002

Russia e-mail: biryuk®mccme.ru

Advances in Mathematical Fluid Mechanics, 31-51 © 2004 Birkhauser Verlag Basel/Switzerland

On the Global Well-posedness and Stability of the Navier-Stokes and the Related Equations Dongho Chae and Moon Lee Abstract. We study the problem of global well-posedness and stability in the scale invariant Besov spaces for the modified 3D Navier-Stokes equations with

the dissipation term, -Du replaced by (-A)"u, 0 < a < s. We prove the unique existence of a global-in-time solution in BZ 1

small BZ

for initial data having

.,in for all a E [0, ). We also obtain the global stability

of the solutions Bz 2,1

a

for a E [1, ). In the case 1 < a < 1, we prove y

+1-2n

the unique existence of a global-in-time solution in By . data, extending the previous results for the case a = 1.

for small initial

Mathematics Subject Classification (2000). 35Q30, 76D03. Keywords. Navier-Stokes equations, global well-posedness, stability.

1. Introduction In this paper, we are concerned with the sub-dissipative or hyper-dissipative Navier-Stokes equations.

(SNS)

div u = 0, u(0,x) = ue(x),

where it represents the velocity vector field, p is the scalar pressure. For simplicity, we assume that the external force vanishes, but it is easy to extend our results to the case of nonzero external force. J.L. Lions [22] proved the existence of a unique

regular solution provided a > 2. If a = 1, then the above system reduces to the usual Navier-Stokes equations. For the Navier-Stokes equations, Kato [20] proved the existence of global solution in C([0, oo); L3(R3)) if IIuo11 V is sufficiently small. After Kato's work [20], there were many important improvements using the scaling

Dongho Chae and Jihoon Lee

32

invariant function spaces. Especially, pioneered by Chemin [12], Cannon-MeyerPlanchon [7], initial value problems of the Navier-Stokes equations in some Besov spaces were extensively studied (see also [3], [4], [6] and [9]). For the Enter equations and compressible or incompressible Navier-Stokes equations, there are many recent

improvements using the notion of the Besov spaces and Triebel-Lizorkin spaces (see [10], [11], [13], [14], [16], [17], [21], [26] and references therein). Recently, Cannone-Karch [5] proved some existence and uniqueness theorems of global-intime solutions with external force and small initial conditions in some Besov type spaces in the case that -0+(-A)a replaces (-A)a. Considering scaling analysis, we find that if u(x, t) is a solution of (SNS)a, then ua(x, t) = \2a-1u(.\x, A2at) Bo9 +1-2a , I < pe Q - oo, are scaling invariant is also a solution of (SNS)a' PThus function spaces. Our first main result of this paper is the global existence and uniqueness result for the initial value problem (SNS)a with the initial data small A -2a in 82,1 norm. Precise statement is as follows.

Theorem 1. Let a E [0, be given. There exists a constant e > 0 such that 4) for any ua E B1 - 2a and IIuoII $-2n < e, the IVP (SNS)a has a global unique 1

82.1

solution u, which belongs to L°°(0, oo; B5 212a) ft L1(0, oo; B2 1) fl C([0, oo); B2 1)

with a =

f

-">

- 2a, if 2 < a - 4

-2a-b1, if0 0. Moreover, for any or > 0,

2°) u also belongs to L' (a, oo; B2,1) fl L1(a, oo; B2,i fl C((0, oo); B2,1), where -y = 1

- 62, if 0 < a < 2 , for any b2 > 0. Purthermore, the solution u satisfies the

1 2, if 2 - a < 4, following estimates

sup IIu(t)II

0 2) is contained in B2 1, our result in the case a = I can be regarded as improvement of the result of the corresponding stability result of the Navier- -Stokes equations in [23].

For the usual Navier-Stokes equations (SNS)1, Chemin [15] proved local in time existence in some critical Besov spaces and Cannone-Planchon [8] proved the global existence in some critical Besov spaces if the initial data has a small Besov norm. Cannone-Planchon [9] also derived various estimates for strong solutions in C([0, T]; L:1 (1R:1)) to the 3-dimensional incompressible Navier Stokes equations. Us-

ing the similar method originated from Fujita -Kato [19] and Kato [20], we can improve parts of Theorem 1 in the case z < a < i as follows. Theorem 3. Let a E

a ]bebe given. Suppose 1 < p < 2a . There exists a constant a}1-2° e > 0 such that for any un E Bpoo and uo a 11 _,, < e, the IVP (SNS),, has 11

II

+1-2° a global solution u E C([0, oo); B, ,0 ).

II

HP P,-

2. Littlewood-Paley decomposition We first set our notation, and recall definitions of the Besov spaces. We follow [24]. Let S be the Schwartz class of rapidly decreasing functions. Given f E S, its Fourier transform .F(f) = f is defined by

fef(x)dx.

f

1 {2,).,/z We consider V E S satisfying Supply C {l;' E W' z < IZ 15 2}, and ;p(t) > 0 if z < IC] < 2. Setting Oj = cp(2-- ) (in other words, 1pj(x) = 2j" p(2'x)), we can adjust the normalization constant in front of cp so that I

VV ER n \ {0}.

jEZ

Given k E Z, we define the function Sk E S by its Fourier transform Sk(i) = 1 -

j>k+1

Dongho Chae and Jihoon Lee

34 We observe

SuppcGflSuppop =0if Ij-j'I>2. Let s E R. p, q E [0, oo]. Given f E S', we denote Lj f = cpj * f . Then the homogeneous Besov semi-norm IIf1IBY ° is defined by IIf1IBy

[E 2j9aIIpi*fllL"]° ifgE[1,00) sup,

°

if q = oo.

f II LP ]

The homogeneous Besov space Bp.q is a quasi-normed space with the quasi-norm given by II II BY For s > 0 we define the inhomogeneous Besov space norm If II R;,° of f E S' as I I f I I s' for the Besov spaces.

IIf 111.P + I I f I I B' 4- Let us now state some basic properties

(i) Bernstein's Lemma : Assume that f E LP, 1 < p < oo, and supp f C {2J-2 < ICI < 2j}. Then there exists a constant Ck such that the

Proposition 1.

following inequality holds: Ck12jkIIfIILP < IIDkfIILP S Ck2'kIIfIILP-

(ii) We have the equivalence of norms DkfII $Pq N

II

IIfIIHp

(iii) Let s > 0, q E [1, oo], then there exists a constant C such that the inequality < C (IIIIILPL II9IIB;

IIf9IIB.

+ II9IIL'i IIf 11 B.

)

I

holds for homogeneous Besov spaces, where pl, r, E [1, oo] such that -

- + r2

+ -'

v

=

.

Let si, s2 < 'P such that s, + 32 > 0, f E Bp, and g E Bpj, B. Then f g E s, +a2- IV

Bp I

P and IIf9IIB,t,2-' p>.

We provide the proof of Proposition 2 in the appendix. Taking the divergence operation on the first equation of (SNS), we have the formula

-Op =

(9j (ok(u3uk).

j.k

Dongho Chae and Jihoon Lee

36

This enables us to define the general sub-dissipative Navier-Stokes type equations

J 8=u - A2°u = Q(u. u),

0 6a, gf,.(ba)

where C = C(I852I). We refer to the function behind the brace as bounding function, see Figure 4 for a sketch in a special situation.

FIGURE 4.

Bounding function of Ca (x), d = 2, 8S2 = B(0,1),

6=0.1,a=0.99,k=1,C=21r

Let C(t) = { (z, t) Id(z, 09D) < y, t = g6 (y), 6' < y < oo) be the cross section of the bounding function at the function value t and A(t) = IC(t) I the area of the cross section. Then 9a(6^)

J C6 (x) dx < C J Rd

U

A(t)dt.

Commutation Error of the Space Averaged Navier-Stokes Equations

65

From Lemma 4.1, we know A(t) < C(yd + y), with C depending only on Sl. Using ga(y) = t, changing variables and integrating by parts yield 9(6°)

f

r9(6)

A(t)dt

C

J

(yd + y)dt = C

0

J

d (yd + y) y (9a (y))dy

00

C C(bd° + 6°)9s (6°)

-d

yd-'9a (y)dy

- I 96 (y)dy

00

00

The integrals on the last line will be estimated using the change of variables y = 6/t and by monotonicity considerations of the arising integrand. For 6 sufficiently small, one obtains f C6 (x)dx < C 1 6d(°-k) + 6°-Rd) exp

\

Rd

(-

6k 62(1-o)

/

from what follows, since a < 1,

limJ Cs(x)dx=0.

6-0 Rd Now we will bound the second term in (19). The function B,kk (x) can be estimated from above in the following way Ba (x) < { Ian n B(x, 6°) I ° 9s (d(x, a11)) 1 0

if d(x, &I) < 61, if d(x, O fl) > 6°,

see Figure 5 for an illustration of the bounding function in a special situation. The bounding function is discontinuous, having a jump from the value 0 to the value Cg6(6°) at {x E !l I d(x,00) = 6°}.

FIGURE 5.

Bounding function of B6 (x), d = 2, asz = B(0,1),

6=0.1,a=0.99,k=1,C=6°

A. Dunca, V. John and W.J. Layton

66

Since 8i2 is smooth, we have I8 fl B(x, b")I < Cb(d-')" if 6 is small enough. It follows

f B6 (x)dx < C

b

A-1 ^k 4

g6 (d(x, 8 Z))dx.

{d(x.dsa)n/a}

+f

(1 + IIXII2)1/211 - 9a121VI2dx, IIxII2 0, IIxII2)-1/2 which does not depend on 6 and v, such that (1 + < Co for IIx112 > Tr/b. From (22) follows the pointwise estimate 11 -gd(x)I < 1 for any x E Rd. Thus, the first integral can be bounded by

f

IIxII2>*/a}

(1 + IIxII2)/211 - 9a121vI2dx


,r/d}

< CS f

(1 + IIxIl2)Ivl2dx.

(23)

{IIxII2>r/a}

A Taylor series expansion of (22) at IIxII2 = 0 and for fixed 6 gives

9a(x) = 1 -

a2a4II2

+ O(6'Ilxlli),

such that we have the pointwise bound 11

- ga(x)12 < C611X112

for any IIx112 < 7r/6 where C does not depend on S or x. In addition, IIxI12 (1 + IIx1I2)1/2 and consequently the second integral can be bounded as follows:

f

IIxII2:5

/h)

(1 + IIxIl)/211 - oI2II2dxCb f

(1+IIxl)II2dx. (24)

{IIxII2 1,

and C and a depend on a, k and IOill. Proof. Analogously to the begin of the proof of Proposition 4.3, one obtains

f v(x) Rd

k

g6(x - s),i(s)dsI dx in

C(k) [jd Iz(x)B6(x)Ik dx + f Iv(x)C6(x)Ik dx] II

IILP(as)),

Rd

where B6(x) and C6(x) are defined in the proof of Proposition 4.3. The terms on the right hand side are treated separately. In Proposition 4.3, it is proven that Ca E Ls (W') for every k E (0, oo). This implies

(Ch )p=C6°=Ca EL'(Rd),

A. Dunca, V. John and W.J. Layton

70

since k' E (0, oo). That means C6 E LP(Pd) for P E [1, oc). From the bounding function of C, it is obvious that C6 E L' (Rd), too. Using Holder's inequality for convolutions, see Adams [1, Theorem 4.30], and 11961ILI(R°) = 1, it follows IlVI1L9(Rd) 1 I1v"IIL9(R°) = IIvIILuk(Rd) < IIVIILvk(Rd)

By the regularity assumptions on v, it follows v E C°(Rd). This implies, together

with v = 0 outside 52, that v E LP(Rd) for every I < p < oo. Consequently, IIVIIL.k(Rd) < oo. Applying Holder's inequality, we obtain JR Rd

Iv(x)C6(x)Ikdx < IIvilLQk(R°)IIC6(x)IILr(Ra).

For the second factor, we can use the bound obtained in the proof of Proposition 4.3, replacing k by kp. Thus if 6 is small enough, we obtain

J

l626kQ)1 IIvIILak(Rd)

d

Jl

(25)

for every test function v which satisfies the regularity assumptions stated in Lemma 6.1.

The estimate of the second term starts by noting that the domain of integrar tion can// be restricted to a small neighbourhood of 85I

Iz(x)B6(x)lkdx =

J Rd

I7,(x)B6(x)Ikdx

J

IlzllL ({d(x,8S1) 0 such that lv(x) - v(y)I < CH llx

- yll2° for all x, y E S2.

By the Sobolev imbedding theorem, this constant can be estimated by CH < C(H)IIvilH2(sz) We fix an arbitrary x E {d(x,852) < 6°} and we take y E 852 with IIx - y112 = d(x, y). Since v vanishes on 852, we obtain Ilv(x)112 < CHd(x, On);'. It follows

IIvllL=({d(x.an) 0

if C is large enough. That means, also for the rational LES model, the kinetic energy of u can be estimated in form (34) and (35) if C is chosen sufficiently large and 6 sufficiently small.

Acknowledgment We thank Prof. G.P. Galdi for pointing out the result of Giga and Sohr in Remark 2.1.

References [1] R.A. Adams. Sobolev spaces. Academic Press, New York, 1975.

[2] A.A. Aldama. Filtering Techniques for Turbulent Flow Simulation, volume 56 of Springer Lecture Notes in Eng. Springer, Berlin, 1990. [3] L.G. Berselli, G.P. Galdi, W.J. Layton, and T. Iliescu. Mathematical analysis for the rational large eddy simulation model. Math. Models and Meth. in Appl. Sciences, 12:1131-1152, 2002.

[4] R.A. Clark, J. H. Ferziger, and W.C. Reynolds. Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mech., 91:1-16, 1979. [5] P. Coletti. Analytic and Numerical Results for k - e and Large Eddy Simulation Turbulence Models. PhD thesis, University of Trento, 1998. [6] A. Das and R.D. Moser. Filtering boundary conditions for LES and embedded boundary simulations. In C. Liu, L. Sakell, and T. Beutner, editors, DNS/LES - Progress and Challenges (Proceedings of Third AFOSR International Conference on DNS and LES), pages 389-396. Greyden Press, Columbus, 2001. [7] Q. Du and M.D. Gunzburger. Analysis of a Ladyzhenskaya model for incompressible viscous flow. J. Math. Anal. Appl., 155:21-45, 1991.

Commutation Error of the Space Averaged Navier--Stokes Equations

77

[8] G.B. Folland. Introduction to Partial Differential Equations, volume 17 of Mathematical Notes. Princeton University Press, 2nd edition, 1995. 191 C. Fureby and G. Tabor. Mathematical and physical constraints on large-eddy simulations. Theoret. Comput. Fluid Dynamics, 9:85-102, 1997. [101 G.P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I. Linearized Steady Problems, volume 38 of Springer Tracts in Natural Philosophy. Springer, 1994. [11] G.P. Galdi and W.J. Layton. Approximation of the larger eddies in fluid motion 11: A model for space filtered flow. Math. Models and Meth. in Appi. Sciences, 10(3):343 350, 2000.

[12] S. Ghosal and P. Moin. The basic equations for large eddy simulation of turbulent flows in complex geometries. Journal of Computational Physics, 118:24-37, 1995. [13] Y. Giga and H. Sohr. Abstract L° estimates for the Cauchy problem with applications to the Navier Stokes equations in exterior domains. J. Funct. Anal., 102:72 -94, 1991.

[14] L. Hormander. The Analysis of Partial Differential Operators I. Springer - Verlag, Berlin, ..., 2nd edition, 1990. [151 T.J. Hughes, L. Mazzei, and K.E. Jansen. Large eddy simulation and the variational multiscale method. Comput. Visual. Sci., 3:47--59, 2000.

[16] V. John and W.J. Layton. Approximating local averages of fluid velocities: The Stokes problem. Computing, 66:269-287, 2001.

[17] O.A. Ladyzhenskaya. New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Proc. Steklov Inst. Math., 102:95-118, 1967.

[18] O.A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 2nd edition, 1969. [191 A. Leonard. Energy cascade in large eddy simulation of turbulent fluid flows. Adv. in Geophysics, 18A:237-248, 1974. [20] M. Lesieur. Turbulence in Fluids, volume 40 of Fluid Mechanics and its Applications. Kluwer Academic Publishers, 3rd edition, 1997.

[21] S. B. Pope. Turbulent flows. Cambridge University Press, 2000. [221 W. Rudin. Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 2nd edition, 1991. [231 P. Sagaut. Large Eddy Simulation for Incompressible Flows. Springer-Verlag Berlin Heidelberg New York, 2001.

[24] J.S. Smagorinsky. General circulation experiments with the primitive equations. Mon. Weather Review, 91:99 164, 1963.

[25] O.V. Vasilyev, T.S. Lund, and P. Moin. A general class of commutative filters for LES in complex geometries. Journal of Computational Physics, 146:82-104, 1998.

78

A. Dunca, V. John and W.J. Layton

A. Dunca Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260 U.S.A.

e-mail: ardst2l+Qpitt.edu V. John Institut fiir Analysis and Numerik Otto-von-Guericke-Universitat Magdeburg PF 4120 D-39016 Magdeburg

Germany e-mail: johnQmathematik.uni-magdeburg.de

Homepage: http://vw-ian.math.uni-magdeburg.de/home/john/

W.J. Layton Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260 U.S.A.

e-mail: vjlQpitt.edu Homepage: http: http://www.math.pitt.edu/-vj1;

Advances in Mathematical Fluid Mechanics, 79-123 © 2004 Birkhiiuser Verlag Basel/Switzerland

The Nonstationary Stokes and Navier-Stokes Flows Through an Aperture Toshiaki Hishida Abstract. We consider the nonstationary Stokes and Navier-Stokes flows in

aperture domains Sl C R",n > 3. We develop the L'?-I.' estimates of the Stokes semigroup and apply them to the Navier-Stokes initial value problem. As a result, we obtain the global existence of a unique strong solution, which satisfies the vanishing flux condition through the aperture and some sharp decay properties as t -. oo, when the initial velocity is sufficiently small in the L' space. Such a global existence theorem is up to now well known in the cases of the whole and half spaces, bounded and exterior domains. Mathematics Subject Classification (2000). 35Q30, 76D05.

Keywords. Aperture domain, Navier-Stokes flow, Stokes semigroup, decay estimate.

Dedicated to the memory of the late Professor Yasujiro Nagakura

1. Introduction In the present paper we study the global existence and asymptotic behavior of a strong solution to the Navier-Stokes initial value problem in an aperture domain 11 C R" with smooth boundary 851:

=AU-VP

Vu =0

=0 ult=o = a ulr)sl

(xESZ, t>0), (x E 9, t> 0), (t > 0),

(1.1)

(x E S2),

where u(x, t) = (uu (x, t), , u"(x, t)) and p(x, t) denote the unknown velocity and pressure of a viscous incompressible fluid occupying 1, respectively, while a(x) = (ai (x), , a (x)) is a prescribed initial velocity. The aperture domain S2 On leave of absence from Niigata University, Niigata 950-2181, Japan (e-mail: hishidaeng.niigatau.ac.jp). Supported in part by the Alexander von Humboldt research fellowship.

T. Hishida

80

is a compact perturbation of two separated half spaces H+ U H_, where H± = {x = (x1, ,x,) E R"; fx > 1}; to be precise, we call a connected open set S2 C R" an aperture domain (with thickness of the wall) if there is a ball B C R° such that S2 \ B = (H+ U H_) \ B. Thus the upper and lower half spaces Hf are connected by an aperture (hole) M C S2nB, which is a smooth (n-1)-dimensional manifold so that S2 consists of upper and lower disjoint subdomains 12f and M: SZ = S2+ U M U S2_.

The aperture domain is a particularly interesting class of domains with noncompact boundaries because of the following remarkable feature, which was in 1976 pointed out by Heywood [34]: the solution is not uniquely determined by usual boundary conditions even for the stationary Stokes system in this domain and therefore, in order to single out a unique solution, we have to prescribe either the flux through the aperture M

b(u)=Jt If

or the pressure drop at infinity (in a sense) between the upper and lower subdomains Sgt [p] =

lim

IxI-oo,xEf2,

p(x) -

lim

lxl-oo,xEft .

p(x),

as an additional boundary condition. Here, N denotes the unit normal vector on M directed to 12_ and the flux 0(u) is independent of the choice of M since V u = 0 in Q. Consider stationary solutions of (1.1); then one can formally derive the energy relation

L IVu(x)]Idx = [p]cb(u),

from which the importance of these two physical quantities stems. Later on, the observation of Heywood in the Lz framework was developed by Farwig and Sohr within the framework of LQ theory for the stationary Stokes and Navier-Stokes systems [21] and also the (generalized) Stokes resolvent system [22], [18]. Especially, in the latter case, they clarified that the assertion on the uniqueness depends on the class of solutions under consideration. Indeed, the additional condition must be required for the uniqueness if q > n/(n- 1), but otherwise, the solution is unique without any additional condition; for more details, see Farwig [18, Theorem 1.2].

The results of Farwig and Sohr [22] are also the first step to discuss the nonstationary problem (1.1) in the LQ space. They, as well as Miyakawa [53], showed the Helmholtz decomposition of the LQ space of vector fields LQ(Sl) = L; (S2) e R' (Q) for n > 2 and I < q < oo, where Lo (1) is the completion in LQ(S2) of the class of all smooth, solenoidal and compactly supported vector fields, and Lq(fl) = {Op E LQ(Il); p E L The space LQ(1) is characterized as ([22, Lemma 3.1], [53, Theorem 4]) I,? (S2) = {u E L°(S2); V V. u = 0, v u]asi = 0, q(u) = 0},

(1.2)

where v is the unit outer normal vector on 852. Here, the condition 0(u) = 0 follows from the other ones and may be omitted if q < n/(n - 1), but otherwise,

Navier-Stokes Flow Through an Aperture

81

the element of Ls(1l) must possess this additional property. Using the projection Pq from L9()) onto Lo(1) associated with the Helmholtz decomposition, we can define the Stokes operator A = AQ = -PQO on Lo (1?) with a right domain as in section 2. Then the operator -A generates a bounded analytic semigroup a-'A in each Lo(ft), 1 < q < oc, for n > 2 ([22, Theorem 2.5]). Besides [34] and [21] cited above, there are some other studies on the stationary Stokes and Navier-Stokes systems in domains with noncompact boundaries including aperture domains. We refer to Borchers and Pileckas [8], Borchers, Galdi and Pileckas [3], Galdi [28], Pileckas [55] and the references therein.

We are interested in strong solutions to the nonstationay problem (1.1). However, there are no results on the global existence of such solutions in the L9 framework unless q = 2, while a few local existence theorems are known. In the 3-dimensional case, Heywood [34], [35] first constructed a local solution to (1.1) with a prescribed either O(u(t)) or [p(t)], which should satisfy some regularity assumptions with respect to the time variable, when a E H2(St) fulfills some compatibility conditions. Franzke [25] has recently developed the L9 theory of local solutions via the approach of Giga and Miyakawa [32], which is traced back to Fujita and Kato [26], with use of fractional powers of the Stokes operator. When a suitable O(u(t)) is prescribed, his assumption on initial data is for instance that a E L9(1l),q > n, together with some compatibility conditions. The reason why

the case q = n is excluded is the lack of information about purely imaginary powers of the Stokes operator. In order to discuss also the case where [p(t)] is prescribed, Franzke introduced another kind of Stokes operator associated with the pressure drop condition, which generates a bounded analytic semigroup on the space {u E L9(S2);V V. u = 0, v u1ou = 0} for n > 3 and n/(n - 1) < q < n (based on a resolvent estimate due to Farwig [18]). Because of this restriction on q, the L9 theory with q > n is not available under the pressure drop condition and thus one cannot avoid a regularity assumption to some extent on initial data. It is possible to discuss the L2 theory of global strong solutions for an arbitrary unbounded domain (with smooth boundary) in a unified way since the Stokes operator is a nonnegative selfadjoint one in L2; see Heywood [36] (n = 3), Kozono and Ogawa [44] (n = 2), [45] (n = 3) and Kozono and Sohr [47] (n = 4, 5). Especially, from the viewpoint of the class of initial data, optimal results were given by [44], [45] and [47]. In fact, they constructed a global solution with various decay properties for small a E (when n = 2, the smallness is not necessary). Here, we should recall the continuous embedding relation D(A214-'/2) C L. For the aperture domain St their solutions u(t) should satisfy the hidden flux condition O(u(t)) = 0 on account of u(t) E L2 (Q) together with (1.2). We also refer to Solonnikov [62], [63], in which a theory of generalized solutions was developed for a large class of domains having outlets to infinity. In his Doktorschrift [24] Franzke studied, among others, the global existence of weak and strong solutions in a 3dimensional aperture domain when either O(u(t)) or [p(t)] is prescribed (the global existence of the former for n > 2 is covered by Masuda [51] when O(u(t)) = 0). As

82

T. Hishida

for the latter, indeed, the local strong solution in the L2 space constructed by himself [23] was extended globally in time under the condition that both a E H0(52) (with compatibility conditions) and the other data are small in a sense, however, his method gave no information about the large time behavior of the solution. The purpose of the present paper is to provide the global existence theorem for a unique strong solution u(t) of (1.1), which satisfies the flux condition O(u(t)) = 0 and some decay properties with definite rates that seem to be optimal, for instance, Ilu(t)I]Lmcsf + IIVu(t)IIL.. u) _ 0 (t-1/2) ,

as t - cc, when the initial velocity a is small enough in Lo(5)), n > 3. The space L" is now well known as a reasonable class of initial data, from the viewpoint of scaling invariance, to find a global strong solution within the framework of LQ theory. We derive further sharp decay properties of the solution u(t) under the additional assumption a E L' (S1) n L" (Q); for instance, the decay rate given above is improved as O(t-" 12). For the proof, as is well known, it is crucial to establish the L9-L' estimates of the Stokes semigroup

0 and f E L" (Q), where a = (n/q - n/r)/2 > 0. Recently for n > 3 Abels [1] has proved some partial results: (1.3) for 1 < q < r < oo and (1.4) for 1 < q < r < n. However, because of the lack of (1.4) for the most important case q = r = n, his results are not satisfactory for the construction of the global strong solution possessing various time-asymptotic behaviors as long as one follows the straightforward method of Kato [39] (without using duality arguments in [46], [7], [48], [49] and [37]). In this paper we consider the case n > 3 and prove

(1.3) for 1 1, where supp u denotes the support of the function u. For a Banach space X we denote by B(X) the Banach space which consists of all bounded linear operators from X into itself. Given R > Ro, we take (and fix) two cut-off functions V)±,R satisfying

E Cx(Rn; [0, 1 ]),

V)±.R( x )

=

in H± \ BR+I, 1

01

in H:F U BR.

In some localization procedures with use of the cut-off functions above, the bounded domain of the form

D±,R={xEH±;R 0 independent off E Co (D±,R) (where Oj denotes all the j-th derivatives); and for all f E Co (D±.R) with f D* x f (x)dx = 0. By (2.2) the operator S±,R extends uniquely to a bounded operator from Wi'Q(D+,R) to W.j+"q(D+.R)°.

Navier-Stokes Flow Through an Aperture

87

For G = S2, H and a smooth bounded domain (n > 2), let Coo (G) be the set of all solenoidal (divergence free) vector fields whose components belong to Ca (G), and L7 (G) the completion of Co (G) in the norm II' Ilq.c If, in particular, G = 52,

then the space Lo(ft) is characterized as (1.2). The space Lq(G) of vector fields admits the Helmholtz decomposition Lq(G) = L9(G) ®Ln(G), 1 < q < oo, with L7 (G) _ {Vp E Lq(G); p E L (G)}; e [271, [60] for bounded domains, [4], [521 for G = H and [22], [53] for G = Q. Let Pq,c be the projection operator

,

from L9(G) onto Lq (G) associated with the decomposition above. Then the Stokes operator Aq,c is defined by the solenoidal part of the Laplace operator, that is,

D(Aq.c) = W2.q(G) n Wo q(G) n L9(G), Aq.G = -Pq.GO, for 1 < q < oo. The dual operator AQ,G of Aq,G coincides with Aq/(q-1),G on Lo (G)' = Lol(q-t)(G). We use, for simplicity, the abbreviations Pq for Pq,n and Aq for Aq,n, and the subscript q is also often omitted if there is no confusion. The Stokes operator enjoys the parabolic resolvent estimate II(. + AG)-' II B(Lo(G)) -< CE/IAI,

(2.3)

for [ arg Al < -7r - e (A i4 0), where e > 0 is arbitrarily small; see [29J, [61] for

bounded domains, [52], [4], [19], [20], [17] for G = H and [22] for G = Q. Estimate (2.3) implies that the operator -AG generates a bounded analytic semigroup {e-tAc; t > 01 of class (Co) in each Lo(G),1 < q < oo. We write E(t) = e-1Ay which is one of E±(t) = e-tA"*. The first theorem provides the Lq-Lr estimates of the Stokes semigroup a-tA for the aperture domain S2.

Theorem 2.1. Let n > 3. 1. Let 1 < q < r < oo (q # oo, r 0 1). There is a constant C = C(12, n, q, r) > 0

such that (1.3) holds for all t > 0 and f E L7 (l) unless q = 1; when q = 1, the assertion remains true if f is taken from V(S2) n Lo(Sl) for some s E (1,00).

2. Let 1 < q < r < n (r

1) or 1 < q < n < r < oo. There is a constant C = C(SZ, n, q, r) > 0 such that (1.4) holds for all t > 0 and f E Lo (1) unless q = 1; when q = 1, the assertion remains true if f is taken from L' (S2) n Ls. (11) for some s E (1, oc).

3. Let 1 < q < oo and f E Lg(Sl). Then as t -* 0 o(t_Q) lie-tA fllr = 1

Iloe- AfIIr = 0(t °-1/2)

if q < r < oo,

ast-oo ifq n even in Theorem 2.2, but we have asserted nothing about their decay rates since they do not seem to be optimal; see Remark 2.1 for the Stokes flow. On the other hand, in Theorem 2.3 the decay rates of Vu(t) in Lr(1l) for r > n are better than t-"/2 for exterior Navier Stokes flows shown by Wiegner [66]. Taking Theorem 5.1 of [17] for the Stokes flow in the half space into account, we would not expect u(t) E L'(9) in general. Thus the decay rates obtained in Theorem 2.3 seem to be optimal; that is, for example, IIu(t)II,o = 0(t-"/2) would not hold true. Concerning the exterior problem, Kozono [42], [43] made it clear that the Stokes and/or Navier-Stokes flows possess L'-summability and more rapid decay properties than (2.11) only in a special situation. Remark 2.7. In Theorem 2.2 one could not define a pressure drop (see Farwig [18, Remark 2.2]) since the solution does not always belong to Lr(S2) for r < n. Due to the additional summability assumption on the initial data, we obtain in Theorem 2.3 the pressure drop written in the form

[P(t)] = P+ (t) - P- (t) = if at+ u Vu - u)(t) wdx, t

where w E W2-4(S2), n/(n - 1) < q < oo, is a unique solution (given by [221) of the auxiliary problem

w-Ow+O7r=0. in 0 subject to wlasa = 0 and t¢(w) = 1. In fact, the formula above is derived from the relations

jw' Vp(t)dx = -[P(t)]O(w) = -[P(t)]. f u(t) Virdx = -[tr]¢(u(t)) = 0. 3. The Stokes resolvent for the half space The resolvent v = (A + AH)-' PH f together with the associated pressure Jr solves the system

Av - Ov+Vir= L in the half space H = H+ or H_ subject to vl8H = 0 for the external force f E L`+(H)l1 < q < oo, and A E C \ (-oo, 0]. In this section we are concerned with the analysis of v near A = 0. Our method is quite different from Abels [1]. One

Navier-Stokes Flow Through an Aperture

91

needs the following local energy decay estimate of the semigroup E(t) = e-tAH, which is a simple consequence of (1.3) for S2 = H.

Lemma 3.1. Let n > 2,1 < q < oo, d > 1 and R > 1. For any small c > 0 and integer k > 0 there is a constant C = C(n, q, d, R, e, k) > 0 such that IIo'ae E(t)PHf II q.)IR 0, f E L',j, (H) and j = 0, 1, 2. Proof. We make use of the estimate u E D(A;/H),

IIV'fhIr.H 1. This completes the proof.

Lemma 3.1 is sufficient for our analysis of the resolvent in this section, but the local energy decay estimate of the following form will be used in section 5.

Lemma 3.2. Let n > 2, 1 < q < oo and R > 1. Then there is a constant C = C(n, q, R) > 0 such that IIE(t)fII2.,I.HR + IIatE(t)fllq.Hn < C(1+t)-n/2giIfIID(A,,H),

(3.3)

fort > 0 and f E D(Aq,H). Proof. The left hand side of (3.3) is bounded from above by C(IIAIIE(t)fIIq.H + II E(t)fIIq,H) 1 it follows from (1.3) for S1 = H with

r = oo that IIE(t)f 11 ,11. < CII E(t)fII oo,H

.n/2g II f II q.H.

The other terms II o'E(t)f 11,11. < CII Aii2E(t)f II r.H S

Ct-j/2IIE(t/2)f IIr.H

(j=1,2)?

IIatE(t)fIIq.11R 0 there is a constant C= C(n, q, d, R, e) > 0 such that

m-1

IAI°Ilaa v(A)112.,,HR + Y, IWaav(A)112,q.H, < CIIfII q,H, k=O

for Re A > 0 (A # 0) and f E m

where

r (n-1)/2 =51 n/2 - 1

ifnisodd, if n is even,

n 1 1/2+e Q=A(e)=1+m-2+e= E

ifnisodd, ifniseven.

Furthermore, we have sup

IIv(A) - w112.q,H IIf IIq.H

f 34 0, f c- LIdi(H) -, 0,

(3.5)

asA-,0 with ReA>0, where

w=/

o

E(t)PH fdt.

Proof. We recall the formula

v(A) = (A + AH)-'PHf = f e-,\'E(t)P,jfdt,

(3.6)

0

which is valid in Lo(H) for Re A > 0 and f E L9(H). In the other region (A E C \ (-oo, 01; Re A < 0) we usually utilize the analytic extension of the semigroup {E(t); Re t > 0} to obtain the similar formula. For the case Re A = 0 (A 34 0) which is important for us, however, thanks to the local energy decay property (3.1), the formula (3.6) remains valid in the localized space Lq(HR) for f E (the function w in (3.5) is well-defined in Lq(HR) by the same reasoning). We thus obtain from (3.1) Il0'a,kv(A)11,,H,, 3, 1 < q < oo, d > Ro and R > R0. For any small e > 0 there are constants Ct = CI (0, n, q, d, R, e) > 0 and C2 = C2(Sl, n, q, d, e) > 0 such that rn- I

Ial°Ilaa T(A)fII2,q,S2R +:IIaaT(.)fII2,q,S:R < CIIIfIIq,

(4.5)

k=0

for Re A > 0 (A # 0) and f E Lq (S2); and m-I JAI" llaa Q(A)fllq+ E II0,k\Q(A)fllq S c'21Ifllq, k=0

for Re A > 0 with 0 < JAI < 2 and f E L' (Q), where m and Q = 0(e) are the same as in Lemma 3.3.

T. Hishida

98

Proof. In view of (4.2), we deduce (4.5) immediately from (3.4) together with (2.2). One can show (4.6) likewise, but it remains to estimate the pressures 7rf contained in (4.4). By (4.1) we have

aairf(x,A)dx=0,

1 0 with 0 < IAl < 2 and f E

This completes the proof.

0

Remark 4.1. In the proof above, we have made use of the inequality (see Galdi [28, Chapter III] ) II9 - 91Iq.G 5 ClIV911-i.q,G

with 9 =

ICI

fg(x)dx,

for g E L(G), I < q < oo, where G is a bounded domain for which the result of Bogovskii [2) introduced in section 2 holds (for instance, G has a locally Lipschitz boundary), although the usual Poincare inequality leads us to Lemma 4.1 because we have (3.4) in Wl-q(H1u). Since the inequality above will be often used later, we give a brief proof for completeness. For each W E Lq/(q-')(G), we put ip = 1 f(3 3, 1 < q < oc, d > Ro and R > R,). Set 4i("`) (s) = 89' (is + A)-1 P

(s E R\ {0}).

For any small e > 0 there is a constant C = C(S2, n. q, d, R, £) > 0 such that

Ixx II-P("')(s + h) f -

II2.y.u,,ds < CIhi'-'IIf Iiq

(4.14)

for lhi < h) = min{i/4, 1/2} and f E

Here, m and,3 = p(e) are the same as in Lemma 3.3, and r) > 0 is the constant such that (4.9) is valid for A E E,,.

Proof. We may assume d > R,) + 2 (as in the proof of Lemma 4.2). Given h satisfying lhi < h,,, we divide the integral into three parts h) f - $(,»)(s)f II2.q.st,,ds

f-:

=11+12+1;3.

+f2jhj2h

W ith the aid of (4.10), we find

I, < 2 f

CIhII-3IIff1q,

Isl21h1

for f E

(S2). Finally, to estimate 13, one does not need any localization. In fact,

since

,t(m)(s+h)f -4("_)(s)f = (-i)-+'(m+1)!

f

+h (ir+A)-(-+2)PfdT,

(2.3) gives

IIVm)(s + h) f - Vm)(s)f II2,q,s2R

< C114,(-)(s + h)f - $(m)(s)f IID(A,) < CIhI IsI-(m+1) IIf IIq,

for IsI > 2h° (> 2IhI) and f E Lq(S2). Therefore, we obtain 13 0 and f E

l,q.O

(ul).

T. Hishida

104

For the proof, the following lemma due to Shibata is crucial since we know the regularity of the Stokes resolvent given by Lemmas 4.2 and 4.4. Lemma 5.2. Let X be a Banach space with norm II ' II and g E L' (R; X). If there are constants 0 E (0,1) and M > 0 such that

fa

B llg(s) II ds + sup h#o IhI

then the Fourier inverse image

x

IIg(s + h) - g(s) Ilds < M,

G(t) = 2

e'stg(s)ds

of g enjoys

JIG(t)II < CM(1 + Itl)-B, with some C > 0 independent oft E R.

Remark 5.1. The assumption of Lemma 5.2 is equivalent to

g E (L' (R; X) W'.'(R; X))e.M , where )g denotes the real interpolation functor (the space to which g belongs is known as a Besov space).

Proof of Lemma 5.2. Although this lemma was already proved by Shibata [56], we give our different proof which seems to be simpler. Since IIG(t)II < M/2ir, it suffices

to consider the case Itl > 1. It is easily seen that if ht $ 2ja (j = 0,±1,±2,... then

etht

G(t)

etst(g(s + h) - g(s))ds,

J

27r(1 - etht) from which the assumption leads us to

a

IIG(t)II 1 and f E L' (S2). This completes the proof.

Remark 5.2. It is possible to show the decay rate t-'1'+£ of the semigroup in W2. (12k) as well. This follows immediately from the proof given above with X = W2-q(fl) for n > 5. When n = 3 or 4 (thus m = 1), as in Kobayashi and Shibata

[40), we have to introduce a cut-off function p E Co (R; [0, 1)) with p(s) = 1 near s = 0; then one can employ Lemma 5.2 with X = W2,q(S2) and g(s) = p(s)z/4(m)(s)f to obtain the desired result since a rapid decay of the remaining integral far from s = 0 is derived via integration by parts. We did not follow this procedure because Lemma 5.1 is sufficient for the proof of Theorem 2.1. The next step is to deduce the sharp local energy decay estimate (1.5) from Lemma 5.1.

T. Hishida

106

Lemma 5.3. Let n _> 3,1 < q < oo and R > R4. Then there is a constant C = C(12, n, q, R) > 0 such that Ie-tAf II1,q.t1n < Ct-nl2gII f

(5.4)

))q,

fort > 2 and f E Lo(ll); and f1ll.gslR II

+ I]ate-tAf Ilvsltt 5 C(1 + t)-"J24IIf IID(Aq),

(5.5)

fort > 0 and f E D(Aq). Proof. We employ a localization procedure which is similar to [38] and [40]. Given

f E Lo (1). we set g = e-'f E D(Ag) and intend to derive the decay estimate of u(t) = e-tAg = e-(t+1)A f in W1'g(SZR) for t > 1. We denote by p the pressure associated to u. We make use of the cut-off functions given by (2.1) and the Bogovskii operator introduced in section 2. Set

9t = V)t.R,+1 9 -

[9 '

and

v±(t) = E±(t)g 0 and that g± E D(Aq,H f) with .

Note that fD}

gV

R

I19±I) 5 C119±112.q.Ht 5 CI19112.q 5 CII9IID(A,) 5 CIIfII9,

(5.6)

by (2.2). We take the pressures zr± in H± associated to vt in such a way that D4.,,

n f (x, t)dx = 0,

(5.7)

for each t. In the course of the proof of this lemma, for simplicity, we abbreviate

'±to 0± and S±.R to S±. We now define {u±, p± I by ut(t) = V'±v±(t) - S±[vt(t)' VV't],

p±(t) = V)t'rt(t)

Then it follows from Lemma 3.2 together with (2.2) and (5.6) that IIu±(t)111.q.sttt 2 and f E L"(Q) on account of n < q < oo. Along the lines of the proof of Lemma 5.4, one can show

IIe`AfIIo.nt\n 0, U

u(t) =

(6.1)

J

by means of a standard contraction mapping principle, in exactly the same way as in Kato [39], provided that IIaIIn 5 b,,, where 6o = 50(fl, n) > 0 is a constant. The solution u(t) satisfies Ilu(t)IIr

Ct-1/2+n/2r IIaIIn

for n < r < oo,

(6.2)

(6.3)

]Iou(t)lln < Ct-112 IIalln,

for t > 0 together with the singular behavior IIU(t)IIr = o (t-1/2+n/2r)

for n < r < 00; IIVu(t)II.. = o (t-1/2)

,

(6.4)

as t 0. Furthermore, due to the Holder estimate (6.9) below which is implied by (6.2) and (6.3), the solution u(t) becomes actually a strong one of (1.1) with (2.6) (see [26], [32] and [64]). We now prove

lim IIu(t)IIn = 0,

(6.5)

for still smaller a E L'(Sl). To this end, we derive a certain decay property of u(t), which is weaker than (2.11) but sufficient for the proof of (6.5), assuming additionally a E L1(5l) fl L'(f) with small IIaIIn. Given ry E (0,1/2), we take q E (n/2, n) so that -y = n/2q - 1/2; then, t

Ilu(t)IIn 0, which together with (6.2) yields IIu(t)IIn < C(1 + t)-'(IIaIII + IIaIIn),

(6.6)

for t > 0 (this decay rate is not sharp and will be improved in Theorem 2.3). From now on we fix ry E (0,1/2) and set 6 = S.(1l,n,-t)/2. Given a E Lo (12) with IIaIIn n (resp. r < n); then, given f E L7 (SZ), we have IIo{e-(t-T/2)A

t-r/2 r/2

'C(St) if and only if there holds compatibility condition (3.8). The solution is unique and II(u,p);

R?'6c(S2)ll.

(3.10)

T. Leonaviciene and K. Pileckas

134

3.2. Modified Stokes problem Let us consider the problem

-µ10u - (lrs + u2)Vdiv u + poV(11/p0) = f

in S1,

div (pou) = g in Q,

(3.11)

tu=h on09D, where p0(x) = p. exp4 (x).'(x) E with ryo > 1. First we deal with the case h = 0 which we regard as the "homogeneous" problem (3.11) and denote (3.11)0. By weak solution of problem (3.11)0 we understand a pair (u, 11) E Do' (n) x L2(S2) satisfying the integral identity

pt J Vu:Vt1dx+(µt+µ2)J divudiv17 dx st

sz

(3.12)

- f Po'Hdiv (Poll) dx = J f tl dx, it

Ytl E DO'(S2).

it

and the equation div (p0u) = g. Theorem 3.5. Let 0 be an exterior domain with Lipschitz boundary. Assume that f E D01(Sl), g E L2(S2). Then there exists a unique weak solution (u, Il) of problem (3.11) and there holds the estimate Ilu; D,(S2)II + IIn; L22(Sl)II 1, then

Hence,

ft f

n

dx1

(3, then D 2V(12) and in the case q = 2 the theorem is proved. If lyo < (3, we continue the iteration process: (u, II) E D3?V(12)

Len=2.3(F, 1

G) E V1-1.2(l) X V1'2 (ft), 3l+1 = min ((1 +

(i)

(u, II)EV ' V(fl)

2 1,2 ... = (F, G) E V3i.;,!-1(12) x V3,+" A Lemma 2.3

Th,,,r< , 9.1 (i)

(u,II)EDs, ..,V(IZ), (3l+_=min ((1+m)'Yo,Q). 1,2

Taking m such that (l + m)-yo > /3, we obtain (u, II) E

Supplying mentioned above inclusions by the corresponding estimates we get for (u, II) the inequality (3.16) at q = 2. Let us consider the case q E (6/5, oo). If q E [6/5,2], /3 > I + 3/2 - 3/q, we have

r

j

f sldx
1, 5 E (0, 1),,6 E (1+&+3/2,1+6+2).

Let us take q = 3. It is easy to compute that then (f,g) E V5'2_3/q+Ea(S1) x Vgi2.i/q+fo(SZ) with 29'o = 3 - l - 6 - 3/2 > 0. It is already proved that problem (3.11)o has a unique solution (u,H) E V2/2_3/Q+E.((l) x Vs%z-3/g+EO(H) and by Lemma 2.2 we get the inclusion (u, II) E A (St) x AV (Q), where Qo = 3/2+6+Eo. 00 00 Now, by Lemma 2.3

V$ u E AV

(II),

HV E

Therefore, (F,G) E A" (0) x Al-a(Sl), where QI = min{13 - I + 1, %3o + rya}. In virtue of Theorem 3.1 (ii) we get

(u,II) E A'a(l) x AJ61!"(n) =VrA(1l). 1

Asymptotic Behavior of Exterior 3D Steady Compressible Flow

139

Repeating the last argument we derive the sequence of inclusions

(u, n) E Lem

2.3(F,

(s2), p2 = min{Q -1 + 2, po + 2-yo l

G) E Apz (s2) x A 132

(u, II) E D'-'s A(Sl)

... Lemur Theorem, 3.1 (ii)

2.:1

(F, G) E AL1-1,6 (II) x AR AL" +..,

1,6

+m

(u, n) E D-" A(Sl), off-

(II)

Qt+,n =min {Q, Q31 + (l + m)yo}.

Taking m so, that 13o + (l + m)yo > Q, we obtain (u, H) E V A(1). The theorem is proved.

Let us consider problem (3.11) with nonhomogeneous boundary condition.

Theorem 3.8. (i) Let (f, g, h) E R0"4 V(1), l > 1, q > 6/5, 0 E (1 + 3/2 - 3/q, 1 + 2 - 3/q). Then problem (3.11) has a unique solution (u, II) E D°V(Sl) and there holds the estimate (u,H);DaQV(52) 0 such that if and let w E (E(i

l.t+z),6

IIw;

(S2) div(wz) E

72+1).a(Q)

Moreover, Oz E

(t-2,t-1).o(52

IIAz;

.(x) II

(4.9)

z+1).a(Q)II)

(4.10)

II + Ildiv (wz); A5(S) II < c IIh; (E(11+1)6ry

(Cryt,a2,t-1)'6(S2),

)II

y.4

div (wz) E Ay 2.l A

At-2,6 (p) I I + Odiv(wz); II ry

(E(ryt,i

< c (IIh;

IIz; (rt

IIw;

Proof. (i) Let h E

(4.8)

II 5 eo,

then problem (4.1), (4.2) has just one solution z with z E A7a(Sl) and IIz;

141

1ZT,(yt.2+1)'Q(S1),

i.e. h admits the representation

h(x) = r-255(0) + h(x). Wt+1,9(S)2 and h E Vy 9(52). For sufficiently small co in [10] with attributes b E is proved the existence of the unique solution z E7t;2t1)'9(St) of problem (4.1),

(4.2) which admits the asymptotic representation

z(x) = r- 2b(O) + i(x), z` E V'"(() (4.11) i.e. the "spherical" attributes of h and z coincide. Moreover, there holds the estimate 1, q > 1, -y E (1+2-3/q, 1+3-3/q). Then problem (4.16) has a unique solution cp which admits the asymptotic representation

p(x) = co(27rlxl)-1 +;i(x), where Cp E

(4.17)

There holds the estimate III; 1'1+' 9(Q)II + Icoi