Contributions to Current Challenges in Mathematical Fluid Mechanics (Advances in Mathematical Fluid Mechanics)

0

We note that in the 2D case the class of periodic solutions with degenerate initial state coincides with the class of periodic solutions with Vp - 0. (For the

Euler equation this is true for any dimension, see Theorem 4) Indeed, if the initial state is degenerate, then Vp =_ 0 due to Lemma 11. To show the converse we apply the identity Ap(t, x) = -212(t, x), (5.5)

Multidimensional Burgers Type Equations

29

where 1k(t,x) are the coefficients of the characteristic polynomial of the matrix

det(g - Al) =(-A)n+(-A)n-)I1(t,x)+...+I,(t,x). The degeneracy of the initial state is equivalent to the condition I1(0, .) = 12(0, ) 0. The coefficient Il = 0 due to (5.2). The identity (5.5) is also valid in higher dimensions. For the proof one can

take the divergence of (5.1), (5.2) and note that div Vuu = -212(t,x). Proof of Theorem 8. As we will show, the assertion of the theorem holds for any u and p that satisfy (5.1) (we note that equation (5.2) is not used in our calculations).

First, from theorem 2 we find T = T(uo). If To < T, then the left hand side of (5.3) is equal to infinity due to Remark 1. Suppose T < T). Then from Theorem 2 again, we find c = c(uo) and rk = rk(uo) (k > 2) such that if "

1

T0

sup IVp(t,x)Idt <

zET"

then for any k > 2 we have T 1

max

j=1...n

0

sup xET"

k

(t, x) dt >

, 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<12<

1

for b1 > 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
1.2. +C 1000 IIu(t)II .

82,1

dt

82,1

<- IIUoIIBJ-2a eXP

(cj°° IIu(t)IIB 2.

.1

,dl)

,

and for any o > 0,

2.o

IIu(t)IIhj 2adt

exp

Ilu(t)II

poIIu(t)IISj1+Cf

a< t<

< IIu(a)II

2,1

(cj 882.1

dt) '

Remark 1. We note that the smallness assumption is made only on the homogeneous norm of the initial data. Our second main theorem below is concerned with the global stability of the solution of (SNS)a in the case a > 2. For the stability of the usual Navier-Stokes equations, Beiriio da Veiga-Secchi [1] and Wiegner [25] obtained LP-stability with p > 3 near the L°°(0, oo; V+')-solution. Ponce-Rscke-Sideris-Titi [23] proved

Well-posedness and Stability of Navier-Stokes and Related Equations

33

the H'-stability of mildly decaying global strong solutions of the Navier-Stokes equations.

Theorem 2. Let a E [2, ) be given. Assume that u' is a solution of the IVP (SNS)° satisfying ul E C([0, oo); B2 12°)nL' (0, oo; B21). Then there exists a posi) such that if Iluo - uoI] i-2 < Eo, tive constanteo = IIu'II 2., .

L1(o.oc;B.i)

Ns.,

1+-2n

there exists a unique global solution u2 E C([0, oo); B2 1 C((0, oo); B2 1) of (SNS)v with initial data up E B2

) n L 1(0, oo; B21) n

12° .

Remark 2. Theorem 2 could be viewed as a generalization of Theorem 1. Since H8 (s > 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-' <_ CIIfIIB'', IIgIIBn2, v

(iv) If s E (-'7 - 1,

then we have

II [u,Oq]wIILP
P.1

with K EZ Cq < 1. In the above, we denote [u, Oq]w = uAgw - A,(uw).

(v) (Embedding) Bp., (R') is an algebra included in the space Co of continuous functions tending to 0 at infinity. Let s E lit, e > 0, and suppose p, q E [1.00]. Then it holds Ba

y,

a

HP

s `' Bn.oo+

Well-posedness and Stability of Navier-Stokes and Related Equations

35

and

(vi) (Interpolation) We have the following interpolation inequalities for sl, 82 E R, 0 E [0,1]: II uII Be., +(1-o)..2 G_ CIIull y, lull r.l

and

Ilull[Ly",.it

<- CIIuilejp,

11u1111;.-'!

y.,

Proof. The proof of (i)-(vi) is rather standard and we can find the proof in many references. The proof of (i) can be found in [14] Lemma 2.1.1. (ii) is straightforward

from (i). The proof of the first part of (iii) can be found in [11] and the second part of (iii) can be found in [13]. We can find the proof of (iv) in [17). The fact that I3.1 is embedded in Co can be found in [2). We can find the proof of the last part of (v) and (vi) in [24]. ty, in the following we denote For simplicity, respectively. If (p, p, r) E [1, oe], we denote

2. by b,, and BL

and

IIuIII..(J.. ) = II(2q'IIOgUIILv(o.1;!,P))gEZII1>(z).

We denote briefly L°°(0, oo; By,,) by L'D(Bp ,.). We denote (-0) by A for notational simplicity. For the proof of Theorem 3, we need the following extended version of Proposition 2.1 of [15].

Proposition 2. Let a > 0 be given. There exists a constant C > 0 such that

., 2 <

IIe-eA="vo II L,_p

r

.(l3,,.,. n )

If u is a solution of

.

(1)

CIIuollt3...

a,u - A'-"u = f, Rs x R+, u(O,a) = 0,

then we have

IIullf (p;f-)
)'

(2)

and

IIull, /. ,. (Bn

(3) )

where pl > 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
Rs X 1R+,

u(0, x) = uo,

with Q(u,u) _ -div(u 0 u) + E;,k VA-'8jOk(uluk). This general equations of the usual Navier-Stokes equations was studied by Chemin [15]. The following proposition is more or less standard (see [15]).

Proposition 3. Let -L+ a2 = p. Set sl-2 = s1 +s2 - p. Ifs, < 2, ands, +82 >0, then we have 1) <

(4)

Proof. The proof can be found in [13] and [15].

3. Proof of Theorems Proof of Theorem 1. If we try to follow the classical Fujita-Kato method ([19] and [20]) to prove Theorem 1, we need to prove the continuity of the bilinear operator in C([0, oo); 2°),

B(u, u) = J

Pe-(t-s)(-o)" V V. (v

0 v)(s)ds,

where P is a Leray projection operator, and we found there are technical difficulties

for such a continuity estimate when 0 < a < Z. Thus we could not adapt FujitaKato's argument to prove Theorem 1. Step 1. A priori estimates Taking the operator Aq on the first equation of (SNS),, we have

8tAqu + (u V)Aqu + A2°Aqu + V qp = [u, A,] Vu.

(5)

Multiplying Aqu on the both sides of (5) and integrating over R'', we obtain 2dtllogUIIL2

+C22°9IIDgUIIL2 < II[oq,U] VUIIL2llAgUIIL2

(6)

Dividing the both sides of (6) by Il.qull L2 and using the commutator estimates of Proposition 1 (iv), we have dtllogUllL2

+C22ng11AgUIIL2 < Ccg2-(2-2°)QIIUIIB z,

Ilulle -2"

(7)

2.1

Multiplying 2(z -2°)`+ on both sides of (7) and taking summation over q E Z, it reduces to

d llu(t)IIBo

2

1

2° +Cillull 8. 2,1

<- C211ullB. z,,

. 2-2"llullB

z .l

Well-posedness and Stability of Navier-Stokes and Related Equations

37

By the Gronwall inequality, we obtain that oo

sup IIu(t)II

0
g-2a +Cl f IIu(t)II H2,1

o oo

<_ Iluoll'§ rj-2,. eXP (C2 f 2.,

dt

$

B2,1

IIu(t)II' dt)

(8)

2,

()

We also note that sup IIu(t)IIL2 < IIuOIIL2

Then we have the following a priori estimates. sup

0
Ilu(t)II 3-2. 8,1

<_

IluollBF-2o eXP

+Cl

100

IlutII

8dt 2.1

foIlu(t)III

dt)

(C2 2.

2,1

(9)

1

Step 2. Iteration and Uniform estimates We define the following iteration sequences ( un+1 pn+1)

(I)

ai,un+1 + (un Q)un+1 + A2oun+1 + Vpn+l = 0 div un+l = 0, ,untl(x,0) u'0+1(y) = Eq
R3 X R+,

0 < 2a < 2,

-

Setting uo = 0, we can find un and pn for all n by solving the linear system. Similarly to a priori estimates, we have dtlluntl(t)IIB-2.

+C1llun+IIIB


2,1

Ilun+'IIBg-2a 2,,

2,1

By the Gronwall inequality, we obtain SUP

lluntl(t)ll

O
. g-20

B2.,

+C1 f

g dt 82,1

Ilun(t)Il',dt)

<_ lluo+'IlBF-20 exp (C2 0

2,1

llunt1(t)ll

0

(10)

2.1

Choose a so small that exp (c-'4;`) < M (this is possible if we take M > 1). Assume that Ilunll (10), we have

+Clllunll

< Me. From the estimate of G 1 (O,T;

Dongho Chae and Jihoon Lee

38

sup

Ilun+1(t)II

O
+C1

.

B2.1 Iluo+1IIj-2a

<_

exp

Ilun+1(t)II

J0

(c2J

Ilu3 }111

dt

Ilun(t)IIB 2,

2,1

<

$

Bz.1

dt`

8 j - 2. eXP 2,1

E)

(CC

1

< ME.

Thus we obtain that if 1luoll-20 <_ E, then for all n, 62.1

sup Ilun+1(t)II

0
j-2. +Cl J B2,1

,} dt < ME,

11un+1(t)1)

(11)

62.1

0

for some M > 0. Step 3. Equations of dif'erences and Existence of solution.

Define bun+l = un+1 _ un and bpn+l = pn+l -pn We obtain the equations of the differences,

(I.)

atJun+1 + (.un . V)6un+l + (bun . v),un +A2a6un+l + V6pn+1 = 0, R3 x 1{I;+, div 6un+l = 0

0 < 2a <

-

,

n+l

Similarly to a priori estimates, we have for 77 satisfying q = max{0,1 - 2a}, 2

<

Wt IIOg6un+1IIi2 + C22agllog6un}11122 Ct92-9(5 -2a-n)IIu"11B,1II6un+1 llBz 2a

+CIIoq((6un .

lIlog6un+l IIL2

V)un)IIL2IIAg6un+1

11L--

Dividing both sides of the above inequality by IIt'. Sun+1IIL2, multiplying with and taking summation over q, we infer that 116un+1

dt

C3II6un+11163

IIB -2o-n + 2.1

2,1


20

2,1

+CSII6unII61- 2a-, IIunII611.

In the above inequality, we used Proposition 1 (iii), e.g., II((6un). V)unll8y 20

s

- 2n-,Ilounll6,1 CII6unII612.1

Well-posedness and Stability of Navier-Stokes and Related Equations

39

Gronwall's inequality gives us that

sup I16u}1(0II 8-20-.+C3

o
< IlbuOn+tll

exp

J-2o

.

f

82., Ilbun(t)II

O
2.1

(C4 f Ilu"(t)II 0

+C,, sup

Ilbu+1(tII 8-dt

o

2.1

. _zo f

B2.1

oo

dt)

j

2.,

11011 . 3 dtexp (C4 B2.,

0

f Ilun(T)II . dr)

.

B2.,

0

(12)

From the uniform estimates (11), we can choose a so small that exp (C4 and

L°° Ilu(t)II 82.E

dt) < 2

oo

Cr, f Ilun(t)II . dt < 81 82.,

for all n. Then we have sup

Il6u"+1(t)II .

0
j-2o-n +C3

B2.1

+

2llbuo}1IIBi -z 2.1

f

oc

Ilau+l(t)II

0

sup 1160(011

4 o
B2,

.

-ndt

B2.,

(13)

B3-2o

By iterating, we have for any positive integer N > 0, N

N

E sup N

1

llauo+lll n=1 N

-sue 2.1

II6un+'(t)II.,_,dt

n=1 0

82.,

n=10
o0

4

82.,

!v

sup Ilbun(t)II 3-2. 82., n=20
2CI:2("+1)(-2«-°)IIon+1u01IL2

<

+

n=1

sup Ilbun(t)II J-2a-

1

4

n=20
82.,

N

< 2Clluoll

SUP Ilbun(t)II -2a_ + 4 n=2o
B2.1

< 2CIluill ., 2.1

5

8C

82.,

4N-I IIUOII BZ, 20-n

IIuoIIH1 2a-

Multiplying both sides of the first equation of (I') by bu"}1 and integrating over R', we have Osup

Ilbun+l(t)IIL2 < IltSuo+IIIL2 +C6OSUPo

Ilbun(t)IIL2 F.

Ilun(t)II

.

dt.

Dongho Chae and Jihoon Lee

40

6dt < a , then we obtain

If we choose a so small that Cs f,00 Ilu" (t) II 2.,

sup 1160+1(t)IIL2 <- 116uo+111L2 + 1 sup II60(t)IIL2. 4 o
(14)

o
2°-'t) By iterating, we obtain that u" converges to u in L°° (0, oo; B2 n L'(0, oo; B2. °). From the uniform estimates we have u E L°°(0, oo; B212°) n 1

L' (0, oo; Bz.1). Uniqueness can be proved similarly. unt1 satisfies aL

,un+1 = -(un . v)un+l - A2°un+1 - vpn+1

Clearly we have (u" - V)u"+', A2au"+' E L1(0,oo; B?

2°). 1

Since pn+1 satisfies

-divVpn+1 = div((un . v)un+l), B2.-2,

) from the Calderon Zygmund estimate [181. we have Vp"+' E L' (0, oo; Thus we readily have that u"+' is continuous with values in B2 -2* . By a 3E argument, u is continuous with values in BZ, 2°-`7 By the interpolation, we have C([0, oo); BZ 1) with

=

r

- 2a, if 2 < a < a

for 61 > 0. We still have

-2a-dl, if0a<

l

to prove that u also belongs to L°°(v,oo; B21)nL'(Q, oo; B. 2°)nC((0, oo); Bz 1), - 6 2, if 0 < a < 2 for any 62 > 0. We provide only a where a > 0 and -y = l 2,if2
j

priori estimates. Since we already know that u belongs to L' (0, oo; B2.1), we can

choose a > 0 such that u(a) E B2.1. Similarly to step 1, we have dt

IloquIIL2 + C22ng11oquIIL2 <- Ccg2-5"I1uII

,11ull13

.

2.

Multiplying 21 , and taking the summation over q, we have

d IIullQj, +CIIuII' 1

j 12n <- C11U11i2

B2.1 -

2.1

Using Gronwall's inequality, we have

sup Ilu(t)IlBj +CJ 2,

o
exp

Ilu(a)Il 2.1

0

(Cf

Ilu(t)11ij

2.,

IIu(t)IIB

2.,

dt)

.

2.1

Well-posedness and Stability of Navier-Stokes and Related Equations

41

12").

By the interpolation theorem, we have u E C([oo.,oo);B2. 62) for any b2 > 0 if 0 < a < 2. If a > 21 u satisfies atu = -(u 0)u - A2au - Op. Since (u 0)8, A2ru, Vp E L' (a, oo; B2.1)+ Thus u belongs to I' (o,, oo; B2.1) n L' (a, oo; B21

u belongs to C([a, oo); B2,1). This completes the proof of Theorem 1. Proof of Theorem 2. We derive a differential inequality for 11u1(t) - u2(t)II ' -2.

2.1

The difference U = u1 - u2 satisfies the system

(A)

+A2aU + VP = 0, div U = 0,

1 < 2a < 2,

1R:; x IR+,

where P = p' - p2. Step 1. Iteration and Uniform estimates We define the following iterating sequences (Un+1 Pn+1) a'Un+1 + (U11+1 , V)u' - (Un . V)Un+l + (ul . V)Un+l +A2aUn+1 + VPn+l = 0, R i x R+, 2 < a < a, (II) div Un+1 = 0

U(0, x) = Eq
= _[Un,Og] VUn+1 + [u',Oq] . VUn+1 _

pq((Un+l

. V)ul)

(15)

Multiplying with Ig6Un+1 on both sides of (15), integrating over R3, and dividing both sides by IlogUn}111L2, we have IIOgU'+111 <-

II[Ur1

L2 +C22agII0gUn+'11L2 VUn+111L2

Aq] - VUn+' IIL2 + II[u', oq] .

+

IIAq((U"+1

. V)u')IIL2.

(16)

Using commutator estimates, multiplying with 20- -2a)q on both sides of (16), and taking summation over q, we obtain that d 1IUnt111B

-?.. +C7IIUrl+l11

2.1

2.1

< C1IUn1Ift,IlUn`lliz 12a +CIIu'II

I1Un+'II1J .

IIUn+111,,i-2,.+C41"


HUn-~' IIH -2n

11

2.1

2.1

12..

2.1

2.1

In the above last inequality, we used Proposition 1 (iii).

+CII(U4+1

.

2

2<.

(17)

Dongho Chae and Jihoon Lee

42

We show that if el> is chosen to be so small that

toexp(Cg

(18)

then it follows

NO

supIIU"(t)IIA

+ 2.1

IIUnIIA ldt

2- J.

< flt exp (C9 f °° IIu' II6 dt)

.

/

0

Suppose that (Qn) is satisfied and let us prove that we obtain

is also true. From (17)

2.1x2.12.1

d IIUn+111

sup . l-2 +(C7- C8 ll
6

6

)IIUn'}111.

-2.,

6

1.2u.

IIUn+111

< Cgllu' II

IIUn(t)II

62.1

j

(19)

2.1

Since co was chosen to satisfy (18), we have dtlIU"+1116

12,.+

IIUn+11161

27

2.


2.

Using Gronwall's inequality, we have IUn+'(t)1I6:.,'°

+ 27

osup

Jl, 11 U` 1(t)II

2.1

dt).

IIUo+'IIBg-2.exp(CglxIIu'IIA 0

2,1

Since we have IIUo +' II

.

2n 5 II U0II .

dt

.

(20)

2

_ 2 <_ fo, (Qn+ 1) is true. Multiplying with

62.1

62.1

Un+' on both sides of the first equation of (II) and integrating over R3, we infer

that IIUn+1II2

L2 <-

at

IIUn+111L2110u'II

Using Gronwall's inequality, we have sup

IIUn+'IIL2


Iu' I.

dt).

(21)

62,1

From (20) and (21), we obtain sup

x

IIUn+1(t)II

o
i20 + C'2

62.1

eXP CIIUU+'IIB23 -2'

.1

IIUn+1(t)II

J0

. j dt

62.1

(CJ x llu'IIA dt) 2.1

.

(22)

Well-posedness and Stability of Navier-Stokes and Related Equations

43

Step 2. Equations of Difference and existence

Setting 5U"+' = Un+1 - U" and bP"+I = P"+' - P", we have the following equations of differences. (U" . V)6U" +1 - (SU" . v)Un

+A2"SU"+l + (SUn+I V)ul + (ul . (II,)

+V6P"+I = 0,

div SU"+' = 0, bU(O,x) = (On+luu1- A>]+1uo2)

Taking Oq on both sides of the first equation of (II'), we infer that BIOg6U"+1 -(Un , V)AgSU"+1+(ul

V)Ag6U"+I+A2"Og6U"+1 +vOg6P,1+1

[u1

_ -[U".O,] vf5U"+I + og] (SUn+I -Oq((bU"+' - V)u') + 4q((bU" V)U").

(23)

Multiplying with Og5U"+1 on both sides of (23) and integrating over R3, we have 2dtIIDgbU,,+11122 +C22°q IA '6U-+1112

S II [U",oq] .

II[u',Og] . VSUn+IIIL2IIog5U"+IIIt.2

+IIog((5U" .

V)U")IIr,2IIogbU11+IIIL2 + IIOq((bU"+l . V)u')IIL2IIog6Un+'111,2.

(24) 201-2,)q on both Dividing both sides of (24) by IIOg5U"+' II i 2, multiplying with sides, taking summation over q, and using (iii) and (iv) of Proposition 1, we obtain

d

+IIIIII6vn+1IIQ3

_z..

Ilbvn+lll,.

2.1

2.,

+CII

CII5U"+'II,j -2.,IIVu'II.

vU" Q IIbUn+Ill

C11IIU"II

j-2n+C1211u'Il.y Ilbvn}111 82.1

83, 1

82, 1

+C,:IIIU"11'

2.1

_.1

2.1

2.1

-20 82.1

11W" IIji 21

,1

Using Gronwall's inequality, we have

2 +Cul f IIsU"+I(t)II

llbU,1+1(t)1

osup

0

2.1

IIbUO

+111.

.

+C13 sup

0
x exp

(f 0

+ CI2IIu' 11dt)

J -zn exp UT CI I IIU,1I1.

82.1

82.1

3 -2,,dt

82.1IIbU"(t)II.3-2o 0 F

CI I lIU" ll

.

B2.1

dt 2.1

+ C12llu' II

/j 2.1

.

82.1

dt)

.

(25)

Dongho Chae and Jihoon Lee

44

From (20) we have

x

O exp (c9j' IIu'IIB tdt)

IIU"(t)II6 1dt <

J

- C7

2,

.

2.

Substituting (26) into (25), we obtain sup IIbU"+1(t)II

O
$-,,.

+CIO1 IIbU"+1(t)II

62.1

dt

.

62.1

O

J0 (iou'ii. + C13 Sup IIbU"(t)II O
IIU"Il/f

2.1

iiiIt= Setting 1111 = exp (

2C1IEO

exp (CoIIu'IIL1(o,o0;62i,1)) +C12IIu'IIL1(o,oo;Bz

t)/

we choose E11 so small that

C7 exp Cs f IIu1 II B 1dt) < 4C1 3M1 Then we have sup O
II5Un}1(t)II j-20 + C10 J 62.,

M1II5UO

t1II69

116U"+1(t)II

+ 4 sup t1

2.1

,1 dt 62.1

o II6U"(t)II61 -20 o

2.1

Multiplying with bU"+' on both sides of (Nt) and integrating over 1R3, we have 2 dt

IIbU,1+1 II2 2

+ IIA"bU'

IIVU" 11 L- II oU" IILl

'U

II6U"+1

2

11 L2 + Ilou' IIL'° IISU"+1 IIL2II5Un+1 111,

IIbU"IIL2II6U"+'11L2 +C15IIulII


II6U"+1IIL2 2.

.

(27)

2.1

2.1

Dividing both sides of (27) by II6U"+'IIL2 and using Gronwall's inequality. we obtain IIbU"

SUP

O
+C14 sup

0
'

IILz _ bUo

II6U"IIL2 exp

t' IL2 exp (c15J (c15J

IIu'II

62.1

IIu' II

dt) fo

IIU"II

dt), choose Eso small that

Setting ILf2 = exp (c15 fo Iu1 II 62.1

C7 exP

dt) 621

(c9j°° IIu' 11 B 1 dt)

<_

4C14M2

621

Well-posedness and Stability of Navier Stokes and Related Equations

45

Thus we have 1 sup

sup IIaU"+t(t)IILZ <

4 o
0
Il

U"(t)IIr.2.

12a)

fl L1(0, oo; B2,1). From By the iteration, U" converges to U in L°°(0, oo; B22 2, the uniform estimates, we have U E L°°(0,oc; B2 )fl L' (0, 00; B2,1). Uniqueness can be proved similarly. We still have to prove that U is continuous with values in Za B2 (so u2 is continuous). U"+1 satisfies atUn+t = _(Un+1 . v)ut + (U" _ V)Un+1 - (ut . V)Un+1 A2aUn+l _ VP"+t 1

-

Since the right-hand side of the above belongs to L1 (0, o0; B2.12a), we readily Bz.',1-2a. have that U+1 is continuous with values in Since U" converges to U in B212a.

L°°(0,oo; B212r), U is continuous with values in

Similarly to the proof

of Theorem 1 and the a priori estimates of solution, we have u2 E C((O,oo); B2.1). This completes the proof of Theorem 2. Proof of Theorem 3. We follow closely [15].

Step 1. A priori estimates If we set u = e-tt2' uo + w, then we have Q(e-tn- uo,

8tw + A2aw = lR3xQt+, w(0, x) = 0.

a-tA 2i' uo) + 2Q(e-t'12T uo, w) + Q(w, w),

z
By using Proposition 2, we have for p > max{ II°IILo°(BP)

CIIQ(w,w)II

<_

2a_ )

, 2},

Lm(Ba

+CIIQ(e_tA2t,U

,w)II_

° LP(BP P +CIIQ(e-tA2.,uo,e-t"2nuo)IIL(BV

)

By using Proposition 3, we have IIQ(w,w)IIr_(Bv +,-,.,) <_ CIIWI12-(BP)'

IIQ(e-t"j'uo,w)II

L°(B,

{,

2

(

)


111 2,,( i) IIwIIL-(BP),

L°(BP

°

°

(28)

and "uo,e_1A2

IIQ(e-t

uo)II

_

LI(R,

5

Clle_tA'°uoIIL L°(Ry

(29) °

)

Dongho Chae and Jihoon Lee

46

Using Proposition 2, we obtain IIQ(e-,A2. u0,w)II_ A+,-2^( L P(BP

)

P

and IIQ(e-'A2'


,e-In2nu{1)IILri

u

ap+, (

P

)

Thus we have IIu'IIL'(BP)

<

CIIwI1Lm(BP) + CIIuoII, IIWIILP(h,) +Clluoll2P

<

C16(IIwIIL-(i3P) + IIuoIIBP)2

(30)

Step 2. Iteration We define the iterating sequence Q(e-tA'^uo, a-tA2D uo) c

R:'xR+,2
2Q(e-,A2

+

uo, w,) + Q(wn, wn),

wn+1(0,x) = 0.

Similar to apriori estimates, we have IIwn+I lIL" (BP) < C1e(IIw,IIL.(BP) +

IItOIIBP)2.

Choose appropriately small a and wl such that II IIIeP < e, IIw1IILm(sD) < e, and 4C16e2 < E. Then we have IIWnhIL=(s,.) < E, for all n.

Step 3. Equations of differences and existence

Let bw,, + =

wn. We have the following equation of the differences: atbwn+1

+A2,bwn+1

2Q(e-t'2'u),bwn)

=

+Q(wn + wn_ 1, Swn ).

Similar to a priori estimates, we obtain Ilbwn+lll L-(BP)

<

C'IInOIIaPIISWnhIL4O(BP)

+C(IIwnhIL^°(BP) + IIwn-1IIL-(&P))IlbwnllL-(BP)'

Choosing a so small that we obtain 1IISWnllL-(BP)'

By iteration, wn is a Cauchy sequence in C([0, oo); Bp) and we have wn --+ w in C([O,oo);Bp). Since u = e-1AZ^uo + w, we obtain u E LOD(Bp). We next prove e-tAz^uo uniqueness. If e-'A 2'ud + w1 and + W2 are solutions of (SNS),,, with W1, W2 E C([O.oc);Bp) and we set bw = w1 - W2, then we have the following estimates for any T > 0 similarly to the above: IISWIILT(BP)

<-

C17IIuohlQPIIbwIIL;.(eP)

+CIK(IIw1IIL;. (s) + IIw2IIL; (eP))IIbwIILT (BD)'

Well-posedness and Stability of Navier -Stokes and Related Equations

47

Choose e so small that CI 7C < a and C, 8c < a . Since w, and w2 E C([O, oo); BP), we can choose T, > 0 so small that IIw1IIL;,(h,)+IIW211LT(H)

e.

IIbWIILT,(t,r), we immediately have bw = 0 on Since we obtain 1IbwI1LT (b,,) < [0, T, ]. Doing the above process repeatedly on [T1 , 2T1 ], [2T1, 3T1 ].... , we have

bw = 0 on (0, oo). Thus we complete the proof of the unique existence.

Appendix For the proof of Proposition 2, we follow the similar ideas in [15]. We first prove the following lemma, which is an extended version of Lemma 2.1 of [15].

Lemma 1. There exists a constant C > 0 such that for any t > 0, 2.uIILP

IIOge-'A

e_Ct22gn

(31)

IIAgUIIL:

Proof. Consider 40 E D(R3 \ {0}). Since we can assume that u has support in an annulus, we obtain that e-tA2"u

_

O(A-ID)e-tA2,,U

=

9a(t, )*u, To prove the lemma, it is enough to

with ga(t,x) = f show that

DA (t, )IIL<

Ce-cta2a

Say, ga(t,x) = A:'ga(t,Ax) with

r

9a(t,x) = J

e`(xe-tae^IE12a de.

By calculation, we have 9A(t,x)

_ (1 +

Ix12)-3

(1 +

Ix12)-3

(1 +

/'(1 +

f(Jd -

x12)3ea(xIE)(Id-

The Leibnitz formula gives us that

(Id-0)';(()e-ta2nIEI2")=

C}a(A

7)()a'(e-ta2nIEI2").

Dongho Chae and Jihoon Lee

48

r

We have also the following formula by algebraic computation. eta2,lfl2..ay(e_ta2..lEl2a)

=

(-tA2a)mfj 18"i

I'tj I>_1

Since 1 has support in an annulus, we have

<

IO'(e-tA2"1E12.,)I

CI71(1+tA2°)1'Y1e-ta2oIE12'.

<

CI71(1+tA2°)I71e-cta2<,

Thus we have Ix12)-3e-Cta2,.

I9A(t,Z)I 5 (1 + The proof of lemma is completed.

0

Proof of Proposition 2. First, we have the following estimate by Lemma 1. e-Ct22q,,

(32)

IIOqU0I11.v.

Taking LP-norm over [0, TI of both sides of (32), we have T

(f

1

n

(jT e-01221*Pdtl I P

Ile-tA2oLquollLrdt

C

IILgUOIILP

J

0

< C2 ?so-' (1 - e-cT22go)) °

and taking jr norm on both sides of (33), we obtain

Multiplying with

22y('+

(33)

)'Ile-tA2nAquollLj.(Lr)


qEZ

Thus we proved the first part of Proposition 2. Taking A. operation on the equation in Proposition 2, we obtain at Aqu + A2n aqu = 1q f

Since ut = 0, we have Oqu(t) =

r te-(t-T)A2o

Oq f (T)d-.

Taking Ln norm, we have

Cf

Iloqu(t)llt

t Ile-(t-T)A2.,ogf(T)

IILrdT

0

<

cf

0

t

e_C(t-r)22pnIIOgf(T)II

LrdT.

Well-posedness and Stability of Navier-Stokes and Related Equations

49

In the above last inequality, we used the estimates of Lemma 1. Taking U' norm over 10, T], we obtain the following by using Fubini's Theorem.

< C `f

Cfu Iloqu(t)Ilirdt)

C


-(t

r)22

IIAgf(T)IIrndT)

f

e-'(t-T)2 ' v,dt

Jo

T

< CJ C1rf

2-

1

2. ..

/

i

II Agf(T)IIGndT

_ e_c7 x ni\ rl ecr2 "'x. IIQgf(T)II /,ndT

ga(1

C2_

dt I

o

o

P2

f 0

n2

IIOgf(T)II°r,ndT

In the above last, inequality, we used Holder's inequality and the following calculation:

(f o

Multiplying with

n2 nx

ecr22°"(11

1

)dT'


/J

t x))

on both sides of the inequality, we have 7

(Jr)

C2`'"IIOgfII1.T (Lp).

Thus we have (3). This completes the proof of Proposition 2. Acknowledgements

This research is supported partially by the grant no.2002-2-10200-002-5 from the

basic research program of the KOSEF, and J. Lee is supported by the BK 21 project.

References [1] H. Beirao da Veiga and P. Secchi, L"-stability, for the strong solutions of the Navier Stokes equations in the whole space, Arch. Rat. Mech. Anal. 98 (1987), 65-70. [2] G. Bourdaud, Redlisations des espaces de Besov homogenes, Arkiv for matematik 26 (1988), 41-54.

[3] M. Cannon, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, 1995.

[4] M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iheroamericana 13 (1997), 515-541.

50

Dongho Chae and Jihoon Lee

[5] M. Cannonc and G. Karch, Incompressible Navier-Stokes equations in abstract Banach spaces, Tosio Kato's Method and Principle for Evolution equations in Mathematical Physics, (2001). [6] M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods and Applications of Analysis 2 (1995), 307-319. [7] M. Cannone, Y. Meyer, and F. Planchon, Solutions auto-similaires des equations de Navier-Stokes in R3, Expose n. VIII, Seminaire X-EDP, Ecole Polytechnique (Janvier 1994). [8] M. Cannone and F. Planchon, On the nonstationary Navier-Stokes equations with an external force, Advances in Differential Equations 4 (1999), 697-730. [9] M. Cannone and F. Planchon, On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations 16 (2000), 1- 16. [10] D. Chae, On the Well-Posedness of the Eider Equations in the 7Yiebel-Lizorkin Spaces, Comm. Pure Appl. Math. 55 (2002), 654-678. [11] D. Chae, Local Existence and Blow-up Criterion for the Euler Equations in the Besov Spaces, RIM-GARC Preprint no. 01-7. [12] J.-Y. Chemin, Remarques sur l'existence globale pour is systeme de Navier-Stokes incompressible, SIAM J. Math. Anal. 23 (1992), 20-28. [13] J.-Y. Chemin, About Navier-Stokes system, Prr publication du Laboratorie d'analyse numerique de Paris 6, R96023, 1996. [14] J.-Y. Chemin, Perfect incompressible fluids, Clarendon Press, Oxford, (1998). [15] J.-Y. Chemin, Theoremes d'unicile pour le systeme de Navier-Stokes tridimensionnet, Journal d'Analyse Mathematique 77 (1999), 27-50. [16] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. math. 141 (2000), 579-614.

[17] R. Danchin, Local theory in critical spaces for compressible viscous and heatconductive gases, Comm. Partial Differential Equations 26 (2001), 1183-1233. [18] M. brazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, AMS-CBMS Regional Conference Series in Mathematics 79 (1991). [19] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal. 16 (1964), 269-315. [20] T. Kato, Strong LP solutions of the Navier-Stokes equations in Item with applications to weak solutions, Math. Zeit. 187 (1984), 471-480. [21] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Mathematics 157 (2001), 22-35. [22] J.L. Lions, Quelques methodes de resolution des problemes aux limites non linearies, Dunod, Paris, 1969. [23] G. Ponce, R. Racke, T.C. Sideris, and E.S. Titi, Global stability of large solutions to the 31) Navier Stokes equations, Commun. Math. Phys. 159 (1994), 329-341. [24] H. Triebel, Theory of Function Spaces, Birkhiiuser, 1983. [25] M. Wiegner, Decay and stability in LP for strong solutions of Cauchy problem for the Navier-Stokes equations, The Navier-Stokes equations. Theory and numerical methods, Proceedings Oberwolfach (1988), eds. J.G. Heywood, et al., Lect. Notes Math. 1431, Berlin, Heidelberg, New York, Springer, 1990, 95-99.

Well-posedness and Stability of Navier-Stokes and Related Equations

51

[26] M. Vishik, Hydrodynamics in Besov spaces, Arch. Rational Mech. Anal 145 (1998), 197-214.

Dongho Chaet and Jihoon Lee't School of Mathematical Sciences Seoul National University Seoul 151-747

Korea e-mail: dhchaeQtmath.snu.ac.krt

[email protected]

Advances in Mathematical Fluid Mechanics, 53-78 Q 2004 Birkhauser Verlag Basel/Switzerland

The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain A. Dunca', V. John2 and W.J. Layton3 Abstract. In Large Eddy Simulation of turbulent flows, the Navier-Stokes equations are convolved with a filter and differentiation and convolution are interchanged, introducing an extra commutation error term, which is nearly universally dropped from the resulting equations. We show that the commutation error is asymptotically negligible in LP(Rd) (i.e., it vanishes as the averaging radius 3 - 0) if and only if the fluid and the boundary exert exactly zero force on each other. Next, we show that the commutation error tends to zero in H "" 1 (Sl) as S -. 0. Convergence is proven also for a weak form of the commutation error. The order of convergence is studied in both cases. Last, we study the influence of the commutation error on the energy balance of the filtered equations. Mathematics Subject Classification (2000). 35Q30, 76F65. Keywords. Large eddy simulation, commutation error.

1. Introduction The space averaged Navier Stokes equations for the space averaged fluid velocity u and pressure p are the basic equations for large eddy simulation (LES) of turbulent

flows. They are derived in many papers and in nearly every book on turbulence modeling, e.g. Aldama [2], Lesicur [20], Pope [21] or Sagaut [23], from the Navier Stokes equations as follows :

1. One chooses a filter g(x) and an averaging radius b > 0. The large eddies u (of size > O(S)) are defined by filtering the underlying fluid velocity u :

I partially supported by NSF grants DMS 9972622, INT 9814115 and 1NT 9805563 2partially supported by the Deutsche Akademische Austauschdients (D.A.A.D.) {partially supported by NSF grants DMS 9972622, INT 9814115 and INT 9805563

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

54

2. To derive the equations for u, the Navier-Stokes equations are convolved

with 3. Ignoring boundaries and commuting convolution and differentiation leads to the space averaged Navier -Stokes equations, given by

i,-V S(u,p)+V (uul)=f,

(1)

where the stress tensor associated with the velocity and pressure averages

(u, p) is given by S(u, p) := 2vDD (u) - pI where DD(u) =

Vu + vu"' (2)

2

is the velocity deformation tensor.

One central problem in LES is the closure problem of modeling V. (uuT) in terms of u, see, e.g., Sagaut [23]. We shall show herein that there is in fact another possibly serious closure problem in steps 2 and 3 above leading to the incorrect space filtered equations (1). It is often reported in the LES literature that difficulties exist for simulating turbulence driven by interaction of flows with boundaries. In this report, we will show one reason: when the flow is given in a bounded domain with typical noslip boundary conditions and the strong form of the space averaged Navier-Stokes

equations is used, steps 2 and 3 lead to an 0(1) error near the boundary. A correct derivation of (1) (Section 2) reveals that an extra commutation error term A6(S(u, p)), see Definition 2.1, must be included in (1). We show, Proposition 4.2, that IIA6(S(u,P))IILP(Rd) - 0 as 6 0 if and only if the traction or Cauchy stress vector of the underlying flow is identically zero on the boundary of the domain ! In other words, the equations (1) are reasonable only for flows in which the domain's boundary exerts no influence on the flow. An inspection of the proof of Proposition 4.3 reveals that the commutation error A,s(S(u, p)) is largest at the boundary and decays rapidly as one moves away from the boundary. If the commutation error term is simply dropped and then the strong form of the space averaged Navier Stokes equations is discretized, as by, e.g., a finite difference method, the results of Section 4 show that the error committed is 0(1). On the other hand, variational methods, such as finite element, spectral or spectral element methods, discretize the weak form of the relevant equations. These methods are known to depend on the size of the H-1--norm of any omitted terms. We show in Section 5 that variational methods are possible since the H-' (1)-norm of the dropped commutation error does approach zero as 6 - 0, Proposition 5.1. Section 6 studies the weak form of the commutation error, (As(S(u, p)), v) for v fixed. The third main result, Proposition 6.1, is that the weak form of the commutation error tends to zero as 6 0. The order of convergence in two dimensions is at least O(61-E) with arbitrary e > 0.

Commutation Error of the Space Averaged Navier-Stokes Equations

55

The issue of the commutation error has appeared occasionally in the engineering community, e.g. see Furehy and Tabor [9], Ghosal and Moin (12], or Vasilyev et al. [25]. Its critical importance is beginning to be realized, see Das and Moser

[6]. One approach, [12, 251, has been to shrink the averaging radius 6(x) as x tends to the boundary of the domain; the correct boundary conditions are then clear : u = 0. This approach requires extra resolution and another commutation error due to the non constant filter width occurs. This other commutation error is usually ignored in the engineering literature on the basis of a one-dimensional Taylor series estimation of it for very smooth functions. Interesting and important mathematical challenges remain for this approach as well. Other special treatments of the near wall regions, such as near wall models, see (23, Section 9.2.2) for an overview, are common in LES to attempt to correct for the error. Recently, there are new approaches to LES without modeling, such as post processing [16] and the variational multiscale method by Hughes and coworkers (15].

2. The space averaged Navier-Stokes equations in a bounded domain To derive the correct space averaged Navier Stokes equations in a bounded domain, we will extend all functions to Rd and derive the equations satisfied by these extensions. Then, the new equations will be convolved. We will always use standard notations for Sobolev and Lebesgue spaces, e.g. see Adams [1]. For vectors and tensors (matrices), we use standard matrix-vector notations. Let 52 be a bounded domain in Rd , d = 2.3, with Lipschitz boundary 852 with

outward pointing unit normal n and (d - 1) dimensional measure 1851 < oo. We consider the incompressible Navier-Stokes equations with homogeneous Dirichlet boundary conditions

in(0,T)x52. in [0, T] x 52,

u = U It=o

0

= uo

.f, p dx = 0

in [0, T] x 852,

(3)

in 52,

in (0, T],

where v is the constant kinematic viscosity. It will be helpful to recall that the stress tensor S(u, p) is given by

S(u, p) = 2v® (u) - ph,

where II is the unit tensor, and that the normal stress / Cauchy stress / traction vector on 852 is defined by S(u, p) n. Our analysis will require that solutions (u, p) of (3) are regular enough such

that the normal stress has a well defined trace on the 852 which belongs to some

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

56

Lebesgue space defined on 851. We assume that

/

d

u E I H2(c) n Hi](i))

p E H1(c) n L2(S2)

for a.e. t E [0,T], (4)

UE

(H1((oT)))d

for a.e. X E 5 2.

Lemma 2.1. If (4) holds then S(u,p)n belongs to (H1/2(8c))d. In particular, for a.e. t E (0,T], S(u,p)n E (L9 (851))d with 1 < q < oo if d = 2 and 1 < q < 4 if d = 3 and IIS(u,p)nhI(L9(as:))d < C (vIIuII(jn(st))d + IIPIIH'(st))

(5)

Proof. This follows from the usual trace theorem and embedding theorems, e.g., see Galdi [10, Chapter II, Theorem 3.1].

Remark 2.1. The result that S(u,p)n E (L9(8Sl))d for 1 < q < 4 suffices for our purposes but it can be sharpened considerably. For example, Giga and Sohr [13, Theorem 3.1, p. 84] show that provided f is smooth enough and the initial condition (W2-2/s,s(S2))d, s > 0, holds, then for a.e. t > 0, ut and V V. (uu1) belong to U() E (LQ(51))d and further S(u, p)n E (L2 (4911)) for a.e. t > 0 when 3/q + 2/s = 4.

In writing down an equation like (1), f must be extended off 0 and then (u, p)

must be extended compatible with the extension of f. For f to be computable, f is extended by zero off 11. Thus, (u, p) must be extended by zero off 11, too. This extension is reasonable since u = 0 on 851. An extension of u off SZ as an (H2(Rd))d function exists but is unknown, in particular since u is not known. Using this extension, instead of u - 0 on Rd \ 51, would make the extension of f unknowable and hence f uncomputable in (1). Thus, define

u=0, u0=0, p=0 f=0 ifxv51. The extended functions possess the following regularities: /

\d

u E I Ho (Rd) I,

for a.e. t E [0, T],

pE

//

uE

(6)

d

(H1((o'r)))

for a.e. x E Rd .

From (4) and (6) follow thatthe first order weak derivatives of the extended velocity ut, Vu V u and V (uuT) are well defined on Rd, taking their indicated values in 0 and being identically zero off Q. Since u (H2(Rd))d, p §t H1(Rd), the terms V D (u) and Vp must be defined in the sense of distributions. To this end, let ,p E CC°(Rd). Since p = 0 on Rd \ S2, we get

(Vp)(v)

-

f

Rd

f, c (x)Op(x)dx - f P(s)p(s)n(s)ds an

(7)

Commutation Error of the Space Averaged Navier-Stokes Equations

57

In the same way, one obtains V . D (u) (v,)

:=

- JR

(u) (x)

(8)

Rd

,(x)V D (u) (x)dx - J o(s)D (u) (s)n(s)ds. lose

st

Both distributions have compact support. From (7) and (8) it follows that the extended functions (u, p) fulfill the following distributional form of the momentum equation:

f+

j

Jft

(2iiD(u) (s)n(s)-p(s)n(s) I o(s)ds. (9)

The correct space averaged Navier-Stokes equations are now derived by convolving (9) with a filter function g(x) E C°°(lR'). Let H(p) be a distribution with compact support which has the form

H(

)_-

JR't

f (x)& p(x)dx.

where 0. is the derivative of ,p with the multi-index a. Then, H * g E C°°(Rd), see Rudin [22, Theorem 6.351, where

H(x) = (H: g)(x) := H(g(x - )) _ - f f(y)aag(x - y)dy

(10)

Applying the convolution with g to (9), using the fact that convolution and differentiation commute on lRd, Hormander [14, Theorem 4.1.11, and convolving the extra term on the right hand side accordingly to (10), we obtain the space averaged momentum equation

f+ J

in g(x - s) [2vD (u) (s)n(s) - p(s)n(s)l ds

in (0, T] x Rd. (11)

Remark 2.2. If the viscous term in the Navier-Stokes equations is written as vAu instead of 2vV 1D (u), the resulting space averaged equation is given by replacing 2v11D (u) in (11) by vVu.

Definition 2.1. The commutation error A5(S(u,p)) in the space averaged NavierStokes equations is defined to be A6(S(u,p)) := ft g(x - s)(S(u,p)n)(s)ds. Jf

The correct space averaged Navier-Stokes equations arising from the NavierStokes equations on a bounded domain thus possess an extra boundary integral, A6(S(u, p)). Omitting this integral results in a commutation error. Including this integral in (1) introduces a new modeling question since it depends on the unknown normal stress on 8S1 of (u, p) and not of (u, p).

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

58

3. The Gaussian filter We will present the results in the following sections for the Gaussian tiller. Tliis filter fits into the framework of Section 2. We shall briefly present the filter's properties that are used in the subsequent analysis in this section.

- S=t 6=o.5

2

FIGURE 1. different 6

The Gaussian filter function in one dimension for

The Gaussian filter function has the form

9(x) =

6

(-)

d/2

exp

6 (_IIxII)

see Figure 1, where II ' 112 denotes the Euclidean norm of x E Rd and 6 is a user chosen positive length scale. The Gaussian filter has the foll(,%ving propert ies, xhich are easy to verify: regularity: g6 E C"'(IRd), 6Y, positivity: 0 < g6 (x) < ()' ,

integrability: IIg6IILn(Rd) < oc, I < p < oo, II96IILI(Ra) = 1,

symmetry: g6(x) = 96(-x), monotonicity: g6(x) ? 96(Y) if IIXII2

IIY112

Lemma 3.1. i) Let c 5E L°(1Rd), then for 1 < p < oc,

lii

1196 *'P -'PIILU(Rd) = 0.

Commutation Error of the Space Averaged Navier-Stokes Equations

59

ii) Let

li?n 1196 * W - iPII

Proof. The proof of the first two statements can be found, e.g. in Folland 18, Theorem 0.13]. The third statement is an immediate consequence of the first one.

For convenience, the Gaussian filter function with a scalar argument x is understood in the following to he 372-

9h (x)

62) .

exp (

4. Error estimates in the (IF(lRd))d-norm of the commutation

error term In this section, it is shown that the commutation error A5(S(u,p)) belongs to (LP(R'))d. We show that Ab(S(u, p)) vanishes as 6 -+ 0 if and only if the normal stress is identically zero a.e. on OSZ. As noted earlier, this condition means the wall

have zero influence on the wall-bounded turbulent flow. Thus, it is not expected to he satisfied in any interesting flow problem ! In view of Definition 2.1 and Lemma 2.1, it is necessary to study terms of the form gs(x - s)y?,(s)ds

(12)

Jasa

with V) E Lw(acZ), 1 < q < oc. We will first show, that (12) belongs to LP(lRd),1 <

p
Proposition 4.1. Let O E LQ(a%Z), 1 < q < oo, then (12) belongs to LP(!Rd), I <

p
IJast

s)V,(s)dsl <

(f ga (x - s)ds) 6

62rr)

rd/2 exp

II0II

(_6r

1/r

- SII2) ds I

As 2IIx - sll22 ? IIxII22 - 2IIsII2 it follows that exp

(_6rIIx - sII2) < J2

+q- = 1, q > 1,

1/r

2IIsIIz ) exp (3r - IIxIIz+ a2

J

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

60 and

119b(x - sMs)ds I

2

s)

II?PIIiq(ao)

(jau exp

J

Texp (--5

ds I

/

2

(13)

J

< 00.

since 8S2 is compact and the exponential is a bounded function. This proves the statement for L°°(Rd). The proof for p E [1, oo) is obtained by raising both sides of (13) to the power p, integrating on Rd and using

jexp (_37) dx < oo. 62 d

Ifq= 1 , we have for 1
110 96

(x - s)V,(s)dsI dx <

f

d

Eept ga (x - s)dx

(au)

IILIIL'(au).

fRd

We choose a ball B(0, R) with radius R such that d(x, 8S2) > 11x112/2 for all x ¢ B(0, R). Then, the integral on Rd is split into a sum of two integrals. The first integral is computed on B(0, R). This is finite since the integrand is a continuous function on B(0, R). The second integral on Rd \ B(0, R) is also finite because

J

d\K(o.K)

96 (d(x, 852))dx <

f

d

9a

(I 12)dx 2

and the integrability of the Gaussian filter. This concludes the proof for p < oo. For p = oo, we have esssup IJ g6(x - s)z/i(s)ds xERd

S2

< esssupesssup96(x-s)IIOIII.t(asl) <-96(0)IhlIIL'(arl)
8EaO

O

In the next proposition, we study the behaviour of the LP(Rd)-norm of (12)

for6-+0. Proposition 4.2. Let ' E LP(852), 1 < p < oo. A necessary and sufficient condition

for lim

= 0,

9 6(x - )V,(s)dsll

I

i

I.o(Rd)

1 < p < oo, is that 0 vanishes almost everywhere on 852.

(14)

Commutation Error of the Space Averaged Navier Stokes Equations

61

Proof. It is obvious that the condition is sufficient. Let (14) hold. From Holder's inequality, we obtain for an arbitrary function E C,°, (Rd),

JI,, w(x) ( fusi96(x - s)z(s)ds I dx < slim f gs(x - s) b(s)ds

slim

fix!

=0

(15)

L P(R11)

where pJ' + q`' = 1. By Fubini's theorem and the symmetry of the Gaussian filter, we have

/

\

aim fn ((x) I f gs(x - s)y5(s)ds I dx R

\\\ JQ

lim f V'(s) (I'd gs(x - s);p(x)dxds = f

/

s-e as!

.-(s);p(s)ds. Q

The last step is a consequence of Lemma 3.1 since :p E L'°(Rd) and tip is uniformly continuous on the compact set W. Thus, from (15) follows 0 = f w(s)v%(s)ds an

for every (p E C, ,(Rd). This is true if and only if zy(s) vanishes almost everywhere on d52.

We will now bound the L"(1Rd)-norm of (12) in terms of 6. The next lemma proves a geometric property which is needed later.

Lemma 4.1. Let 52 C Rd, d = 2,3 be a bounded domain with Lipschitz boundary 852. Then there exists a constant C > 0 such that

{x E IRdId(x,852) 0, where

I

I

denotes the measure in

(16)

Rd.

Proof. For simplicity. we present the proof for S2 being a simply connected domain.

The analysis can be extended to the case that 852 consists of a finite number of non-connected parts. We will start with the case d = 2. We fix a point x0 on 852 and an orientation of the boundary. Next, we construct x, such that the length of the curve between xO and x, is y. Continuing this construction, we obtain a sequence (x;)oo<;<,v such that for every 0 < i < N the length of curve between x; and x;+1 is y. The length of the curve between XN and xf) is less or equal than y, see Figure 2. The number of intervals is N + 1 with N < IOS2I/y < N + 1. Obviously, we have

{x E ]Rdld(x, 852) < y} = U B(x. Y). XEas!

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

62

X2

FicURE 2. Mesh on 812 for d = 2

But for every x in 812, there exists an i such that x is on the part of the curve from

xi to xi4.1 or from XN to x0. By the triangle inequality, this implies B(x, y) C B(xi, 2y). Thus

{x E ]Rdld(x, 812) < y} C U B(xi, 2y), O
from which N

I{x E Il2dld(x,812) < y}I

<

E IB(xi, 2y) I <

II

+ + 1 1 47ry2

1=0

4irlestly + 47ry2 follows.

In the case d = 3, 812 is a compact manifold. Then, for every x E 812, there exists a neighborhood Ux C 812 such that its closure Ux is homeomorphic to a closed square V. C 1R2 through the homeomorphism 4 : Vx - U. The homeomorphism is Lipschitz continuous with a constant L. We cover the manifold by

812= U Ux xEJ52

and, because 812 is compact, we can choose a finite cover (Ux;)u
fixed. Let the length of the sides of V, be equal to ai. We create a mesh over on Vx, of cells of size y/L (or smaller). On this mesh, there are less than (aiL/y+2)2 vertices and we denote them by (zj)n<j
IIZk - 0-'(z)II2 < L and the Lipschitz continuity of 0 gives

Ilm(Zk) - ZII2 S LIIZk - V'(0I2 S y

Commutation Error of the Space Averaged Navier--Stokes Equations

63

y/L

VX,

FiCURE 3. Homeomorphic map to the square V.,, d = 3

By the triangle inequality follows now B(z, y) C B(O(zk), 2y).

(17)

Because z E U., was chosen arbitrary, for every z E U,, there exists zk E V; such that (17) holds. Combining (17) for U.,, 0 < i < N, gives {x E ll

Id(x, all) < y} c U U B(Ozk), 2y). O
By the sub additivity and monotonicity of Lebesgue measure, we obtain

{x E R''Id(x,O) 5 y}I

S

II F I B(O(zk),2y)I 5

i=0 k = 0

<

i_ 0

( -+2) Z 3xy:j

C(y'i + y)

for an appropriately chosen positive constant C. Note, the quadratic term in y can be absorbed into the linear term for y < 1 and into the cubic term for y > 1. Proposition 4.3. Let S2 be a bounded domain in IRd with Lipschitz boundary OS2,

0 E Lr'(a1l) for some p > I and p-1 + q-' = 1. Then for every a E (0, 1) and k E (0, oo) there exist constants C > 0 and e > 0 such that k

gb(x - s)zv(s)dsl dx < Cb'+k(

2 -d)jjOjjk "

(ail)

for every b E (0, e) where C and a depend on a, k and I0SZI.

Proof. We fix an a E (0, 1). From Holder's inequality, we obtain

f

R

I f ga(x - s) 8S2

kdx (s)ds

< fR(111 ga(x-s)ds / k/4dx a

IIVII p(ast)

(18)

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

64

Let B(x,6Q) be the ball centered at x E Rd and with radius S'. Then, the term containing the Gaussian filter function can be estimated by the triangle inequality k/v

(fa,

g6(x - s)ds)

/ r

(x)dx\

dx < C(k) I J B6(x)dx + Jd C6

1

(19)

d

where 1/v

B6(x) = I J

\ OPfB(x,6^)

96(x - s)dsl

/

(

, C6 (X) _

II4

96(x - s)dsl (J \ 8Si\!3(x.6^)

with the constant C(k) depending only on k. We estimate the terms in (19) separately.

Using the monotonicity of the Gaussian filter, one can obtain the following inequality if d(x, 81l) < 60, C6(x) < C { ll 96 (d(x, OS2)) if d(x, 8Sl) > 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)<6^ }

RA

We will estimate the integral by integrating over the cross sections of the function

in the integral. For the function values t, 0 < t < g00), all cross sections have the same form. For function values t = g6(y), 0 < y < b°, the cross section is {x E IRd d(x, 8SZ) < y}. We denote the area of the cross sections by A(t). I

Integration of the areas gives r9a(6°)

f

g6(d(x, %I))dx =

4 (0)

Jg6(6^)

JU

J{d(x.as1)<6^}

A(t)dt

A(t)dt +

ge(0)

A(t)dt.

A(96 (b°))96 (b") + 9a(6^)

We will use now the estimate of the areas of the cross sections given in Lemma 4.1. If y is small enough, the term yd can be absorbed into the term y in this estimate.

Thus, if b is small enough, we have I{x E Rd I d(x,8S2) < y}I < Cy, 0 < y < 6°. We obtain, changing variables and applying integration by parts

JO)

A(t)dt < -C 1 ydy96(y)dy = C (_S0g(50) + J.

96 (y)dy

The last integral can be estimated further with the substitution y = bs:

6^

g6(y)dy <

Cb'-kd

(20)

v

with C depending on k. Collecting estimates, using A(g6(6")) < Cb°, which results in the cancellation of the terms b°g6(b"), we obtain

B6(x)dx <

Cbl-kd+ (AI)ok

Rd

The estimate for C, (x) converges exponentially for b -+ 0. Thus, for b sufficiently small, the estimate of B6(x) will dominate. This proves the proposition. 0

5. Error estimates in the (H-1(S2))d-norm of the commutation

error term The main result of this section is that the commutation error tends to zero in H-' (f2) as b -. 0, see Proposition 5.1. The order of convergence is at least 0(b1/2).

Lemma 5.1. There exists a constant C, which depends only on d, such that IIv - VIIHI 2(R.i) < Cb"2IIv1I Ht(RA)

for any v E H' (Rd) and any 6 > 0.

(21)

Commutation Error of the Space Averaged Navier Stokes Equations

67

Proof. By using the definition of II . IIHli2(R), we have

IIv -

f(i

vl121/2(Rd)

+ IIxI12)1/211 - 9a12'I2dx

Rd

(1 + IIxIIz)1/2I1 {IIxII2>n/a}

+f

(1 + IIXII2)1/211 - 9a121VI2dx, IIxII2<_n/a}

where v denotes the Fourier transform of v and the Fourier transform of the Gaussian filter is given by

((52

9e(x) = exp ` - 24IIXIl2)

(22)

First, we prove a bound for the first integral. There exists a constant C > 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

<

(1 + IIxII2)({ + IIxII2)-1/2IvI2dx {IIxII2>,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<_n/a}

Combining (23) and (24) gives

IIv-vIIW/2(Rd)
C1

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

68

Let H-' (R) be the dual space of Ho (52) equipped with the norm (v, w)

IIwIIN''(S2) =

vEH,1(R) IIv11H'(S2)

denotes the dual pairing. Proposition 5.1. Let 7P E L2(8S2), then there exists a constant C > 0 which depends where

only on SZ such that

f8 g6(x - s)O(s)ds R

< C61121101IL2(as2)

IH-'(R)

for every 6 > 0.

Proof. Let v E Ho (f2). Extending v by zero outside 52, applying Fubini's theorem, using that v vanishes on 852, applying the Cauchy-Schwarz inequality, the trace theorem and Lemma 5.1, give

I v(x) dx =

gb(x f2 \ 1,-, (?

s

aJR J8R

<-

i(s)v(s)ds 'b(s) (v(s) - v(s)) &

IIv -

Clly -vii H'(R) and using the definition of the H-1(Q) norm gives the desired

0

result. Let

N = {v E H'(Rd)

:

vlast = 0}

and let the assumption of Proposition 5.1 be fulfilled. An inspection of the proof shows that then also sup

g6 (x - s),O(s)ds H.H'(Rd)

E?{

vr aR

ga(x - sms)ds ) IIvUH'(Rd)

< C61121101IL2(as2)

6. Error estimates for a weak form of the commutation error term In this section, we consider a weak form of the commutation error term, A&(S(u, p)),

multiplied with a suitable test function v(x) and integrated on lRd. The following proposition shows that this weak form converges to zero as 5 tends to zero for fixed v(x). For d = 2, Corollary 6.1 and Remark 6.1 show that the convergence is (at least) almost of order one if 0 is sufficiently smooth.

Commutation Error of the Space Averaged Navier -Stokes Equations

69

Lemma 6.1. Let v E H1(Pd) such that visa E H11(52) n H2(1) and v(x) = 0 if < oo. Then x S2, and let 0 E LP(8152), 1 p
J

96(x - s)?i(s)dsdx = 0,

where U(x) _ (96 * v)(x).

Proof. By Fubini's theorem and the symmetry of 96, we obtain aim J 6~0 Rd

v(x)

aim

(fail

6~0 Jas:ON

96(x - s),O(s)ds dx

/

(J'd

g6 (s

- x)v(x)dx

/

ds.

By a Sobolev imbedding theorem, we get v E L'''(S2) from what follows by the construction of v that v E LO0(]Rd). Holder's inequality for convolutions gives v E L°°(]Rd). In addition, v is uniformly continuous on the compact set 852. The same holds for v since v E C°(52) by a Sobolev imbedding theorem. Applying twice Lemma 3.1 gives

lim 1 v(x) I J

96(x - s)V,(s)ds 1 dx = J *(s)v(s)ds = 0,

0

since v vanishes on 85l.

With the result of Proposition 4.3, we want to study the order of convergence with respect to 6 of the weak form of the commutation error term.

Proposition 6.1. Let v and O be defined as in Lemma 6.1 and let the assumption of Proposition 4.3 be fulfilled. Then, there exists an e > 0 such that for 6 E (0, e)

fR" v(x) aa g6(x - s)a('(s)dsl A dx < C61+(-d+

II OIIkP(asa)IIVIIk (s:),

where kE [1,oo),)3E (0,1) ifd=2 and,3=1/2 if d = 3, p-1+q-1

1, p> 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) <_ IIg611LI(Rd)IIv11La(Ra) = IIv11L9(Rd)

where 1 < q < oo. With the same argument, we get for qk > 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)<61})J

B'k (x)dx {d(x,8Q)<6^ }

pillkOD({d(x,dSt)<6^})61+(-d+(d vl)°)k

(26)

where a c (0, 1) and p-1 +q-z = 1. The last estimate is taken from the proof of Proposition 4.3. It remains to estimate the norm of v. By the triangle inequality, we obtain IIvII1.°°({d(x.9S2)<6^}) < Ilv - VIIL-({d(x,on)<6^}) + IIvIIL=({d(x,dS2)<6^})

Since z; E H2(52), we have v E

(27)

CO'. I3(?!) with 3 E (0,1) if d = 2 and 8 = 1/2 if

d = 3. That means, there exists a constant CH > 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)<6^}) <_ C,, °d.

Commutation Error of the Space Averaged Navier-Stokes Equations

71

The first term on the right hand side of (27) is, using that the L' (Rd) norm of the Gaussian filter is equal to one, IIv - v1lLo.({d(x.au)<6°}) = ess

sup

f 96(x -

xE{d(x.Ou)<6°}

y)(v(x) - v(y))dy

R°

Since v vanishes outside 11, it can be easily proven that Iv(x) - v(y)I 5 CHllx - Y112 holds for all x, y E Rd. We obtain, using the symmetry of the Gaussian filter, J

g6(x - y)(v(x) - v(y))dy

R°

< CH f ° g6(x - y)Ilx - yllzdy = CH f ° CCH6 f

Rd R°

The last integral is finite. Thus, we can conclude 1Iv - v11L-({d(x

l)<6°}) -< CCHO'

Since 5 decays faster for small 6 than 63', we obtain the estimate IlvllL° ({d(x.au)<6°}) - CHb3ok

Combining this estimate with (26) and using the estimate for CH, we get Iv(x)B6(x)lkdx <

C61+(-d+(d y,)°+3n)kIIv11;j2(u).

JRd

This dominates estimate (25) for small 5. Collecting terms, gives the final result. An easy consequence of Proposition 6.1 is the following

Corollary 6.1. Let the assumptions of Proposition 6.1 be fulfilled. Then, the weak form of the commutation error is bounded :

f -v(x) f 96(x - s)+/,(s)ds dx < d

Cb1-d+

+:3n lJ

IILa(ou)IIvlIH7(n)

(28)

il

Remark 6.1. Let d = 2 and p < oo arbitrary large. Then q is arbitrary close to one. Choosing a and Q also arbitrary close to one leads to the following power of 6 in (28): 1 + (-2 + (1 - ( 1) + (1 - E2)) = 1 - (e + e2) = 1 - E3 for arbitrary small e I, E2, e3 > 0. In this case, the convergence is almost of first order. The result of Proposition 6.1 does not provide an order of convergence for d = 3. Lemma 2.1 suggests choosing p = 4, i.e. q = 4/3. Then, the power of 6 in (28) becomes 2(a - 1), which is negative for a < 1.

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

72

7. The boundedness of the kinetic energy for u in some LES models The space averaged Navier Stokes equations

ur - vO-u + V (uuT) + Op

(29)

f + J g(x - s) [vVu(t, s)n(s) - p(t, s)n(s)] ds in (0, T) X Rd

Vi = 0 u It=o

in (0,T) x Rd

(30)

in Rd

(31)

= uo

are not yet a closed system since the tensor uuT is a priori not related to u and p. One central issue in LES is closure : modeling this tensor in terms of u and p. In this section, we will apply the results of Section 6 to show that the kinetic

energy of u is bounded for a number of LES models including the previously omitted A,(S(u, p)) commutation error term if 6 is sufficiently small. 7.1. The Smagorinsky model We consider first one of the simplest LES model which goes back to Smagorinsky [24]. This model is obtained by

u uT -

uuT

VUIIFVU + terms absorbed into Vp,

where C > 0 and IIVUIIF = (Vii: Du)1/2 is the Fhobenius norm of VU. The existence and uniqueness of a weak solution of the Smagorinsky model posed in a bounded domain, with homogeneous Dirichlet boundary conditions and without the boundary integral in (29) has been proven by Ladyzhenskaya [17, 18) and Du and Gunzburger [7]. The momentum equation of the Smagorinsky model has the form

u

((v+C SZIIDuIIF)Vu)+D (uuT)+np f + J g,5 (x - s)O(t, s)ds

in (0, T) x Rd

(32)

dst

with ap(t, s) = vVu(t, s)n(s) - p(t, s)n(s). We assume, there exists a solution (u, p) of (32), (30), (31) Multiplying (32) by u and integrating on Rd gives IIu11(L2(Rd))d

at

2

-f,V R

+ R'

fit '

J Ra

f Udx +

f

U(t, x)

(j a s:

\

gb(x - s),O(t, s)ds) I dx.

(33)

Commutation Error of the Space Averaged Navier Stokes Equations

73

Next, the terms on the left hand side are studied. Using the definition of u, changing variables, noting that u vanishes outside 0 and applying Fubini's theorem, we obtain f VP(x) - U(x)dX ^

f u(Y)

VP (Y + z)ga(z)dz dY

.

\/

(fRd

fR-

9s(z) (f VP(Y + z) . u(Y)dy/ I dz s

t

\

f0,dga(z) I - f5(y + z)(0 u)(y)dy + fi(s + z)u(s) n(s)ds) dz

\

111

since u is divergence free and vanishes on 0912. With an index calculation, using that

U is divergence free in Rd, and applying the same arguments as for the pressure term, one obtains

f

2f Rd

By applying Fubini's theorem in the same way as before, it follows that

- fd V. ((u +

f >

C,6211 VUII, )ou) . udx

d((v + C bzllouII,)vn) : v-ndx

0.

The first term on the right hand side of (33) is estimated by the Cauchy-Schwarz inequality and Young's inequality

_

f e Udx < ^

II"II(L2(Rd))d

IIfIl(L2(Rd))d 2

+

2

We obtain a at

IIuII(L2(Rd))d

2

IIfI.(L2(Rd))d 2

+

IIUII(L2(Rd))d

2

u(t, x) (L g,5 (x - s)zb(t. s)ds111 I dx. + f^ R an

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

74

Assuming f E L'(0, T; (L2(lR'))d), so that Gronwall's lemma can be applied, gives II u11 )II(lr'(Rd))' 2

2 (R"))"

< exp(T) +

J

f

+

f exp(T - t) f

IIf(t)II(L(Rd))d

dt

(34)

2

exp(T - t) ( fit u(t, x) U? g a (x - s)V,(t, s)dsl dxl dt. d

Thus, the kinetic energy of u at time T is bounded by the kinetic energy of the initial data, by the right hand side and by a third term, which vanishes as 5 -+ 0 by Proposition 6.1. For d = 2, it follows from Remark 6.1, (5) and Young's inequality I1u(T)II(1.2(R2))2 2

<

exp(T)11%)11(22(W))

+C51

'f

0

r

+ f exp(T-t)llf(t)IQ 27(RZ))Zdt

exp(T - t) (IIu(t)IItH2(f2))2 +

Ilp(t)112

(35)

dt

with arbitrary r > 0. Remark 7.1. One obtains the same result for the deformation tensor formulation of the momentum equation and a Smagorinsky model of the form

uj uu' -

terms absorbed into Vp.

7.2. The Taylor LES model The second model which we will discuss is called variously gradient method of the Taylor LES model developed by Leonard [19] and Clark et al. [4]. In the mixed Taylor LES model, a Smagorinsky model for the turbulent fluctuations is included in the model of the non-linear convective term given by

uuu u'' - C652II VUIIFVU + b2ouou'

.

The existence and uniqueness of a weak solution of the Taylor LES model in a bounded domain, equipped with homogeneous Dirichlet boundary conditions and without the boundary integral in (29) has been proven for C. large enough by Coletti [5]. We study the energy balance of the Taylor LES model including the term A6(S(u,p)) Inserting the Taylor LES model into (29), multiplying by u and integrating by parts, the non-linear convective, term and the pressure term vanish as in the Smagorinsky model. The bilinear viscous term is obviously non-negative. The nonlinear viscous term is treated in connection with the third term in the Taylor LES

Commutation Error of the Space Averaged Navier Stokes Equations

75

model. Using the same arguments as in the Smagorinsky model, one finds z

V. (_C,62jjVUjj,V- + 12v-vu'') Udx fd

r

VU

z

12(VuvuT)

: VUdx.

JR°

(36)

By norm equivalence in Rd x Rd, \3

d

Iai,I

d

IIAIIF < C(d)

Iaiil3)

we get fRd

C, a2IIVUIIFVU : vudx

f C b2IIvu1lFdx >

1,3 (Rd)

d

The second term in (36) is estimated in a similar way. Using twice the Cauchy one obtains 62

JRd

12(vuvuT) :vudx -- jd

62 121lvuIL.dx

- c(d)1211VUI1:3(R.i).

We get finally

-f

z

(_co2IIvuIIFvU + 12vuvu'') vudx

Rd 2

> C 6211vu11r (Rd) - C(d)12IIVUII,,3(Rd) -if C, is sufficiently large.

Now, we can continue as for the Smagorinsky model and obtain also the estimates (34) and (35) for the kinetic energy of u if C is chosen large enough and if 6 is sufficiently small.

7.3. The rational LES model The rational LES model was proposed in (11). Including the Smagorinsky model for the effects of turbulent fluctuations, there are two variants of this model, one of them has the form

uu'

z

u u' - C 52II VUII Fvu + 1296 * (vuvu') .

The existence and uniqueness of generalized solutions of the rational LES model in the space periodic case for small data and small times has been proven by Berselli et al. [3]. Proceeding as, in the Taylor LES model, the only difference is the term

f

1296

* (vuvu') : vudx.

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

76

The application of Fubini's theorem and the symmetry of the Gaussian filter yield

I g, * (Vuou") : VUdx = j V1 u' : (g, * VU)dx. d

,1

It follows with the same arguments as in the estimate for the Taylor LES model, using in addition Holder's inequality for convolutions,

I'

gh * (vuvut) : VUdX < d

Jd

I]VuII' Ilg6 * ouIIi.dX

IIVUIIL3(Rd)IIg6*vuIIL3(Rd) IIVull'L3(Rd)IIg6IIt.'(Rd) = lIVUUL3(Rd).

This gives the estimate 62

Jd

C'.b2II VU11F

- 12 go * (VUV i ')

(c1o' - CKC(d)12)

: Vudx

IIouII'L3(Rd) > 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

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[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


(1.3) (1.4)

for all t > 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
oo, r 54 1),

and

(1.4)forI
(1.3) and (1.4) are well established for these four types of domains. Let us give a brief survey on the literature concerning the L9-L' estimates. For the whole space the Stokes semigroup is essentially the same as the heat semigroup because the Laplace operator commutes with the Helmholtz projection. For the half space Ukai [65] explicitly wrote down a solution formula of the Stokes system and derived

(1.3) and (1.4) for n > 2 and I < q < r < oo. See also Borchers and Miyakawa [4] for (1.3) with 1 < q < r < no and the following literature concerning marginal

Navier-Stokes Flow Through an Aperture

83

cases, that is, (1.3) for q = r = oo and (1.4) for q = r = 1 or oo: Giga, Matsui and Y. Shimizu [31], Y. Shimizu [59], Desch, Hieber and Priiss [17] and Shibata and S. Shimizu [58]. For bounded domains (1.3) and (1.4) are deduced from the result of Giga [30] on a characterization of the domains of fractional powers of the Stokes operator. In this case, moreover, an exponential decay property of the semigroup

for large t is available. For exterior domains with n > 3, based on (1.3) for q = r due to Borchers and Sohr [9], some partial results were given by Iwashita [38], Giga and Sohr [33] and Borchers and Miyakawa [5]; in particular, Iwashita proved

(1.3) for 1 < q < r < oo and (1.4) for 1 < q < r < n, which made it possible to construct a global solution. Later on, due to the following authors, (1.3) for

n>2, 1 2, 1
were also derived: Chen [13] (n = 3), Shibata [57] (n = 3), Borchers and Varnhorn (11] (n = 2, (1.3) for q = r), Dan and Shibata [15], [16] (n = 2), Dan, Kobayashi and Shibata [14] (n = 2,3) and Maremonti and Solonnikov [50] (n > 2). In the proof of the L"-L' estimates, it seems to he heuristically reasonable to combine some local decay properties near the aperture with the L9-L" estimates of the Stokes semigroup for the half space by means of a localization procedure since the aperture domain Sl is obtained from H+ U H_ by a perturbation within a compact region. Indeed, Abels [1] used this idea that had been well developed

by Iwashita [38] and, later, Kobayashi and Shibata [40] in the case of exterior domains. We should however note that the boundary aQ is noncompact; thus, a difficulty is to deduce the sharp local energy decay estimate He-'"fllw v(nR) <_ Ct-,,12q ilfllf. (n)i

t > 1.

(1.5)

for f E L,(5l),1 < q < oc, where S1R = {x E S2; Ixi < R}, but this is the essential part of our proof (Lemma 5.3). Estimate (1.5) improves the local energy decay given by Abels [1], in which a little slower rate t-"/2q+= was shown. In [1], similarly to Iwashita [38], a resolvent expansion around the origin A = 0 was derived in some weighted function spaces. To this end, Abels made use of the Ukai formula of the Stokes semigroup for the half space ([65]) and, in order to estimate the Riesz operator appearing in this formula, he had to introduce Muckenhoupt weights, which caused some restrictions although his analysis itself is of interest. On the other hand, Kobayashi and Shibata [40] refined the proof of Iwashita in some sense and obtained the LQ-L' estimates of the Oseen semigroup for the 3dimensional exterior domain. As a particular case, the result of [40] includes the estimates of the Stokes semigroup as well. In this paper we employ in principle the strategy developed by [40] (without using any weighted function space) and extend the method to general n > 3. This paper consists of six sections. In the next section, after notation is fixed, we present the precise statement of our main results: Theorem 2.1 on the L"-L" estimates of the Stokes semigroup, Theorem 2.2 on the global existence and decay properties of the Navier-Stokes flow, and Theorem 2.3 on some further asymptotic behaviors of the obtained flow under an additional summability assumption on

84

T. Hishida

initial data. We obtain an information about a pressure drop as well in the last theorem. Section 3 is devoted to the investigation of the Stokes resolvent for the half

space H = H+ or H_. We derive some regularity estimates near the origin \ = 0 of ()1+A11)`PH f when f r= Lq(H) has a bounded support, where AH = -PHA is the Stokes operator for the half space H (for the notation, see Section 2). Although the obtained estimates do not seem to be optimal compared with those shown by

[40] for the whole space, the results are sufficient for our aim and the proof is rather elementary: in fact, we represent the resolvent (A + in terms of the semigroup e` ` A" and, with the aid of local energy decay properties of this semigroup, we have only to perform several integration by parts and to estimate AH)_1

the resulting formulae. One needs neither Fourier analysis nor resolvent expansion. In Section 4, based on the results for the half space, we proceed to the analysis of the Stokes resolvent for the aperture domain Q. To do so, in an analogous way to [38], [40] and [1], we first construct the resolvent (A+A)-1 P f near the origin A = 0

for f E Lq(11) with bounded support by use of the operator (A+ AH)_1PH, the Stokes flow in a bounded domain and a cut-off function together with the result of Bogovskil [2] on the boundary value problem for the equation of continuity. And then, for the same f as above, we deduce essentially the same regularity estimates near the origin A = 0 of (A + A) `P f as shown in Section 3. In Section 5 we prove (1.5) and thereby (1.4) for q = r E (1,n] as well as (1.3) for r = cc, from which the other cases follow. Some of the estimates obtained in Section 4 enable us to justify a representation formula of the semigroup a `APf in W 1 q (5211) in terms of the Fourier inverse transform of e9" (is + A) -1 P f when

f E L9(S2) has a bounded support, where n = 2m + 1 or n = 2m + 2 (see (5.3); we note that the formula is not valid for n = 2). We then appeal to the lemma due to Shibata ([561; see also [40] and a recent development [58]), which tells us a relation between the regularity of a function at the origin and the decay property of its Fourier inverse image, so that we obtain another local energy decay estimate (1.6) t> 1, for f E Lq(12),I < q < oc, with bounded support, where e > 0 is arbitrary

IIe-`APIIIWI..,(fH)
(Lemma 5.1). Estimate (1.6) was shown in [1] only for solenoidal data f E Lg(12) with bounded support, from which (1.5) with the rate replaced by t-"/29+e follows through an interpolation argument. But it is crucial for the proof of (1.5) to use (1.6) even for data which are not solenoidal (so that the support of Pf is unbounded). In order to deduce (1.5) from (1.6), we develop the method in [38] and [40] based on a localization argument using a cut-off function. In fact, we regard the Stokes flow for the aperture domain S2 as the sum of the Stokes flows for the half spaces Ht and a certain perturbed flow. Since the Stokes flow for the half space enjoys the Lq-L"° decay estimate with the rate t-"/2q (Borchers and Miyakawa [4]), our main task is to show (1.5) for the perturbation part. In contrast to the case of exterior domains, the support of the derivative of the cut-off function touches the boundary 812; indeed, this difficulty occurs in all stages of

Navier-Stokes Flow Through an Aperture

85

localization procedures in the course of the proof (Sections 4 and 5) and thus we have to carry out such procedures carefully. Furthermore, the remainder term arising from the above-mentioned localization argument involves the pressure of the nonstationary Stokes system in the half space and, therefore, does not belong to any solenoidal function space. Hence, in order to treat this term, (1.6) is necessary for non-solenoidal data, while that is not the case for the exterior problem. Once Theorem 2.1 is established, one can prove the existence part of Theorem 2.2 along the lines of Kato [39] (see also [26] and [32]) and therefore the proof may be omitted. Thus, in the final section, we derive various decay properties of the

global strong solution as t -. oo to prove the remaining part of Theorem 2.2 and Theorem 2.3. This will be done by applying effectively the L9-L' estimates. Recently Wiegner [66] has discussed in detail sharp decay properties of exterior Navier-Stokes flows. Our proof is somewhat different from his and seems to be elementary. When a E L' (SI)nL" (S2), some decay rates are better than those shown by [66] since, unlike exterior Stokes flows, (1.4) is available for 1 < q < n < r < oo.

Finally, we compare the result on De-°' with that for exterior Stokes flows from the viewpoint of coercive estimates of derivatives. For the proof of (1.4) there is another approach based on fractional powers of the Stokes operator. When 11

is an exterior domain (n > 3), Borchers and Miyakawa [5] developed such an approach and succeeded in the proof of IIVUIIL9(s:) <_ CIIA'"2UIIL9(s2),

u E D(A1/2),

(1.7)

for 1 < q < n (this restriction is optimal as pointed out by themselves [6]), which implies (1.4) for q < r < n. Independently, as mentioned, Iwashita [38] derived (1.4) for q < r < n and, later, Maremonti and Solonnikov [50] showed that the restriction r < n cannot be improved for exterior domains. In our case of aperture domains, we have (1.4) for q < n < r < oo, which is a consequence of the estimate due to Farwig and Sohr ([22, Theorem 2.5]) IIV UIILq(u) <_ CIIAuIILq(u),

u E D(Aq),

(1.8)

for 1 < q < n together with an embedding property ([22, Lemma 3.1]); we mention that (1.8) holds true for n = 2 as well. This argument does not work for the exterior problem because (1.8) is valid only for 1 < q < n/2 (n > 3) as shown by Borchers and Sohr [9] (the restriction on q is again optimal by, for instance, [6]). Thus, as

for (1.8), we have the better result. We wish we could expect (1.7) for every q, which would imply (1.4) for 1 < q < r < oo; however, so far, no attempts have been made at the boundedness of purely imaginary powers of the Stokes operator (see [30] and [33] for bounded and exterior domains) and, unless q = 2, estimate (1.7) remains open.

2. Results Before stating our main results, we introduce notation used throughout this paper. We denote upper and lower half spaces by Hf = {x E R"; ±Xn > 1}, and

T. Hishida

86

sometimes write H = H+ or H_ to state some assertions for the half space. Set B« = {x E Rn; IxI < R} for R > 0. Let S2 C Rn be a given aperture domain with smooth boundary 852, namely, there is Ro > 1 so that SZ\BR

in what follows we fix such Re. Since Sl should be connected, there are some apertures and one can take two disjoint subdomains SZ± and a smooth (n - 1)dimensional manifold M such that 52 = 52+ U M U 52_, Stf \ BR = Hf \ BR° and Al u 8M = 852+ n 852_ c RIZ. We set OR = 52 n BR and HR = H n BR, which is one of H±.R = H± n BR, for R > 1. For a domain G C Rn, integer j > 0 and 1 < q< oo, we denote by Wj.9(G) with the standard L9-Sobolev space with norm II ' ll j,,,c so that L9(G) = is the completion of CO (G), the class of C°° norm II ' Ilq.G. The space functions having compact support in G, in the norm II' Il j.q.C, and W-j!9(G) stands for the dual space of Wo'g1(q-1) (G) with norm II ' ll _ j,q,c. For simplicity, we use ' Ilj,q.si when G = 12. We often the abbreviations II ' I1, for II ' Ilq,n and II ' use the same symbols for denoting the vector and scalar function spaces if there is no confusion. It is convenient to introduce a Banach space II

L1Rj (C) = {u E L9(C);supp u C GR}, G = S2 or H,

for R > 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
operator St.R from Co (D±.R) to C00°(D±.R)' such that for 1 < q < oo and integer j > 0 CIIV'fllq,D+.tt,

(2.2)

with C = C(R, q, j) > 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
J as t -* 0

(2.4)

if q:5 r < oo,

,l ast-*oo ifq
where o = (n/q - n/r)/2. Furthermore, for each precompact set K in Lo(SZ) every convergence above is uniform with respect to f E K.

T. Hishida

88

Remark 2.1. Estimate (1.4) for large t is not proved in the following cases: (i)

n n. Remark 2.2. Let 1 < q < r < oc (q oo, r i4 1). The Lq-Lr estimate for ate-tA with the rate t-"-' is nothing but a simple corollary to (1.3). In fact, for example,

(late-`Aflla 5 Ct-n/2sllAe-(t/2)Aflls < Ct-n/2-'llfll.,

fort>Oand f EL'(5l)nLo(Sl). By use of the Stokes operator A, one can formulate the problem (1.1) subject to the vanishing flux condition O(u(t)) =

It N u(t)du = 0,

t > 0,

J

(2.6)

as the Cauchy problem

ttu + Au + P(u Vu) = 0, t > O; u(O) = a.

(2.7)

in Lo(S2). Given a E Lo(S2) and 0 < T < oc, a measurable function u defined on SZ x (0, T) is called a strong solution of (1.1) with (2.6) on (0, T) if u is of class u E C([0, T); L" (11)) n C(O, T; D(An)) n C' (0, T; L'(52))

together with lim llu(t) - all, = 0 and satisfies (2.7) for 0 < t < T in L"(1). nextttheorem tells us the global existence of a strong solution with several The decay properties provided that llalln is small enough.

Theorem 2.2. Let n > 3. There is a constant 6 = 6(11, n) > 0 with the following property: if a E Ln(1) satisfies llalln < 6, then the problem (1.1) with (2.6) admits a unique strong solution u(t) on (0, oo), which enjoys llu(t)IIr = o (t

1/2}n/2r)

for n < r < oc,

ilVu(t)lln = O (t-1/2 )

,

llatu(t)Iln + IIAu(t)IIn = 0 (t-') ,

(2.8) (2.9) (2.10)

ast -+oo. Remark 2.3. When one prescribes a nontrivial flux

O(u(t)) = F(t) E

C'.e([O,T]),

with some 0 > 0 and T > 0, there is T. E (0.T] such that the problem (1.1) with the flux condition admits a unique strong solution on (0,T«) provided that a E Ln(c2) satisfies the compatibility conditions V a = 0, v ala1, = 0 and O(a) = F(0). This improves a related result of Franzke [25] and can be proved in the

Navier-Stokes Flow Through an Aperture

89

same manner as the proof of Theorem 2.2 with the aid of the auxiliary function of Heywood [34, Lemma 11], which is used for the reduction of the problem to an equivalent one with the vanishing flux condition (2.6). As is well known, (2.4) and (2.5) as t -. 0 play an important role for the construction of the above local solution.

Remark 2.4. The solution obtained in Theorem 2.2 is unique within the class

Vu E C(0, oo; L'(1)), without assuming any behavior (6.4) near t = 0 as pointed out by Brezis [12]. For the proof, one needs the final assertion of Theorem 2.1 on the uniform behavior of the semigroup as t -i 0 on each precompact set K in L; (12) together with the theory of local strong solutions mentioned in the previous remark (with O(u) _ u E C([0, oo); L"(Sl)),

F = 0). In fact, it follows from the above property of the semigroup that the length of the existence interval of the local solution can be taken uniformly with respect to a E K and that the convergence (6.4) of the local solution as t 0 is also uniform with respect to a E K. These two facts combined with the classical uniqueness theorem of Fujita-Kato type [26] (assuming some behaviors in (6.4) near t = 0) imply the desired uniqueness result. Remark 2.5. Consider the 3-dimensional stationary Navier-Stokes problem in SZ subject to wait = 0 and a nontrivial flux condition O(w) = y E R. When IryI is small enough, there is a unique solution to such that to E LQ(12) for 3/2 < q < 6

and Ow E Lr(S1) for 1 < r < 2 with IIVwIJ = -Y[lr]; see Galdi [28]. By use of Theorem 2.1 it is possible to show the asymptotic stability of the small stationary solution to of the class above for small initial disturbance in Lo (St) in the sense that the disturbance u(t) decays like (2.8) and (2.9) as t oo. In fact, the above summability properties of Vw allow us to deal with the term P(w Vu + u Ow) as a simple perturbation of the Stokes operator, as was done by Chen [13, Lemma 3.11 and Borchers and Miyakawa [7, Theorem 3.131; see Remark 6.1. The final theorem shows further decay properties of the global solution when we additionally impose L'-summability on the initial data. Theorem 2.3. Let n > 3. There is a constant r) = n(11, n) E (0, b] with the following property: if a E L'(Sl) fl L"(fl) satisfies IIail,, < n, then the solution u(t) obtained in Theorem 2.2 and the associated pressure p(t) enjoy

IIt(t)Ilr = 0

/ 1

t-(n-n/r)/2)

for 1 < r < oo,

IIVU(t)Ilr = 0 (t\-(n-n/r)/2-t/2) Ilatu(t)IIr + IIAu(t)IIr = 0

Ilo2u(t)IIr + Ilop(t)llr =

(t-(n-n/r)/2-1)

O\(t-(n- /r)/2-1)

for 1 < r < oo,

(2.11) (2.12)

for 1 < r < oo,

(2.13)

for 1 < r < n,

(2.14)

90

T. Hishida

as t - oo. Moreover, for each t > 0 there exist two constants p±(t) E R such that P(t) - P±(t) E L' (Q±) with

IIP(t) - P+(t)II r.?} = O (t-("-"/' )/2-'/2) for n/(n - 1) < r < oo,

IP+(t) - P-(t)I = O as t

(t-n/2-1/2+a)

.

(2.15)

(2.16)

oc, where e > 0 is arbitrarily small.

Remark 2.6. Indeed Vu(t) E L" (Q) for r > 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 <-

Ct-j/2-k(1 +

IIq.H,

(3.1)

fort > 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 <_ CIIAi12UIIr.H,

(3.2)

for 1 < r < cc and j = 1, 2 (Borchers and Miyakawa [4]). For 1 < p < q < r < oo it follows from (1.3) for S2 = H, (3.2) and a property of the analytic semigroup that IID'aiE(t)PHIIIv,11n


qH, +

for t > 0, f E Lq(dj (H) and j = 0, 1, 2. This estimate with p = q = r implies (3.1)

for 0 < t < 1. We may assume that 0 < e/n < min{1/q, 1 - 1/q}; and then one can take p and r so that 1 - 1/p = 1/r = e/n and p < q < r. Then the estimate above yields (3.1) for t > 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) <_ CIIfIID(Aq.H),

which implies (3.3) for 0 < t < 1. For t > 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
T. Hishida

92

We next employ Lemma 3.1 to show some regularity estimates near A = 0 of the Stokes resolvent in the localized space W2,q(HR).

Lemma 3.3. Let n > 3, 1 < q < oo, d > 1 and R > 1. Given f E

(H), set v(A) _

(,\+A,,) 'PH f . For any smalls > 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,, <-

fW t" IIO'E(t)PHfIIq.H,,dt
provided that

j=0,1 ifk=0; j=0,1,2 ifn>5, 1

Navier-Stokes Flow Through an Aperture

93

The remaining case k = m is the most important part of (3.4). Since 5 Cm!II(A+AH)-(,»+1)PHfIlD(A,.H) IaI-(,"+1)}IIII1,11, < Cm!{IAI "` +

Ilaa v(A)Ilz.q.lln

we have the assertion for lAl > 1. For 0 < IAl < 1 and odd n (resp. even n), we have already shown the estimate as above when j = 2 (resp. j = 1,2). Thus, let j = 0 or 1 for n = 2m + 1 and j = 0 for n = 2m + 2. We divide the integral of (3.6) into two parts 1/lal

a v(A)

1

+f

e-at(-t),r,E(t)PHfdt

= w1 (A) + w2(A)

/lai

Then (3.1) implies IlV3W1('1)Il9.HR < CIAI-3+,/2IIfilq.H,

On the other hand, by integration by parts we get

for f E w2(A) =

a

IaI J

E

IaI PHf + f

at [(-t)mE(t)PHf] dt,

in Lq(HR) since (3.1) implies tlim t"'liE(t)PHfllq.lln = 0. With the aid of (3.1) again we see that C.C

llV w2(A)IIq.H <

Ilo38t [t-E(t)PHf] IIq.HHdt I IIo'E(I/IaUPHfllq,H,, + 1 f Ial 1/la1 CIAI-'1+"2IIfiiq.H,

for f E L' (H). Collecting the estimates above leads us to (3.4). We next show (3.5). Since let - 11 < 21'e1ai°te for Re \ > 0 and 0 E (0,1], we have llV (v(A) - w)IIq.11R < 21-01A1°

J0

t0 IIV'E(t)PHfIIq.Hndt,

for j = 0, 1, 2. From (3.1) together with a suitable choice of 0 (for instance, 0 < 1/2 for n = 3), we conclude (3.5).

Remark 3.1. When n = 2, one can show IAI llvll2.q.11n <_ Cllf llq,H (with )3 = e) which corresponds to (3.4) with m = 0. However, this will not help us since our key formula (5.3) is not valid for m = 0.

Remark 3.2. The Green tensor associated with the Stokes semigroup E(t) for the half space (as well as the projection PH) was explicitly given by Solonnikov [61], Maremonti and Solonnikov [50, section 2]. In view of the simple relation fo°(41rt)-"12e-I21'/4tdt = r(n/2)/2(n - 2)7r"/2lxl"-2 for n > 3, the function w in (3.5) is the solution written by the Green tensor for the stationary Stokes problem

in H and, thereby, we know the class of w (for the latter Green tensor, see for instance [28]).

T. Hishida

94

Finally, we derive further information on the regularity of the resolvent along the imaginary axis.

Lemma 3.4. Let n >3,1 1 and R > 1. Set

44 (s)=89(is+AH)PH (sER\{0}, k=mor m-1), where i = %f--1. Then, for any small c > 0, there is a constant C = C(n, q, d, R, e) > 0 such that Il4i 7 (s + h)f - 41(1f) (s)f II2.q.HR < CIhI

ISI-s-'

(3.7)

IIf J1q.H,

II4?H1-1)(s + h)f - -tti-')(s)f 112,q,HR < CIhI

(3.8)

IsI-,3IIf l1q.11,

where m and ,C3 =,3(e) are the same as in

for h E R, Isl > 21h1 and f E Lemma 3.3.

Proof. Estimate (3.8) is a direct consequence of (3.4). In fact, we see that re+n

1)(S+h)f

')(S)fII2,9Hr,

5 J

II4'(Hn)(T)fII2.9.HRdT

9

9+h


ITI

L

3dT

,

which together with the relation Is + hl > Isl - IhI > IsI/2 implies (3.8). We next show (3.7). By (3.6) with Re A = 0 in LQ(HH) we have

-DH )(s+h)f -4'tii)(s)f {j1111 +

f W e-ist(et - 1)tE(t)PH fdt = (-i)m(w1 +w2). 1

For the convenience we introduce the function

Fk(t) = 8,k[t-E(t)PHf],

k > 0.

We then deduce from (3.1) 5Ct-k+m-1(1+t)-'12+1+e

IIFk(t)1I2,9,HR

(3.9)

IIfII,.H,

for t > 0 and f E Lq (H). Taking Ie-'ht -11 <- hit into account, we see from (3.9)

that 1/IAI IIw1112.q,HR

5 Ihl f

tII Fo(t)112.q.HRdt

0

5 CIhIIIf II q.ft

<

for f E w21 =

Clhllsl-a-'

f

'/191

t'+

/2+`dt

By integration by parts we split w2 = w21 + w22 + w23, where

i(n+h)1I8IFO (I) s(sz+h)e

ISIse

is/1"I(e-ih/191

- 1)F1

\ISI/

Navier-Stokes Flow Through an Aperture w22 _

ih s(s + h)

-i J

W23 =

95

t/181

oo

e-"t(e-iht - 1)F1(t)dt.

1/181

s

Since 1/ls(s + h)I < 2/1812 for 181 > 21h1, it follows from (3.9) that <

IIw21112,q,HR

3IhIIsI-2IIFo(1/I5I)112.q,HR

CIhIIsl-2-m+n/2-E(1 + 131)-n/2+1+E11Aq,11

CIhII3I-"-' II!IIq.H,

and that < 21h1 1s1-2 f

IIFt(t)112.q.Hndt

IIw22I12,q.HR

x/181

< CIhIISI-2IIfIIq,H f

t-t+m-n/2+Edt

t /181

Clhllsl-"-' Ill II

for f E

We perform integration by parts once more to obtain w23

w231 + w232 + w233 with

w232 =

W233 =

h

°°

2(8 + h)

1/181

-1

°°

2J

S2

e-t(s+h)tF..2(t)dt,

e-,st(e-tht - 1)F2(t)dt.

/181

By the same way as in w21 + w22 we find IIw231 + W232112.q.HR

<

3IhII3I-3 {I1F11/1sI112Q.HR

+

f

l IIF2(t)112.q.HRdt } J

1/181

<

for f E

Clhllsl-"-' IllIIq.H, Finally, we use (3.9) again to get

11w233 II2.q, Hn

S 1hIISI-2 f

tIIF2(t)II2.q.Ht:dt

1/181

< CIhIIsI-21If IIq.H f

t-1+m-n/2+8dt

1/181

< Clhllsl-"-' IIf IIq,H,

for f E

We gather all the estimates above to conclude (3.7).

T. Hishida

96

Remark 3.3. Estimate (3.7) together with (3.4) implies II$(m) (s + h)f -t'-) (s) l`II2,v.H ds < CIhI'-0IIf II q.H,

for h E R and f E

(see Lemma 4.4 and its proof), which is related to the assumption of Lemma 5.2. In Lemma 4.4 we will deduce the same regularity

of 8q'(is + A)- Pf for an aperture domain St as above when f E Lq (fl) has a bounded support. For the Oseen resolvent system in the 3-dimensional whole space, Kobayashi and Shibata [40, Lemma 3.6] showed a sharper estimate; indeed, IhI1-f1 can be replaced by Ih]1/2. Their method is different from ours.

4. The Stokes resolvent In this section, based on the results for the half space obtained in the previous section, we address ourselves to analogous regularity estimates near A = 0 of the Stokes resolvent u = (A + A) -'Pf, which together with the associated pressure p satisfies the system

Au-Du+Vp=f, in an aperture domain ) subject to ulas2 = 0 and q(u) = 0, where f E Lq(i2), I < q < oo and A E C \ (-oo, 0]. To this end, as in [38], [40] and [1], we start with the construction of the resolvent near A = 0 for f E Lq(l) with bounded support. We fix a smooth bounded subdomain D so that flRo+3 C D C Q. Given f E Lq(f1), we set vo = AQ 'Pq,D f and take a pressure ao associated to vo; they solve the Stokes system

-Lvo + O7ro = f,

V vo = 0, in D subject to volat, = 0, where f is understood as the restriction off on D. We further set

v±(x,A) _ (A+Aq,Hf)-1Pq,Ht[)3.Rf], whereV)f.Ro are the cut-off functions given by (2.1). One needs also the case A = 0

vf(x,0) = JE±(t)Pq.H±ttb±,Rof]dt, which is the solution written by the Green tensor for the Stokes problem in Hf (see Remark 3.2). We take the pressures 7rf in H+ associated to of so that J-± (x, A) - ao(x)}dx = 0,

(4.1)

D±. n'+,

for each A. In this section, for simplicity, we use the abbreviations aPf for the cutoff functions V;±,& +l given by (2.1) and S± for the Bogovskii operators S±,Ro+s

Navier-Stokes Flow Through an Aperture

97

introduced in section 2. With use of {v±, rr±}, {vo, iro} and 7P± together with Sf, we set v = T(A)f

=i(i+v++*-v-+(1-V1+-?P-)vo - vo) -S+[(v+ - v0) VV5+] - S[(v_

7r

(4.2) VV)- 1,

= +a+ + Vi-7r_ + (1- 7 / 1 + - i('_ )Tro.

We here note that (vt - v0) V ±dx = 0 since V v± = V vo = 0. An elementary calculation shows that the pair {v, n} satisfies

Av - Ov+Va= f +Q(A)f,

(4.3)

in Il subject to vlac = 0 and

O(v) = f N ' voda = f

S2+nD

M

V ' vodx = 0,

where

Q(A)f = QI(A)f + Q2(A)f

(4.4)

with Q1(A)f

-(0'+;+)(v+ - VO) - (AV)-)(v- - Vo) +(Dtp+)(i+ - lro) + - 'aa)

-AS+[(v+ - v0) ' V +] - AS-[(v_ - v0) VV)-], and

Q2(A)f = oS+[(v+ - VO) . VV'+] + OS-[(v- - VO) VVI-]-

By (2.2) we have S±[(v± - vo) VV,±] E W0'1(D±_,+I). But one can obtain the regularity of this term only up to WO" (while the WW'q-regularity of the corresponding term is available for the exterior problem). This is the reason why the remaining term Q(A) has been divided into two parts. We first derive the regularity estimates near A = 0 of T(A) and Q(A).

Lemma 4.1. Let n > 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
(4.7)

On the other hand, from the Stokes resolvent system we obtain

AM v, + kak-'v, - Daavf + vaa7rf = 0,

1 < k < rn,

in H. This combined with (4.7) gives II(V ±)aairt(A)IIq < CllV8 77r (A)11-1,q.Df.R,,,.1 <

C11oa1'vt(A)IIq.Hj.RO}2

+CIalllaav±(A)IIq.H=.R0+2

+Ckllaa-'vt (,\) Ilq,rit.R,,, 2,

for 1 < k < m. Similarly, for k = 0, we use (4.1) to get < <

11(V '±)(irt(A) - iro)11q

CIIV(r±(a) - 7r0)II-I.q,D,.R,+, CjjVV±(A)Ilq.11t.RO,2 + C1A111V±(A)IIq.11t.R0,2 + Gill llq.

It thus follows from (3.4) that

,n-i lal`jll(oVvt)aa -t(a)Ilq + > II(oV5t)aa(-±(A) - 710)IIq < CIIfIIq, k=0

for Re A > 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
IIVWIIq/(q-n),G 5 C,12 - Wllq/(q-1),G 5

Therefore,

(9,o'w)I = I(-Vg, w)l < CIIv911-i.q.Gh011,/(q-1),G,

for all i E L9/(q-')(G), which implies the desired inequality.

Navier-Stokes Flow Through an Aperture

99

Let us consider the case A = 0 and simply write v± = v± (x, 0). Since II(v± - vo) vV)+112,q S Cll.f!Iq,

the operator If 1--* (v+ - v0) 00+] : L9(52) - Wo'q(D+.RO+1) is compact, which

combined with (2.2) implies that so is the operator Q2(0) :

LQ(S2)

Lq (11),

where d > Ro + 2. The other part Q1(0) f fulfills IIQ1(0)f111.q <- ClIfl1q,

from which the compactness of Q1(0) : L9(S2)

LIdl (S2) follows; as a consequence,

Q(0) = Q1(0) + Q2(0) is a compact operator from Lqld](1), d > Ro + 2, into itself. We will show that 1 + Q(0) is injective in L[ d) (a). Let f c L[ d] (12) satisfy (1 + Q(0)) f = 0. In view of (4.3), the pair {v, 1r} given by (4.2) for such f should obey -ov+Vir=o,

V v=o,

in S2 subject to vlssz = 0 and 0(v) = 0. Since f E L'[d) (Sl) for 1 < r < min{n,q}, we have ,V2v, Vir E L' (S2), Vv E LR''/("-')(S2), v, 7r E L10

especially, the summability of Vv at infinity is implied by the boundedness of the support of f. It thus follows from Theorem 1.4 (i) of Farwig [18] that v = Vir = 0; here, it should be remarked that the uniqueness holds without any radiation condition (unlike the exterior problem discussed in [38] and [40]). We go back to

(4.2) to see that v± = V7r± = f = 0 in H± \

and that vo = Da0 = f = 0

in 11Ro+1. Set U+ = (D U BRO) fl H. Both {v+,'rr+} and {vo, 1rO} then belong to W2,4(U+) x W1.9(U+) and are the solutions of the Stokes system in U+ with zero

boundary condition for the external force f. They thus coincide with each other and, in view of (4.2) again, we have vo = OTro = f = 0 in D; after all, f = 0 in Q. Owing to the Fredholm theorem, 1 + Q(0) has a bounded inverse (1 + Q(0))-1 on Lid] (52).

Set E,1 = {A E C; Re A > 0,0 < IAI < ill for tj > 0. By (4.1) we have {ir+(x,A) - 7rt(x,0)}dx = 0, which together with the Stokes system in Ht yields

fD±.Ro+l

<

II(VTG±)(ir+(a) - ir+(0))IIq

CIIV(i+(A) -

Ir+(0))11-1.q.Df.RO+,

CIIV(v+(A) - v+(0))IIq.Ht.RO+2 + CIAIIIv+(A)IIq,Ht.RO+z'

Therefore, we get <

IIQ(A)f - Q(0)fllq

-

CIIy+(A) v+(0)lil.q.ii+.RO+2 + CIIv-(A) - v-(O)II1.q.H_.RO+2 +CI.I{IIy+(A)IIq,H+.RO+2 + IIy011q.D}. Ily-(A)IIq,H_.RO+2 +

From (3.5) we thus obtain

IIQ())-Q(O)IIa(fq,"](u))

0,

100

T. Hishida

as A - 0 with Re A > 0, which implies the existence of a constant rI > 0 such that 1 + Q(A) has also a bounded inverse (in terms of the Neumann series) on Le(st) with uniform bounds

II(1 +Q(A))-lltDt d,s:
(4.8)

for A E E, U {0}. Since the resolvent is uniquely determined, one can represent it (1), d > Ro + 2, as for A E Eq and f E

(A+A)-'Pf =T(A)(1+Q(A))-'f.

(4.9)

We are in a position to show an analogous result for the resolvent to (3.4).

Lemma 4.2. Let n > 3,1 < q < oo, d > Ro and R > Re. Given f E L q (11), set u(A) = (A+ A) -' P f . For any small e > 0 there is a constant C = C(1l, n, q, d, R, e)

> 0 such that IIaau(A)II2.q.::R < Cllfllq,

IAI';IIaa U(A)II2.q.uR +

(4.10)

k=O

for Re A > 0 (A 36 0) and f E

(Q), where m and,3 = /(e) are the same as in

Lemma 3.3.

Proof. The problem is only near A = 0 because we have (2.3) for G = Q. We may

for such d. It thus suffices to also assume d > R + 2 since L[R1(Sl) C show (4.10) for A E E,1 by use of (4.9). For such A and 0 < k < m we see that aa(1 +Q(A))-' E B(L'K(Sl)); furthermore, m-1

lAI'3Ilaa (1 +Q(A))-'fllq + E llaa(1 +Q(A))-'fllq s Cllfllq,

(4.11)

k=0

for f E Lq (Q). In fact, we have the representation

aa(1 + Q(A))-'f -(1 +Q(A))-'[aaQ(A)1(1 +Q(A))-'f + Lk(A)(1 +Q(A))-'f,

(4.12)

fork > 1 and f E

where LI(A) = 0 and Lk(A) with k >_ 2 consists of finite , aa-'Q(A). Consequently, (4.6) sums of finite products of (1+Q(A)) ', ajQ(A), together with (4.8) implies (4.11). In view of aau(A) = ( 8,k\ _j T(A) aa(1 + Q(A))f, i=0

we conclude (4.10) from (4.5) and (4.11).

In the last part of this section we will complete the regularity estimate of the resolvent. To this end, we employ Lemma 3.4 to show the following lemma.

Navier-Stokes Flow Through an Aperture

101

Lemma 4.3. Let n > 3,1 < q < oc, d > Ro and R > RU. Set. T(k)(s) = &.T(is), Q(k)(s) = 88Q(is) (s E R\ {0}, 0 < k < m). For any small e > 0 there is a constant C = C(12, n, q, d, R, e) > 0 such that IIT1k) (s + h)f - V') (3)f 11 2,9J1,, + IIQ(k) (s + h)f - Q(k) (s)f IIq CIhIIsi-3-1IifIIq

if k = m,

if k = m - 1,

CIhIIsi-3IIf1Iq CIhIIIfIIq

ifn>5.0
for 2Ihl < Isi < 1 and f E

(4.13)

where m and,3 = (3(e) are the same as in

Lemma 3.3. Concerning the first term of the left-hand side, (4.13) holds true for

hERandlsl>2lhl. Proof. Set

vfk)(s) = 8s v±(is),

7r

(k-)(s)

= 8 7rf(is)

(s E R \ {0}, k = m or m - 1).

It then follows from (4.2) together with (2.2) that IIT(m)(s+h)f -T(m)(s)f(I2,gJ1,, <

Cllv+"`)(s + h)

- v(m)(3)II2,q.H,..r + CIIv(m)(s + h) - v(n')

In order to estimate Q("'), let us investigate the pressures 7r("'). Similarly to the proof of Lemma 4.1 with the aid of (4.7), one can show II(V '±){7r('')(s + h) - 7r(m) (s)}IIq

C11 V7rf')(s+h) - V7r )(s)II <

Cllov(m)(s+h) - Vv+'")(3)II4 Ht.rep+2

+CII(s + h)vf')(s+h) - 37Jfm'(3)II q,Ht.rta+2

+Cmlly. 1) (s + h)

- v±

This combined with estimates on the other terms by use of (2.2) yields IIQ(m)(s + h)f - Q(m)(s)fIIq CIIv(")(s + h) v(m)(s)II I,q.H+.,,+2 +Cllv(m)(s + h) -V c,n)(3)II1.q,H_.r:o+,

-

+CIsIIIv+m'(3+h)-v± +CIsIIIv(n')(s + h)

-

(Y)IIq.H;..R0,2

+Cmlly(r"-1)(s + h) - v(74 -1)(8)IIq.H+.rt(,+2 v(rn-1)(s)IIq,H_,R,,..,. +Cm1lv(ni-1)(s + h)

-

Hence (3.7), (3.8) and (3.4) imply (4.13) for the case k = m. For 0 < k < m - 1 we have s

IIT(k)(s+h)f -7(k)(s)fII2.q,11k < f

+hIT1(T)fI2,q,iIRdr,

T. Hishida

102

$+h

IIQ(k)(s + h)f - Q(k)(s)f II9 < f

IIQ(k+))(T)fIigdT ,

A

which together with (4.5) and (4.6) respectively lead us to (4.13). The proof is thus complete.

The regularity of the resolvent along the imaginary axis given by the following lemma plays a crucial role in the next section.

Lemma 4.4. Let n > 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.

+f2jhj<jsj52ho + Js I >2h

W ith the aid of (4.10), we find

I, < 2 f

CIhII-3IIff1q,

Isl<<31hI

for f E

In order to estimate

we use the representation

(»`)(y)f = E (j ) T(--j)(,) V(j)(s) f, j=e

where

V(j)(s) = dy(1 +Q(is))-1 E B(Ljdl(I))

(0 < IsI <

71,

0 < j < m).

Then,

h)f - V-) (s) f

(m )

[T('"-j)(s+h)-T(",-j)(s)) V(j)(s+h)f

j=))

+[ (nt ) j=()

7

T("i-j)(s) [V(j)(s+h) - 0-7)(s)]f

Navier-Stokes Flow Through an Aperture

103

We first show

IIV(j)(s+h)f -V(i)(s)f11, if j = m, CIhIIsI-'IIfiiq if j = m - 1, CIhIIsI-a-'IIfIIq

<

(4.15)

if n > 5, 0 < j < m-2,

CIhIIIf IIq

Similarly to the proof of (4.13) for 0 < k < 2h° and f E m - 1, (4.11) implies (4.15) for 0 < j < m - 1. As in (4.12), we have for 2IhI < IsI

V(m)(s) = -v(°)(s)Q(m)(s)v(0)(s) +Wm(s)v(0)(s),

where Wi(s) = 0 and, for m > 2, Wm(s) = i"`L,,,(is) consists of finite sums of ,Q(m-')(s). Therefore, we collect (4.6), (4.8), finite products of V°(s),Q(')(s) (4.13) and (4.15) for j = 0 to arrive at (4.15) for j = m. It thus follows from (4.5), (4.11), (4.13) and (4.15) that II,P(m)(s + h)f - D(m)(s)f II2.q.fzR < CIhIIsI-3-'IIf IIq, As a consequence, we are led to for 2IhI < IsI < 2h° and f E

12 < CIhIIIfIIq f

IsI-3-1ds < CIhI1-3IIfIIq

I91>21h1

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 <_ CIhIIIfIIq f

IsI-("'+')ds < CIhIIIfIIq,

IsI>2ho

for f E Lq(SZ). Collecting the estimates above on 1,,12 and 13, we conclude (4.14).

0

5. Lq-Lr estimates of the Stokes semigroup In this section we will prove Theorem 2.1. As explained in Section 1, the first step is to derive (1.6) for non-solenoidal data with bounded support.

Lemma 5.1. Let n > 3,1 < q < oo, d > R° and R > Ro. For any small E > 0 there is a constant C = C(S2, n, q, d, R, E) > 0 such that ApfII (5.1) <_ Ct-'12(1 + t)-"/2+1/2+e IIf IIq, He-

fort > 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 <_

Ml hl 27rj1 - etht I

Taking h = 1/t immediately implies the desired estimate. Proof of Lemma 5.1. Since )lle_tApfll,V2 <
Ile-tAPIII'.9
(5.2)

for 0 < t < 1 and f E LQ(Sl), we will concentrate ourselves on the proof of (5.1) for t > 1, namely (1.6). Given R > RD, we set 4' 1= 1 - 0+,R - zk_.R, where the cut-off functions '+Vt.R are given by (2.1). One can justify the following representation formula of the semigroup for f E We

00

to p f

2at'"

(5.3)

where c("')(s) = a (is + A)-'P and m is the same as in Lemma 3.3. In fact, starting from the standard Dunford integral representation, we perform rn-times

Navier-Stokes Flow Through an Aperture

105

integrations by parts and then move the path of integration to the imaginary axis but avoid the origin . = 0, so that

W00 1 r a+ (

.,tt

e-tAPf =

J

7rtm

s

4

eatrbi (A+A)-'PfdA, +2aitm IF, for any 6 > 0, where I'b = {be'B; -ir/2 < 0 < 7r/2} (this formula is valid for

f E Lq(SZ) without Vi). Owing to (4.10), the last integral vanishes in Lq(1) as 6 -, 0 for f E thus, we arrive at (5.3). Now, it follows from (4.10) and 5 CIIb(m)(S)fIID(Aw)IIb("')(S)f119/1 together with (2.3) that

f II+G'P(m)(s)f11I.gds
IIfl3gds+C f

"IS1

00

ISI

IIf1k

ds
ISI

Further, (4.14) and the estimate above respectively imply that

ff oghpna lhl'-3 1

°O

+h)f


and that 1

hupo

Ihl'-p

f f

h'?0 33

11V4("') (s + h)f -(s)fII,,gds II

("`)(S)f 111,ds < CII f [I,.

Hence, we can apply Lemma 5.2 with X = W',9(11) and g(s) = t,14("')(s) f to the formula (5.3); as a consequence, we obtain II e-tAPf 1I1,"0. <_ JIV a-LAPf IIi q < Ct-m(1 + t)-'+aIIfIIq,

for t > 0, which implies (5.1) for t > 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 <- Cllv±(t)111.q.Ht.t. 5 C(1 +t)-12g11f11"

(5.8)

for t > 0, where L = max{R, RD + 1}. Thus, in order to estimate u(t), let us consider

v(t) = u(t) - u+(t) - u_(t),

ir(t) = p(t) - p+(t) - p-(t),

which should obey

atv-Av+Vir=K, in 1 subject to vlast = 0, q5(v) = ¢(u) = 0 and vlt_o = vo = 9 - 9+ - 9- E

n D(Aq),

where

K = 20W+ Vv,. + 2V _ Vv_ + (AO+)v+ + (0'V'-)v-AS+[v+ . VV'+] - AS- [v_ ' VV'_ J +S+[atv+. V7P+1 + S_[ety- ' V'_] - (VV ,+)7r+ - (0'O-)-

Navier-Stokes Flow Through an Aperture

107

we here note that V - K # 0 as well as KI v1134 0 and we can obtain the regularity of K only up to Lq (in contrast to the exterior problem discussed in [381 and [40]). By (5.7) and in view of the Stokes system in Hf we have

II(V ±)nf(t)IIq

CIIVir (t)ll-1.q.Dt.Ra CII VV±(t)II9,f1.,.,, +, + CIIOtV±(t)Ilq,H4.RO+,,

which together with (2.2) implies K(t) E L?,+1(52) and IIK(t)IIq <_

CIIV+(t)II1,q,H+.RO+ +CIIv-(t)lll,q.tr_.,,0+, +CiIaty+(t)IIq.H+,RO+, +

CIIOty-(t)IIq.H_.R0+,

Therefore, Lemma 3.2 and (5.6) yield IIK(t)ll9 < C(1 + t)n/2gllfllq,

(5.9)

for t > 0. In order to estimate e-(t-T)APK(T)dT,

y(t) = e-tAvO + J n

we employ Lemma 5.1. By (5.1) with a suitable e > 0 and (5.6) we find Ile-tAVOII1,q,s1R

lit-n/2+fllyollq for t > 1. We next combine (5.1) with (5.9) to get J

<

t

Ct-n/2qIlfll9,

Ile-(t-T)APK(T)II1.gs1RdT

CIIfIlq

J0

t (t-T)-1/2(l+t-T)

n/2+1/2+e(1+T) n/2gdr

CII fIlq(I1 + 12),

where I, =

fo/2

and 12 = ft/2. An elementary calculation gives

Ct-112(1 + t/2)-n/2-n/2q+3/2+E Ct-1/2(l + t/2)-n/2+1/2+e log(1 + t/2) Ct-112(1 + t/2)-n/2+1/2+E for t > 1 and

1, <

if q > n/2 if q = n/2 if q < n/2

< Ct-n/2q

12 < (1 +t/2)-n/2q fW T-1/2(1 +T)-n/2+1/2+edT < C(1 + t/2) -n/29, 0

for t > 0. We collect the estimates above to obtain Ily(t)Il1,9,1)R < Ct-n/241lfllq,

(5.10)

for t > 1. From (5.8) and (5.10) we deduce llu(t)II1,9,oR = IIt(t) + u+(t) + u-(t)II1,9s1R < Ct n/2gllf ll9,

for t > 1 and f E L9(1), which proves (5.4). Let f E D(Aq). Then we easily observe Ile-"f 111.9siR +llate-t"fIl9,ilR S Clle-tA flID(A,,)

CIIflID(A,),

T. Hishida

108

for t > 0 and also we can estimate Ote-tA f for large t; in fact, by virtue of (5.4) just proved we get IIate-t'4fIIq.cH

= Ile-tAAfll.,.nH < Ct-n/2'IIAfIIq,

for t > 2. This implies (5.5).

We are interested in the Lg estimate of De-tA for large t, in particular, the Ln estimate is quite important for us. Lemma 5.4. Let n > 3 and 1 < q < oo. Then there is a constant C = C(Q, n, q) > 0 such that Ct-min{1/2.n/2q}IIf1Iq,

IIVe-1Afllq <-

(5.11)

fort > 2 and f E Lo(ll). Proof. We fix R > Ro + 1. Since we have already known the decay rate t-n/2q of Iloe-tAfllq.lt5 by Lemma 5.3, it suffices to derive the estimate outside S2R, that is,

IIDe

rnfllq.tt*\nH


(5.12)

for t > 2 and f E Lo(S2). In an analogous way to [381, [40] and [1], we make use of

the decay properties of the semigroup E±(t) for the half space. Given f E La(I), we set g = e-A f E D(Aq) and then u(t) = e-tAg = e-(t+1)A f. We choose two pressures pf in 11 associated to u in such a way that Dt.H-,

p±(x,t)dx = 0,

(5.13)

for each t (p+ and p_ will be used independently). With use of the cut-off functions given by (2.1) and the Bogovskii operator introduced in section 2, we define {vt,

r±}

by

vf(t) = V'±u(t) - S±[u(t) . VO±1, 'r±(t) = V,fpf(t) Here and in what follows, we use the abbreviations +b± for 0±,R-1 and Sf for S±.R_ 1. Since vt = u for x E Sgt \ OR = H± \ BR, we will show IIVV±(t)Ilq.H.


(5.14)

for t > 1, which combined with II9IID(Aq) S CIIf IIq implies (5.12) for t > 2. It is

easily observed that {vf,rf} satisfies atv±-Av±+Vr±=Z±,

in Hf subject to vjI,H± = 0 and

ofIt_o =af where

Zt = -2V ,± Vu - (OO±)u+OSf[u. V' -S±[atu VV'±1 + (VV)pt.

Navier-Stokes Flow Through an Aperture

109

Our task is now to estimate the gradient of

v±(t) = E±(t)a± + J t Ef(t - r)PH,Z±(r)dr.

(5.15)

v

By virtue of (5.13) we have II(Vl1)±)P±(t)IIq.Hf

< CIIVP±(t)II-1.q.Df.R-1 < CIIVu(t)Ilq.su, + CIIatu(t)11olR,

from which together with (2.2) it follows that IIZ+(t)IIq.H± < CIIu(t)III.q.uR + CIIatu(t)IIq.s1R.

Hence, (5.5) implies IIPujZ±(t)Ilr.llf <_ CIIZ±(t)IIq.11t s C(1 + t)-n/2gIIgIID(AQ),

(5.16)

for t > 0 and r E (1,qj since Zt(t) E LJR)(Hf) C LIRI(H±) for such r. In view of (5.15), we deduce from (1.4) for 1 = H# together with (5.16) Ilov±(t)119.Ht Ct-'/2ijatll9./J0, t +CII9IID(Aa)

<

(t - r)-'/2(1 + t - r)-(n/' -n/9)/2(1 + r)-nt2gdr

Ct-1/2IIgiiq +CII9IID(Aq)(I1 + 12),

for r E (1,q], where Il = fo/2 and 12 = f}/2. We take r so that 1 < r < min{n/2, q}. Then we see that

Il <

Ct-112(1 + t/2)-'1/2r+1

if q > n/2

Ct-1/2(1 + t/2) -n/2r+1 log(1 + t/2) Ct-1/2(1 + t/2) -(n/r-n/q)/2

if q = n/2 if q < n/2

< Ct-'/2,

for t > 0 and that

+ t/2)-/2q if q > n, I2 0. Collecting the estimates above concludes (5.14). This completes the proof.

The following lemma is concerned with the L°° estimate of the semigroup (the restriction q > n will be removed later). Lemma 5.5. Let 3 < n < q < oc. There is a constant C = C(11, n, q) > 0 such that

lie-'Af1I. 5 Ct n/2gIlfII9, fort > 0 and f E L9 (Q).

(5.17)

T. Hishida

110

Proof. For fixed R > R0 + 1, estimate (5.4) together with the Sobolev embedding property implies II e

Af II..uR < Ct-"129II f IIq,

for t > 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 <- Ct-"/2gIIIIIq,

(5.18)

fort > 2. In fact, given f E Lo(Sl), we take the same g, {u, pi} and {vi, Tri}, and apply the Lg-L°° estimate (1.3) for S1 = Hi to (5.15). Then, taking (5.16) into account, we get <

Ilvi(t)II=.H± Ct-"12gllaillq,H± t

f (t - r)-"/29(1 + t -

r)-("/r-"/q)/2(l +.r)-"/2gdr,

0

for r E (1, q]; we now choose r E (1, n/2) to find IIvi(t)IIa.H± < Ct-"/2'IIgIID(AQ),

for t > 1, which proves (5.18) for t > 2. We thus obtain (5.17) for t > 2. For 0 < t < 2, we recall (5.2) to see IIe-`14fll_ <- CIIe-1Aflli

q°Iie-`AfII9-"/q < Ct-"12gIIIIIq

0

The proof is complete.

We are now in a position to prove Theorem 2.1. Abets [1] showed (1.3) for 1 < q < r < cc; when we use this result, the first step of the following proof will become shorter. However, in order to make the present paper self-contained, we do not rely on any result of [1]. We emphasize that our proof is based on (5.11) and (5.17), in other words, the other estimates follow from them. Proof of Theorem 2.1. The proof is divided into four steps. Step 1. First of all, we observe (1.4) for q = r E (1, n]. Indeed, it follows from (5.2)

forO2that Ct-'12IIf IIq,

IIVe-1Af IIq

for t > 0 and f E Lo (ft) provided 1 < q

(5.19)

n. In this step we accomplish the proof

of(1.3)for1 n in Lemma 5.5. In view of (5.19) and the Sobolev embedding property we have Ile-

IAfllr

< Ct-112IIfIIq,

(5.20)

for t > 0 and f E Lo(Q) when 1 < q < n and 1/r = 1/q - 1/n. Let n/(k + in such a way that 1) < q < n/k with k = 1, 2, , n - 1. We put {qj

Navier-Stokes Flow Through an Aperture

111

gt)= q. Since n
Ct-n/2qk IIe-a/2)Af

Ilgk <

Ct-,3/2q,.-k/2 II f llq,

, n/(n - 1). But the excepfort > 0, which proves (5.17) except for q = n, n/2, tional cases can be also deduced via interpolation. Thus the L'I-I,°° estimate (5.17) has been established for all q E (1, oo). This together with the L`t boundedness (namely, (1.3) for q = r) immediately gives (1.3) for 1 < q < r < oc, from which combined with (5.19) we further obtain (1.4) for 1 < q < r < n.

Step 2. In this step we prove (1.4) for 1 < q < n < r < oo, making use of (1.8) due to [22]. Given r E (n, cc), we take s c (n/2, n) so that 1/s = 1/r + 1/n. When 1 < q < s, an embedding relation given by Lemma 3.1 of [22] together with (1.8) implies IIVe-taflh_ < CIIV2e-tAflls <-

CIIAe-Afll6

<

Ct-'Ile-(t/2)Afll9,

for t > 0, from which together with (1.3) we obtain (1.4). If s < q < n, which implies r < q. with l/q. = 1 /q - 1/n, then by the same reasoning as above IIoe-tnfllr

<

IlVe-tAfIlq.Blloe-tAfIlq < CIIAe-``%fII

for t > 0, where 1/r = (1 - 8)/q. +8/q = 1/q - (1 - 0)/n. Therefore, (5.19) yields (1.4).

Step 3. Let f C L' (St) n L' (Sl) for some s C (1, oo). This step is devoted to the case q = 1, namely Lt-L' estimate. Let 1 < r < oo. We apply a simple duality argument; in fact, the Lq-L°° estimate implies

I(e-tAf,9)I =

I(f,e-tA9)I <-

Ilf11,Ile-tAgII,


for g E L /(r-1) (Q), which gives (1.3) for q = 1 < r < oc. Combining this with (5.17) and (1.4), respectively, we obtain (1.3) for q = 1 < r = oo and (1.4) for

q=1
Step 4. Once the Lq-L' estimates (1.3) and (1.4) are established, (2.4) and (2.5) can be proved by means of a standard approximation procedure. We show only

the behavior as t -, or, (which is the main concern in the present paper). Let 1 < q < oo and f E Lo (Q). For any e > 0 we take ff E such that II ff - f lI q < e. It then follows from (1.3) that

Ile-tAfIlq 0, which immediately yields tli IIe-tAfIlq

= 0,

(5.21)

since e > 0 is arbitrary (one can give another proof by use of ker(Aq) = {0}). Let K be a precompact set in Lo(ft). For any q > 0 there is a finite set {fj IT 1 C K so

T. Hishida

112

that {B,1(fj)} 1 is a covering of K, where B,(fj) denotes the open ball centered at fj with radius n. Then we have

sup lle-`AfIIq 5 Cq+ lmax Ile-`Afjllq. Hence, from (5.21) we deduce

lim sup

IIe-tA f II q

`JOIEK

(5.22)

= 0.

All the other decay properties as t - oo follow from (5.22) combined with (1.3) and (1.4). We have completed the proof.

6. The Navier-Stokes flow In this section we apply the developed Lq-L' estimates of the semigroup to the Navier-Stokes initial value problem. In the proof of Theorems 2.2 and 2.3, we will not cite (1.3) and/or (1.4) if the application is obvious. We first prove Theorem 2.2.

Proof of Theorem 2.2. One can construct a unique global solution u(t) of the integral equation

r t e-(t-r)AP(u Vu)(r)dr, t > 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 <- Ct-7IIallq + C

f (t - r)-1/2IIu(r)IInIIVu(r)Ilndr, U

Navier-Stokes Flow Through an Aperture

113

which together with (6.3) implies t' IIu(t)IIn < CIIaII9 + CIIaIin sup r IIu(T)IIn, O
fort > 0. Hence, for any ry E (0,1/2) there are constants 6. = b.(l,n,ry) E (0,6o]

and C = C(fl,n,-y) > 0 such that if IIaIIn < 5., then IIu(t)lln < Ct-'Ilallq for t > 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 <- 6 and any e E (0,S], we take aE E C07.(0) so that Ila, - all. < E. Since Ila,lln <- 5., the corresponding global solution fulfills (6.6). We combine this fact with the continuous dependence: Lo(12) E) u(0) '--* u E BC([0, oo); Ln(fl)), where BC denotes the class of bounded continuous functions. As a consequence,

the global solution u(t) with u(0) = a satisfies IIu(t)lln < Ce + CO + t)-7, which proves (6.5) (although the method above was mentioned in [391 and is well known, we gave the proof for completeness; see also Theorem 3 of Wiegner [661 for another

proof). Combining (6.5) with (6.2) for r = oo immediately leads us to (2.8) for n < r < oo. We next prove (2.8) for r = oo and (2.9). As is standard, we rewrite the integral equation (6.1) in the form t

u(t) = e-(t/2)Au(t/2) -

e-(t-')AP(u Vu)(r)dr,

1/2

t > 0.

(6.7)

Then we obtain

llu(t)II- + Ilou(t)lln

/2(t Ct-112 llu(t/2)Iln + C 1/2 (t

-

r)-3/4

JIu(r)I12nllou(t)llndr,

from which together with (6.3) we at once deduce

t'/2(Ilu(t)II. + llou(t)lln) <-

CIIu(t/2)lln +CIIalln

Sup

t/2
T'/'IIu(T)II2n,

for t > 0. Obviously, (6.5) and (2.8) for r = 2n conclude both (2.8) for r = oo and (2.9). These immediately yield

IIP(u. ou)(t)IIn <- CIIu(t)II-IIVu(t)II, = 0(t-'),

(6.8)

as t -+ oo, which will be used to show (2.10) below. Fix 0 E (0, 1/2) arbitrarily. We then observe Ilu(t) - u(r)ll. + IIDu(t) - Vu(T)Iln < C(t - T)BT-'/2-AIIall n,

(6.9)

T. Hishida

114

for 0 < T < t. Given f E L?(St), 1 < q < n, we have IIe-tA f - e-rA film +

f

Ilve-t-4 f

-

Ve-TA flln

(lie-(4/2)AAe-('/2)Afllx +

t

C(t

r) T

Ilve-(A/2)AAe-(9/2)Afiin) ds

n/2q-1llf

llq,

which implies Re-tA f

- e-TAf II. + live-tAf

- V,-rAf lln <- C(t - T)0T-n/2q-0 II f II q,

for 0 < r < t, where 0 < 0 < 1. Using this estimate together with

u(t) - u(r) = (e-'A _ e-1A )a -

-f r{e-(t-s)A u

f

' e-(t-.,)AP(u Vu)(s)ds

- e-(T-.)A}P(u Vu)(s)ds,

we obtain

IIn(t) - u(r)II. + Ilvu(t) - VU(T)II.

< C(t - r)°r-'/2-011aiin +C f t(t +C f

r(t - r)°(r -

S)-n/2T-1/2-BIIU(S)IIrIIvu(S)IIndS,

n

where we have taken p, r E (n, oo) so that 0 < n/2r < n/2p = 1 /2 - 0. Then (6.2) and (6.3) yield (6.9). From (6.7) and (6.9) one can deduce the representation

Au(t) = Ae-(t/2)A u(t/2) + {e-(t/2)A - 1}P(u Vu)(t) + Az(t),

(6.10)

in La(S2), where e-(t-T)AP{(u Vu)(t)

z(t) = 1/2

- (u Vu)(T)}dr.

In fact, (6.9) implies IIAz(t)lln

< C f t (t -

T)-1+0T-1/2-0(Ilvu(t)IIn

t/2

< Ct1/2llvu(t)lln +Ct-' sup

t/2
+ IIu(r)II.)dr

T'/2IIu(r)II00,

for t > 0. As a direct consequence of (2.8) for r = oc and (2.9), we see that IIAz(t)Iln = o(t-1), as t - oc. In view of (6.10), we collect (6.5), (6.8) and the above decay property of Az(t) to obtain IIAu(t)Iln = o(t-1) as t -, oo, which together with (6.8) again shows (2.10). The proof is complete.

Navier-Stokes Flow Through an Aperture

115

Remark 6.1. Consider briefly the 3-dimensional stability problem mentioned in Remark 2.5. The problem is reduced to the global existence and asymptotic behavior of the solution to

u(t) = e-'A a -

e e.-(1-r)AP(u Vu + w Vu + u Ow)(T)dT, J0

t > 0,

where w is a stationary solution of class Vw E Lr(1). 1 < r < 2, and a e L(Q) is a given initial disturbance. Set

E(t) = sup 7-"2(IIu(T)II. + IIVu(r)113) + SUP rl/a11u(T)II6, 0
li
and fix r E (1,3/2) arbitrarily. Then the integral equation yields the a priori estimate E(t) < CIIa1I3 + CE(t)2 + COVW IIr + 11 Vw112)E(t), for t > 0, which gives an affirmative answer to the stability problem provided

that both IIVwIIr + IIDw112 and 11a[13 are small enough. In fact, by following the argument of Chen [13], the above inequality for E(t) is deduced from lie-tAP(w Vu + 21. Vw)& + Iloe-tAP(w . Vu + u Ow)113 < C(Ilowllr + IIVwI12)(IIuII. + IIouJI:i) t .i/a(1 + t) 3/2r+3/4 and

<

IIe-tAP(w . Vu + U. Ow)II(, C(IIOwllr + IIOw!I2)(IIujI. + IIVuII3) t-'/2(1 + t)-3/2r+3/4

We next assume that a E L' (S2)nL'(S2) with IIa[In < 6. Let u(t) be the global solution constructed in Theorem 2.2. Our particular concern is more rapid decay properties of u(t). Starting from (6.2) and (6.3), we observe that u(t) E W'-'(Q) for 1 < r < n and t > 0 (without going back to approximate solutions). In fact, there is a constant M = M(Sl, n, r. Ilall i, IIal!,,, T) > 0 such that IIV'u(t)ll,.

<

Ct-j12IIaIIr +CJt (t 0

T)-I+n/2r-j/2

lu(T)llnllVu(T-)IIndT

< Mt-j/2, for n/2 < r < n, j = 0, 1 and 0 < t < T, where T > 0 is arbitrarily fixed; and then,

IIV'u(t)Ilr
< Mt-j/2

t

f(t-T)-J/2IIu(T)112rIIDu(T)Il2rdr

0

for n/4 < r < n/2, j = 0, 1 and 0 < t < T. We repeat the process above to get

u(t) E

for I < r < n with sup (Ilu(t)II,.+t'/211VU(t)Ilr) < M.

(6.11)

11
Remark 6.2. Following the argument of Kato [39], we see that the above constant M does not depend on T > 0 if IIaII is still smaller. However, we do not rely on

T. Hishida

116

his procedure because the smallness of initial data depends on r > 1. Note that the constant q in Theorem 2.3 is independent of r > 1. As the first step of our proof of Theorem 2.3, we show the following lemma which gives a little slower decay rate than desired (later on, e > 0 will be removed so that estimates will become sharp).

Lemma 6.1. Let n > 3 and a E L'(S2) fl Ln(SZ). For any small e > 0 there are constants r). = p.(S1, n, e) E (0, b] and C = C(1l, n, IIaII1, Ilalln, e) > 0 such that if hall,, < r]., then the solution u(t) obtained in Theorem 2.2 satisfies Ilu(t)I1n/(n-1) 5 C(1+t)-1/2+E

(6.12)

IIu(t)112n < Ct-114(I + t)-n/2+1/2+E

(6.13)

IlVu(t)lln G Ct-112(1 +t)-n/2+1/2+e

(6.14)

fort >0. Proof. We make use of (1.3) for r = oo to obtain (e-(t-T)AP(u Vu)(T), Iv)l = I((u Vu)(-r), e- (1-r)A1v)j C(t - T)-(n-n/q)/2IIu(T)Iln/(n-I) IIVu(T)IIn1IIvllq/(q-1), I

for all W E C 07,(Q), which gives IIVie-(1-T)AP(u Vu)(T)IIq

C(t - T)-(n-n/q)/2-j/211u(T)Iln/(n-1)IIVu(r)IIn,

(6.15)

for 1 < q < oo, j = 0,1 and 0 < r < t (the case j = 1 follows from (1.4) and the case j = 0). Given e > 0, we take p E (1,n/(n - 1)) so that 1/p = 1 - 2e/n. From (6.15) with q = n/(n - 1) it follows that Ilu(t)lln/(n-1) :5 Ct-1/2+ellallp+C

f

(t-r)-1/211u(r)lin/(n-I)IIVu(T)IlndT.

0

In an analogous way to the deduction of (6.6), one can take a constant 710 = rm(S2,n,e) E (0,6] such that if Ilalln < ro, then Ilu(t)Iln/(n-l) <- Ct-'/2+`Ilallp for t > 0, which together with (6.11) gives (6.12). To show (6.13) and (6.14), we will derive

Ct-(n-n/T)/2-I/2+e for r = n, 2n/3, IIVU(t)IlT < for t > 0. We divide the integral of (6.1) into two parts

f1

e-(1-r)AP(u

jt/2 Vu)(T)dr =

+f/2 = v(t) + w(t);

(6.16)

(6.17)

then we obtain II VU(t) Ilr <

Ct-(n-n/r)/2-1/2+e llallp + 11 + 12,

for t > 0 (p is the same as above) with 1/2

I1 = IIVt(t)IIr <_ Cf (t - T)-(n-n/r)/2-1/2IIu(T)l1n/(n-I)IIVu(T)IIndT, U

Navier-Stokes Flow Through an Aperture I2 = IIVW(t)Ilr <

CJ

117

(t - T)-/2llu(T)IIoIlVU(T)IIrdT, /2

where (6.15) has been used in I. Using (6.12) together with (6.2) and (6.3), we

see that Ct-(n-n/r)/2-1/2+e11alln,

II <

I2

ClIalln

Sup

t/2
Ilou(t)IIr,

for t > 0. Therefore, setting

Er(t) = sup

7-(n-n/r)/2+1/2-Ellou(T)llr

0
for r = n, 2n/3,

we get Er(t) < CIIaIIp + CIIaIIn + CoIIaIInEr(t) for t > 0, where Co > 0 is independent of a. As a consequence, there is a constant rt. = 77. (11, n, e) E (0, 77o] such

that if Ilalln <_ 17., then En(t) + E2n/s(t) < C for t > 0, which proves (6.16). This combined with the Sobolev embedding, (6.2) for r = 2n and (6.3) imply (6.13) and (6.14). 0

Based on Lemma 6.1, we supply the proof of Theorem 2.3, by which we conclude this paper.

Proof of Theorem 2.3. We fix e E (0,1/2) and put n = i(St, n) = ti7. (12, n, e) Assuming Ilalln <- n, we first show (2.11). Since lie-tAallr <

Ct-(n-n/r)/211al1l,

for t > 0, our task is to derive the required estimate of (6.17). By (6.15) together with (6.12) and (6.14) we have t/2

llv(t)llr

< CJ

(t - T) (n-n/r)/2IIu(T)IIn/(n-1)Ilou(T)IIndT


Ct-(n-n/r)/2

7-1/2(1+T)-n/2+2edT

0

for 1 < r < oo and t > 0; here, note that the case r = oo follows from the L9-L' estimate (1.3) together with (6.15). If 1 < r < n/(n - 2), then the same estimate of the integrand as above works well on w(t) too; as a result, we have IIw(t)Iir <

Ct-(n-n/r)/2-n/2+1/2+2e

for t > 0. For r = oo, we make use of (6.13) and (6.14) to get

IIw(t)II. 5 C f /2(t -T) .1/QIIu(r)II2nIlVu(T)llndT < Ct-n+1/2+2e

for t > 0. We collect the estimates above to obtain (2.11) for 1 < r < n/(n - 2) and r = oo; and the remaining case n/(n - 2) < r < oo follows via interpolation as well.

We next show (2.12). Let 1 < r < n. In view of (6.7), we have IIou(t)IIr

Ct-1/2IIu(t/2)IIr + IIVW(t)IIr,

T. Hishida

118

for t > 0, where w(t) is the same as above. By (2.11) the proof is reduced to the estimate of IIow(t)Ilr. If in particular 1 < r < n/(n - 1), then from (2.11), (6.14) and (6.15) we deduce


IIVw(t)IIr

J

r)-(n-n/r)/2-1/211u(r)Iln/(n-1)IIvu(r)Ilndr

(t -

`

t/2

for t > 0. If r = n, then one appeals again to (2.11) and (6.14) to find IIVw(t)iln < C

1/2

(t - r)-1/2llu(r)II.IIVu(r)Ilndr < Ct-n+112+f

for t > 0. We thus obtain (2.12) for 1 < r < n/(n - 1) and r = n; and the case n/(n - 1) < r < n also follows via interpolation. It remains to show the case n < r < oo. From (1.4) for 1 < q < n < r < oo we deduce Ct-(n/q-n/r)/2-1/2Ilu(t/2)IIq

IIVu(t)IIr <

+ Ilow(t)Ilr, for t > 0, and the first term possesses the desired decay property on account of (2.11). We take p in such a way that 1/n < l/p < 1/n+1/r. Since we have already known (2.12) for r = p as well as (2.11), we are led to IIVw(t)llr


(t - r)-(n/p-n/r)/2-1'2llu(r)Il.llVu(r)llpdr 1/2 2

< ct n+.t/2r for t > 0, which proves (2.12) for n < r < oo. Finally, by use of (6.10), we show (2.13) and thereby (2.14), (2.15) and (2.16). From (2.11) and (2.12) it follows that (6.18)

IIAe-(t/2)"u(t/2)llr < Ct-' llu(t/2)Ilr = 0 (t-(n-n/r)/2-11J as t -+ oc and that IIP(u'VU)(t)llr < CIIu(t)II0IIVU(t)IIr = 0

(t-n+n/2r-1/21

,

(6.19)

as t -+ oo. We are thus going to estimate Il Az(t)ll r

< CllVu(t)Ilr f /2(t - r)-' Il u(t) - u(r)Il.dr

+C f t(t - r)-' Ilu(r)II.IIVu(t) -

Vu(r)llydr = Il + I2.

With the aid of (6.9) and (2.12) we observe

Il = 0 (t-(n-n/r)/2-1 1

(6.20)

as t -+ oo. We need also a Holder estimate of Vu(t) in L''(Sl), which implies the decay property of I2 as well as u E C(0, oo; D(Ar))nC'(0, oo; L" (Q)). To this end,

Navier-Stokes Flow Through an Aperture

119

let us consider u(t) - u(T)

{e_(t-T/2)A _ e-(Tj2).4}u(T/2) -

-IT

Vu)(s)ds JT

{e-(t-'s) l - e-(T-s)A}P(u Vu)(s)ds

r/2 w1 (t, T) + w2(t, T) + w3(t, T),

for 0 < T < t. In order to estimate w1, we take q E (1, n) (resp. q E (1, r]) if r > n (resp. r < n); then, given f E L7 (SZ), we have IIo{e-(t-T/2)A

t-r/2 r/2


- e-(r/2)A}fII,.

Ilve-(s/2)AAe-(s/2)4 fll,.ds

t-r/2

J /2

s-(n/q-n/r)/2-1 /2II Ae-(y/2)Af Ilgds

C(t - r)

T-(n/q-n/r)/2-3/2IIf IIq,

which implies IIV{e-(t-T/2)A

-

e-(r/2).4}f

T)07--(n/9-n/r)/2-1 /2-9IIf

II r < C(t -

IIq,

where 0 < 9 < 1. Setting f = u(7-/2) and using (2.11), we get IIow1(t, T)IIr < C(t -

7)0T-(n-n/r)/2-1/2-9.

(6.21)

In order to estimate w2 and w3, we take q E (1, r] so that 0 < 1/q - 1/n < 1/r; then we see from (6.19) in La(S2) that 11VW2(t,T)IIr < C(t

-T)1/2-(n/q-nl*)/2.r-n+n/2q-1/'2

and that UVw3(t.T)IIr

CJ

T

(t -T)B(T - g)-(nfq-n/r)/2-1/2-OIIP(u' Vu)(s)Ilgds

/2

C(t -

(6.23)

T)0,r-n+n/2r-0

where 0 < 6 < 1/2 - (n/q -- n/r)/2. Collecting (6.21), (6.22) and (6.23) together with (2.11) yields

I. = O (t-n+n/2r-1/2)

.

(6.24)

as t --, oo. From (6.18), (6.19), (6.20) and (6.24) we obtain (2.13). Due to (1.8) and in view of the equation (1.1), we deduce (2.14) immediately from (2.11), (2.12) and (2.13). By Lemma 3.1 of [22] there exist p* (t) e R such that 11p(t) -p±(t)Ilr,nt +

Ip+(t) - p_(t)I < CIIVp(t)IIq for 1 < q < n and 1/r = 1/q - 1/n. Hence, (2.14) implies (2.15) and (2.16). The proof is complete.

0

T. Hishida

120

Acknowledgments The present work was done while I stayed at Technische Universitat Darmstadt as a research fellow of the Alexander von Humboldt Foundation. Their kind support and hospitality are gratefully acknowledged. I wish to express my sincere gratitude

to Professor R. Farwig for his interest in this study and useful comments. I am grateful to Professor Y. Shibata who introduced me to the method in [40]. Thanks are also due to Drs. M. Franzke and H. Abels for showing me their manuscripts prior to publications.

References [1] H. Abels, LQ-Lr estimates for the non-stationary Stokes equations in an aperture domain, Z. Anal. Anwendungen 21 (2002), 159-178. [2] M.E. Bogovskii, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), 10371040; English Transl.: Soviet Math. Dokl. 20 (1979), 1094-1098. [3) W. Borchers, G.P. Galdi and K. Pileckas, On the uniqueness of Leray-Hopf solutions for the flow through an aperture, Arch. Rational Mech. Anal. 122 (1993), 19-33. [4) W. Borchers and T. Miyakawa, L2 decay for the Navier-Stokes flow in halfspaces, Math. Ann. 282 (1988), 139-155. [5) W. Borchers and T. Miyakawa, Algebraic L2 decay for Navier-Stokes flows in exterior domains, Acta Math. 165 (1990), 189-227. (6) W. Borchers and T. Miyakawa, On some coercive estimates for the Stokes problem

in unbounded domains, The Navier-Stokes Equations II - Theory and Numerical Methods, 71-84, Lecture Notes in Math. 1530, Springer, Berlin, 1992. (7) W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math. 174 (1995), 311-382. (8] W. Borchers and K. Pileckas, Existence, uniqueness and asymptotics of steady jets, Arch. Rational Mech. Anal. 120 (1992), 1-49.

(9) W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior domains in L°-spaces, Math. Z. 196 (1987), 415-425.

[10) W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990), 67-87. [11] W. Borchers and W. Varnhorn, On the boundedness of the Stokes semigroup in two-dimensional exterior domains, Math. Z. 213 (1993), 275-299. (12) H. Brezis, Remarks on the preceding paper by M. Ben-Artzi "Global solutions of two-dimensional Navier-Stokes and Euler equations", Arch. Rational Mech. Anal. 128 (1994), 359-360.

[13] Z.M. Chen, Solutions of the stationary and nonstationary Navier- Stokes equations in exterior domains, Pacific J. Math. 159 (1993), 227-240. (14] W. Dan, T. Kobayashi and Y. Shibata, On the local energy decay approach to some fluid flow in an exterior domain, Recent Topics on Mathematical Theory of Viscous Incompressible Fluid, 1-51, Lecture Notes Numer. Appl. Math. 16, Kinokuniya, Tokyo,1998.

Navier-Stokes Flow Through an Aperture

121

[15] W. Dan and Y. Shibata, On the Lq-Lr estimates of the Stokes semigroup in a 2dimensional exterior domain, J. Math. Soc. Japan 51 (1999), 181-207. [161 W. Dan and Y. Shibata, Remarks on the L.-L. estimate of the Stokes semigroup in a 2-dimensional exterior domain, Pacific J. Math. 189 (1999), 223-239. [17] W. Desch, M. Hieber and J. Priiss, L° theory of the Stokes equation in a half space, J. Evol. Equations 1 (2001), 115-142. [18] R. Farwig, Note on the flux condition and pressure drop in the resolvent problem of the Stokes system, Manuscripta Math. 89 (1996), 139-158. [19] R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in L'-spaces, The Navier Stokes Equations II Theory and Numerical Methods, 97-110, Lecture Notes in Math. 1530, Springer, Berlin, 1992. [20] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in

bounded and unbounded domains, J. Math. Soc. Japan 46 (1994), 607-643. [21] R. Farwig and H. Sohr, On the Stokes and Navier-Stokes system for domains with noncompact boundary in Lq-spaces, Math. Nachr. 170 (1994), 53-77. (22] R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in Lq-spaces, Analysis 16 (1996), 1-26. [23] M. Ranzke, Strong solutions of the Navier-Stokes equations in aperture domains, Ann. Univ. Ferrara Sez. VII. Sc. Mat. 46 (2000), 161-173. [241 M. Franzke, Die Navier- Stokes- Gleichungen in Ofnungsgebieten, Doktorschrift, Technische Universitht Darmstadt, 2000. [25] M. Ftanzke, Strong Lq-theory of the Navier-Stokes equations in aperture domains, Preprint Nr. 2139 TU Darmstadt (2001). [26) H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269-315. (271 D. Fujiwara and H. Morimoto, An L, theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA 24 (1977), 685-700. [281 G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Vol. II: Nonlinear Steady Problems, Springer, New York, 1994. [29] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L, spaces, Math. Z. 178 (1981), 297-329. [30] Y. Giga, Domains of fractional powers of the Stokes operator in L,-spaces, Arch. Rational Mech. Anal. 89 (1985), 251-265. [31] Y. taiga, S. Matsui and Y. Shimizu, On estimates in Hardy spaces for the Stokes flow in a half space, Math. Z. 231 (1999), 383-396. [321 Y. Giga and T. Miyakawa, Solutions in L,. of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267-281. [33] Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA 36 (1989), 103-130. [34] J.G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math. 136 (1976), 61-102.

T. Hishida

122

[35] J.G. Heywood, Auxiliary flux and pressure conditions for Navier-Stokes problems, Approximation Methods for Navier-Stokes Problems, 223-234, Lecture Notes in Math. 771, Springer, Berlin, 1980. [36] J.G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), 639-681. [37] T. Hishida, On a class of stable steady flows to the exterior convection problem, J. Differential Equations 141 (1997), 54-85. [38] H. lwashita, La-L,- estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in LQ spaces, Math. Ann. 285 (1989), 265-288. [39] T. Kato, Strong L° solutions of the Navier-Stokes equation in R', with applications to weak solutions, Math. Z. 187 (1984), 471-480.

[40] T. Kobayashi and Y. Shibata, On the Oseen equation in the three dimensional exterior domains, Math. Ann. 310 (1998), 1-45. [41] H. Kozono, Global L"-solution and its decay property for the Navier-Stokes equations in half-space Rn, J. Differential Equations 79 (1989), 79-88. [42] H. Kozono, L'-solutions of the Navier-Stokes equations in exterior domains, Math. Ann. 312 (1998), 319-340. [43] H. Kozono, Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains, Math. Ann. 320 (2001), 709-730. [44] H. Kozono and T. Ogawa, Two-dimensional Navier-Stokes flow in unbounded domains, Math. Ann. 297 (1993), 1-31. [45] H. Kozono and T. Ogawa, Global strong solution and its decay properties for the Navier-Stokes equations in three dimensional domains with non-compact boundaries, Math. Z. 216 (1994), 1-30. [46] H. Kozono and T. Ogawa, On stability of Navier-Stokes flows in exterior domains, Arch. Rational Mech Anal. 128 (1994), 1-31. [47] H. Kozono and H. Sohr, Global strong solution of the Navier-Stokes equations in 4 and 5 dimensional unbounded domains, Ann. Inst. H. Poincare Anal. Nonlineaire 16 (1999), 535-561. [48] H. Kozono and M. Yamazaki, Local and global unique solvability of the NavierStokes exterior problem with Cauchy data in the space L°.°°, Houston J. Math. 21 (1995), 755-799.

[49] H. Kozono and M. Yamazaki, On a larger class of stable solutions to the NavierStokes equations in exterior domains, Math. Z. 228 (1998), 751 785. [50] P. Maremonti and V.A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Sup. Pisa 24 (1997), 395-449. [51] K. Masuda, Weak solutions of Navier-Stokes equations, Tohoku Math. J. 36 (1984), 623-646. [52] M. McCracken, The resolvent problem for the Stokes equation on halfspaces in L5, SIAM J. Math. Anal. 12 (1981), 201-228. [53] T. Miyakawa, The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ. 17 (1994), 115-149.

Navier-Stokes Flow Through an Aperture

123

[54] K. Pileckas, Three-dimensional solenoidal vectors, Zap. Nauch. Sem. LOMI 96 (1980), 237-239; English Transl.: J. Soviet Math. 21 (1983), 821-823. [55] K. Pileckas, Recent advances in the theory of Stokes and Navier-Stokes equations in domains with non-compact boundaries, Mathematical Theory in Fluid Mechanics, 30-85, Pitman Res. Notes Math. Ser. 354, Longman, Harlow, 1996. [56] Y. Shibata, On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain, Tsukuba J. Math. 7 (1983), 1-68. 157] Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quart. Appl. Math. 57 (1999), 117-155. [58] Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech. 3 (2001), 213- 230. [59] Y. Shimizu, L°°-estimate of first-order space derivatives of Stokes flow in a half space, Funkcial. Ekvac. 42 (1999), 291-309. [60] C.G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in L°-spaces for bounded and exterior domains, Mathematical Problems Relating to the Navier-Stokes Equation, 1-35, Ser. Adv. Math. Appl. Sci. 11, World Sci., NJ, 1992. [611 V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Zap. Nauch. Sem. LOMI 38 (1973), 153-231; English Transl.: J. Soviet Math. 8 (1977), 467-528. [62] V.A. Solonnikov, Stokes and Navier. Stokes equations in domains with noncompact boundaries, Nonlinear PDE and Their Applications: College de France Seminar IV, 240-349, Res. Notes in Math. 84, Pitman, Boston, MA, 1983. [631 V.A. Solonnikov, Boundary and initial-boundary value problems for the NavierStokes equations in domains with noncompact boundaries, Mathematical Topics in Fluid Mechanics, 117-162, Pitman Res. Notes Math. Ser. 274, Longman, Harlow, 1992.

[64] H. Tanabe, Equations of Evolution, Pitman, London, 1979. [65] S. Ukai, A solution formula for the Stokes equation in Rn, Commun. Pure Appl. Math. 40 (1987), 611 -621. [66] M. Wiegner, Decay estimates for strong solutions of the Navier- Stokes equations in exterior domains, Ann. Univ. Ferrara Sez. VII. Sc. Mat. 46 (2000), 61-79. Toshiaki Hishida Fachbereich Mathematik Technische Universitat Darmstadt D-64289 Darmstadt Germany e-mail: hishidatmathematik.tu-darmstadt.de

Advances in Mathematical Fluid Mechanics, 125 151 © 2004 Birkhauser Verlag Basel/Switzerland

Asymptotic Behavior at Infinity of Exterior Three-dimensional Steady Compressible Flow T. Leonaviciene and K. Pileckas Abstract. Steady compressible Navier-Stokes equations with zero velocity conditions at infinity are studied in a three-dimensional exterior domain. The case of small perturbations of large potential forces is considered. In order to solve the problem, a decomposition scheme is applied and the nonlinear problem is decomposed into three linear problems: Neumann-type problem, modified Stokes problem and transport equation. These linear problems are solved in weighted function spaces with detached asymptotics. The results on existence, uniqueness and asymptotics for the linearized problem and for the nonlinear problem are proved. Mathematics Subject Classification (2000). 76N, 35Q. Keywords. Steady compressible Navier-Stokes equations, three-dimensional exterior domain, weighted function spaces with detached asymptotics, pointwise decay at infinity.

1. Introduction In this paper we study the asymptotic behavior of steady solutions to equations describing an isothermal motion of compressible viscous fluid (gas) in an exterior domain 11 C 1R3. We assume that the flow domain Cl is an open, connected set, exterior to a compact set B C 1R3 with a sufficiently smooth boundary 852. Moreover, we suppose that the interior of B is non-empty and contains the origin of coordinates. In such a domain 52 we consider the classical Poisson-Stokes equations for unknown functions p (density) and v = (Vi, V2, v;l) (velocity):

-µ10v - (µl + µ2) Vdiv v + Vp = pb - p(v V )v, div(pv)=0, xE51,

lv=0,

x E 52,

xE8S1.

This work was supported by Lithuanian State Science and Foundation grant A-524.

(1.1)

T. Leonaviaiene and K. Pileckas

126

Here p 1, p2 are constant coefficients of shear and bulk viscosities satisfying the conditions 2 (1.2) P1 > 0, p2 > -3P1 and b is a density of external forces. We assume that b is a "small" perturbation of a "large" potential force, i.e. that b has the form

b=V$+f,

(1.3)

where 4' is a potential which can be (as well as its derivatives) arbitrary "large" and f is a "small" perturbation of V4'. Since the flow domain Il is unbounded, the equations (1.1) have to be supplied with the conditions at infinity. We assume that the velocity field v tends at infinity to zero and the density p to a constant density and prescribe the following conditions:

v(x) - 0, p(x) - p., p. = const > 0, jxi -' oo.

(1.4)

It is a standard observation that the exact solution (po, vo) of problem (1.1), (1.4) corresponding to the potential force V4i (i.e. f = 0) is the rest state (po,0), where po satisfies the equation

Vpo = po04'.

If4i(x)--'0aslxi-'oo,from (1.4),(1.5)wefind po(x) = p. exp4>(x).

The equations for the perturbation (a = p - po, v) have the form -p1 Ov - (Al + p2) Vdiv v + Da - o V4P = F(a, v), x E 0, div (pov) = -div(av), x E S2,

v=0,

xEaQZ,

v(x) - 0, 0'(X)-' 0,

ixj-' oo,

where

F(a, v) = -(po + a)(v V)v + (po + a)f.

(1.8)

A possible linearization of the system (1.7) near the equilibrium state (po,0) reads

-MIAV-(p'+p2)Vdivv+Va-aV4>=F, xEQ, div (pov) = -div (aw),

v=0,

xE8S1,

(1.9)

x E SZ,

v(x)--'0, a(x)-'0,

lxj -co,

where a, v are unknown, while F, w are given. We will consider a slightly more general system

-piAV -(pi+p2)Vdivv+Va-a04>=F, xES1, div (pov) = -div (aw) + g, x E 11, v = 0, x E a!1, v(x) -' 0, a(x) --' 0, where g is given.

(1.9')

lxi -' oo,

Asymptotic Behavior of Exterior 3D Steady Compressible Flow

127

In order to solve the linearized problem (1.9') and the nonlinear problem (1.7), we apply the decomposition scheme proposed in [15] (see also the references in [15] for the original application of the similar decomposition scheme to steady compressible Navier Stokes equations linearized on a constant density). We represent the velocity field v as a sum

v=u+

(1.10)

where

xEafl,

div(pou)=0, xESl;

(1.11)

and

AP

xEO1.

(1.12)

As it is shown in [15], then the complicated mixed elliptic-hyperbolic system (1.9') splits into the following three simpler problems:

I

il, o(x) - 0, IxI - co,

-div (aw) + g, ap

an

= 0,

x E an,

SC

_P1 Au - (P l + P2) Vdiv u + po V (II/Po) = G, div (pou) = 0, x E Sl, U = - V p, x E asl, u(x) --* 0, II(x) -0, IxI - oo, a + (2p + P2) div

1 13

x E 11,

(1.14)

aw = II,

x c= n,

(1.15)

PO

where A., = div (po(x)V), G = F + (2p + p2)V(Po 2VPo . wa) - (2Pi + p2) V (Po' VPo DAP)

- (2p, +P2)Po'Vpodiv

('wow)

+(2p. +P2)V( -) .

(1.16)

Note that (1.13) and (1.14) are elliptic problem: (1.13) is a Neumann problem for the operator O,, and (1.14) is a Stokes type problem; the last problem (1.15) is the transport equation. Thus, the elliptic character of the original system (1.9') is separated from its hyperbolic character. The hyperbolic effects are concentrated in the transport equation (1.15). The solution (a, v) of the problem (1.9) can be found as follows. We define a linear map

G:'r-'a

in the following way:

(a) for given r we find p by solving the Neumann problem

A ,,w = -div (rw), x E SI;

a

an = 0, x E 490;

(b) next we find the solution (u, II) of the Stokes-type problem (1.14) taking

in(1.16)a=r;

T. Leonaviciene and K. Pileckas

128

(c) finally, we find a from the transport equation (1.15). It is easy to verify that the fixed point a of the map .C and the corresponding (u, 11, gyp) solve the system (1.9').

While the linearized system (1.9) is solved, we find the solution of the nonlinear problem (1.7) in the form (a, v = Ocp + u), where (a, 0, u) is a fixed point of the nonlinear mapping

N: (T, , z) - (a,
(1.17)

with (a, gyp, u) being a solution of the linear problem (1.9) taking in it F = F(rr, V +

z) andw=V + z. The solvability of problem (1.1)-(1.4) was studied in [14] using a different decomposition scheme and the techniques due to Matsumura, Nishida [2]. Moreover, in [14] were obtained decay estimates for the solution (p = po+a, v) of (1.1)-(1.4). In particular, it is shown that the solution is "physically reasonable", i.e.

v(x) = O(1xj-'),

w(x) =

O(lxl-2)

a(x) =

O(IzI-2`.

However, the analysis of [14], based on the integral representation formula for the Stokes problem and estimates of weakly singular integrals appearing in this representation, does not provide optimal decay rates for the higher order derivatives of the solution. Furthermore, with this method one cannot specify the asymptotic behavior of the solution as jxj - oc. In this paper in order to investigate the solution of problem (1.1) (1.4) we employ methods related with the application of weighted function spaces (e.g. [9, 5]). However, already the Navier. Stokes equations of the incompressible fluid motion are not solvable in classical weighted Sobolev (Holder) spaces while considering the convective term (v V)v as a perturbation of the Stokes problem. On the other hand, if we study these equations in function spaces which elements take suitable asymptotic forms (i.e. the elements are defined as a sum of two parts one of which contains the main asymptotic term and another one belongs to the usual weighted space), then the Navier-Stokes problem is well-posed (see [6, 7]). We also refer to [3, 4, 11, 8] where such weighted spaces with detached asymptotics were successfully applied to study Navier-Stokes equations in other unbounded domains and to [10] were the spaces with detached asymptotic were applied to study viscoelastic flows in an exterior domain. In this paper we prove that problem (1.1)-(1.4) is well-posed in appropriately constructed weighted Sobolev and Holder spaces with detached asymptotics and obtain an accurate information about the asymptotic behavior of solutions. The paper is organized as follows. In Section 2, we define the function spaces used in the paper and prove various embedding results for these spaces. In Section 3 we present known results concerning the Stokes problem in weighted spaces with detached asymptotics and study the Stokes-like problem (1.14) in such functional setting. In Section 4, we study other auxiliary problems ((1.15) and (1.13)) appearing in the decomposition scheme. Finally, in Section 5 and 6 we prove existence,

Asymptotic Behavior of Exterior 3D Steady Compressible Flow

129

uniqueness and assymptotics results for the linearized problem (1.9') and the nonlinear problem (1.7) using the results of the previous sections and the fixed point

argument. Let us point out that, although we consider here only the equations of an isothermal perfect gas, the result could be easily generalized to the case of barotropic motion. The authors are deeply grateful to Prof. S. A. Nazarov for the helpful discussions.

2. Function spaces and auxiliary results Here and in the sequel 52 C JR3 is an exterior domain with the smooth boundary on. We assume that the point 0 E J:s\S2. The outer normal to 852 is denoted by n. By c we denote different constants. They may have different values in different formulae.

If we want to fix the value of some constant, we regard it as cj, j = 1, 2,.... The norm of the element u in the Banach space V is denoted by flu; Vi. We do not distinguish in notations between the spaces of scalar and vector valued functions. The difference is always clear from the context. We use the following function spaces: C°°(52) is the set of all infinitely differentiable in 52 functions; Co (S2) is the subset of functions from C°°(0) having compact supports in 52; Co (52) is the set of functions from C°°(52) which are equal to zero in the neighborhood of infinity, i.e. for sufficiently large jx) (but not necessary on 852).

I > 0, q E (1,00), is the usual Sobolev space and W'4(52) is the closure of Co (52) in the norm ii ; Wl.q(S2)11; LQ(f) = W1-1/Q,q(852) is the space of traces on 8S2 of functions from Wl4(52), l > 1. The norm in Wl-'Iq,4(852) is defined by l1';Wt-1"QQ(8S2)11 = inf 1lu;W1.q(H)11

where the infimum is taken over all u E W1"4(S2) such that u = y on on. Let S2 be a unit sphere in JR3. Wt'Q(S2) (resp. CI.I(S2)) is Sobolev (resp. Holder) space of functions defined on S2. D' (f2) is the closure of CO°(52) in the Dirichlet norm 11V-;L2(52)ll. D-1(52) is the dual space to Do(S2) with the usual duality norm. VV'q(S2), l > 0, q E (1,00), 0 E IR, is the closure of C$°(52) in the weighted norm VV,q(f))ll Flu;

_

lIIxl3-1+I"ID°u; L`I(S2)II,

Ialu

where a = (al, a2, 03), D° = el"I/8x1 'ex2aex`33, aj > 0, jal = al +a2+ a3.

T. Leonaviciene and K. Pileckas

130

A'1j6(52) is the weighted Holder space defined as the closure of CC (ft) in the norm (axle-1-6+I0IIDau(x)I)

sup

lIn; Ag6(52)Il =

+

sup {Ixi1 sup (ix -

yilDu(x) - Du(3/)l) }. JJJ

r vi
k)'9(f1), l > 0, q E (1, oc), s > 1, k E Z, y E (1 - 3/q + k, l + 1 - 3/q + k), is the weighted Sobolev space with detached asymptotics, i.e. the space of functions admiting the asymptotic representation u(x) = r- k!1(O) + u(x),

(2.1)

where r = IxI, 0 E S2, U E Wl'q(S2), ii E V,14(0). The norm in 'I1y,k)'q{S2) is defined by IIU; W''q(S2)II + 11u; VY q(cZ)ll.

Functions U and u are called the attributes of u E 2Ty,k)'q(S2).

eSZ is the weighted Holder space of functions with detached asymptotics (2.1) having the attributes U E C8.6(S2), ii E A',6(52), where y E (1+45+k, 1+6 + 1+k), 6 E (0,1). The norm in e: ry.k is given by the (52l`'")6)

formula IIu;Cs.6(S2)II +

llu;A(i6( 1)II'

From the definitions of the norms follows

Lemma 2.1. (i) Let u E V'q(S2) (u E Aa6(52)), m < 1, jal < 1. Then u E

V3-1+m(S2),D°uEV

52

m,d ( ) uEAg_I+mSZ, Du uEA

)J

(ii) Let u E 27I,k) 4(11) (u E C(1,k)'5(f )), m < 1, m < p <_ s, Joel < 1. (u E Lr-,_ +m.k(H), D°u E Then u E (Q) Dau E 'I77`k+lI The following embedding results are proved in [1]. Lemma 2.2. Let u E Vp'q(52).

(i) If qI < 3 with q < s < 3q/(3 - ql), then u E V,-81-3/8+3/q(52) and IIu: V3_1-3/s+3/q(n)II < c llu; V,3 q(1l)lI.

(2.2)

(ii) If qI > 3 and m + 6 < 1- 3/q with 6 E (0,1), then u E A-`51+0-1+3/,(Q) and 11

u' Am+6+,3-,+3/4(H) II

e ll u Va

Il

(2.3)

Asymptotic Behavior of Exterior 3D Steady Compressible Flow

131

Lemma 2.3. Let v E A,,,+6+3, (Q) (i) Ifu E Vy2'Q(11), m < 1 + 1, then vu E V,,"+1(S2) and

Ilvu;V;+(n)II

q(1)II.

(2.4)

(ii) If U E A"26(12), m < I + 1, then vu E A`602(fl) and

-

Ilvu;A-;+32(4)11

cIlv;Ai++6+3, (Q)1IIIu;A 02

Proof. Since

aD

D "(vu) = E


a_

"vD"u

a µ(a -

6(c)II.

(2.5)

a) = a,!a2!a:1!.

,=1,2,3

we get

f ID' (vu) glxIQ(32+9,-M+J I) dx Ial


r

J

JxjQ(,il+laI-I AI) IDn-"vlglxlg(32--+I`I) IDµuj4 dx

INI=0 S2

4(sl)II

clIv;A+

Summing these inequalities over a with Ial

m, we derive (2.4). The estimate

(2.5) can be proved analogously.

Let us formulate other results which give the possibility to estimate nonlinear terms in (1.8), (1.16). Lemma 2.4. (i) Let v E IV,(' .+ 1.5+1)'4(11) with l > 1,

s > 1, 3/2 < q < oc and

y E (1 + 2- 3/q, 1+ 3- 3/q). Then v 0 v E'Xly,2)'Q(12) and IIV 0 V;

,+I.S+l),g(Q)112.

c MI V; T(-Y. I

y.2'

(ii) Let V E y1 Then v 0 v E C7t 2)'6(1l) and 11V 0

Lemma 2.5. (i) Let g E

(2.6)

1 > 1, s > 1, 6 E (0, 1)1 y E (l + 2 + 6,1+ 3 + 6). V;c71,Z),6(n)11

s

(2.7)

h E I7ykI > 2, s > 1+ 1, q >

3/2, y E (l + k - 3/q, l + k + 1 - 3/q), k > 2. Then gh E Vy g(12) and there holds the estimate Ilgh; V7'q(fl)II

c11g;IV('+1.i).4(Q)Il11h;T(rk),q(Q)11

(ii) Let g E y1-h E E*yyk)'6(11), 1> 0, s> 1+ 1, 5 E (0,1),

(2.8) '

E

(l + k + 5, 1 + k + 1 + 6), k > 2. Then gh r= A76(12) and

Ilgh;A1

(2.9)

T. Leonaviciene and K. Pileckas

132

Estimate (2.6) is proved in [10] for s = 1 (see Lemma 3.3 in [10]). Estimate (2.8) is proved in [10] for s = I + 1. k = 2 (see Lemma 3.5). The proof for arbitrary s > I + 1 and k > 2 is completely analogous. Estimates (2.7), (2.9) easily follow from the definition of the norm in the space c7",ki'0(Il).

3. Stokes and modified Stokes problems in weighted spaces 3.1. Stokes problem in weighted Lg and Holder spaces Let us consider in fZ the Stokes problem

-v0u+Vp=f inn, div u = g

inn,

tu=h onou. We associate with problem (3.1) the mapping SSg: V V(SZ) - "'V(Q) defined by

(u,p) - (f,9,h) = S °(u,p),

(3.2)

where D3gV(1l) = V +''g(Il) X Vs'g(Q),

R.3 V(u) l > 1,

V -''q(cl) x V '3'q(cl) x

Wt+J-Uq.q(8Sl),

13

q E (1,00),

,O E IR.

The following results are well known (see [9, 6]).

Theorem 3.1. (i) If

0E(1+1-3/q,1+2-3/q),

(3.3)

then the mapping (3.2) is an isomorphism. (ii) Let (f,g,h) E 7 V(SZ) C 7ZajgV(SZ) with

ry E (l + 2- 3/q, I+ 3- 3/q).

(3.4)

Then the solution (u,p) E D"V(Sl) admits the asymptotic representation

(u,p) = (u°,p°) + where (ii,p) E

(3.5)

V(Q) and (u°,p°) = b. E(') +b.2E(2) +b3E(3),

(3.6)

with EU denoting the j-th column of the fundamental matrix for the Stokes operator in R3 and bj E R. j = 1, 2, 3. Moreover, there holds the estimate

1I(u,P);vyv(u)II+Ib1I+Ib.]+Ib3] sc11(f,9,h);1Z''gV(0)II.

(3.7)

Asymptotic Behavior of Exterior 3D Steady Compressible Flow

133

Remark 3.2. The columns of the fundamental matrix for the Stokes operator in IIV are defined by

E(i)(x)=

I s(b?IlxI2+xI xj, bj2IXI2+x2xj, 5J:4Ix12+x:sx,, 2vxj)T 7 = 1,2,3. 8rvlxi

Let us consider the problem (3.1) in weighted Sobolev spaces with detached asymptotics. Denote

D, 9z(Q) =

v(1+ I.1+2).q(H) X Vl.l I

(52),

x'27}tz+i).v(c)

RY°iXT(it) = `b71,:; i.i).v(S1)

x W1+1- /9.9(an)

with ry satisfying (3.4). It is not difficult to compute that D;9iI1(51) C Di°V(51),

RYgQJ(51) C RIi"V(51)

with 0 taken from (3.3). Let GI',q be the operator of the Stokes problem (3.1) acting on the domain

D',M(51) to the range RYQro(s1).

Theorem 3.3. (see 16, 71.) Let (f,g,h) E IVY-M(Q). Problem (3.1) has a solution (u, p) E DY9V(sl) if and only if there holds the compatibility condition J

j(9)dso = 0,

(3.8)

S2

where a(8), f are the attributes of the function f in representation (2.1). The solution is unique and there holds the estimate I I (u, p); D'.1'V (Q) jj < c 11 (f, g, h); 7ZYQ%1(51) II.

(3.9)

The analogous results are also valid in weighted Holder spaces. Let us fix

l > 1 , S E (0,1), /3 E (1+1+6,1+2+6), ry E (1+2+6,1+3+6) and define the spaces

73 A(Q)

A'+' 6(51) x A,,;, (n),

Re6A(c) = A' '.6(51) x A" (11) x C'i+1.6( D'.6irp (rUi i.1+2)6(51) 13

=

Yl.!)6A x

x C7r:2+1j6(51), (12+')6(51) C1+1.6(an)

x

(SZ). Problem (3.1) has a solution Theorem 3.4. (see (61.) Let (f,g,h) E 1 (u,p) E V>'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 <_ c (IIf; Ds '(Q) 11 + IIg;L2(S2)II).

(3.13)

The proof of Theorem 3.5 is standard and does not differ from the proof of the analogous result in the case of a bounded domain 11 (see [151). Note that the proof is based on the following. Lemma 3.6. (i) Let U E Do'(S2), g = div (p)u). Then g E L2(S2) and IIg:L2(Sl)II <_

cIlu;D1)(S2)II.

(3.14)

(ii) Every function g E L2(9) can be represented in the form g = div (pou), u E D21(S2) and there holds the estimate IIu;D0'(9)II <_ eJig; L2(S1)II.

(3.15)

Proof. (i) Since po 1 Vp0 = V4 , we have g = div (pou) = p. exp 4' (V u + div u) and using the inclusion V4i E A,,+1+s+.,.(Sl), we derive the estimate (3.14). (ii) Let v E D(,(S2) be a solution of the problem

rdiv v = g, Sl

v=0. xEafl,

satisfying the estimate

IIv;Do(S2)1I
Asymptotic Behavior of Exterior 3D Steady Compressible Flow

135

Such a solution v exists for each g E L2(1l) (see [16]). Taking u = po'v we get div (pou) = g and the estimate (3.15) holds true. 0 Let us consider problem (3.11)0 in weighted function spaces.

Theorem 3.7. (i) Let f E VQ-1'4(12), g E V4(12),

I > 1, q > 6/5, /3 E (I +

3/2 - 3/q,I + 2 - 3/q). Then problem (3.11)o has a unique solution (u, [1) E V)31+1'4(S2) X VI'q(12) satisfying the estimate

x V,q(l)II

II(u,IT)


(3.16)

(ii) Let (f, g) E A'3 1'6(12) xA"6(12), 1 > 1, 8 E (0,1),/3 E (1+8+3/2,1+b+2). Then problem (3.11)0 has a unique solution (u, 11) E AR 1'6(12) x A"6(1l) and II(u,II);A3 1'6(12) x A 6(s2)II s cII(f,g);A 1,6(H) x A"6(p)II. 0 Y3

113

(3.17)

Proof. (i) Let first consider the case q = 2. For each u E D(,(1l) there holds the estimate f IxI-2Iu12 dx < c f IVu12 dx. n sZ V1'2(S2). Moreover, if .8 > 1, then

Hence,

ft f

n

dx1


1/2

1/2

(f

Ix12If I2 dxl

IxI-211712 dx)

n

/


1/2

\f

1/2

I Vsil2

dx)

Vt E D0'(12).

n

11

Therefore, for,3 > 1, V131 -1'2(12) C Do 1(12) and problem (3.11)0 has a unique weak

solution (u, II) E Do' (Q) x L2(S2) = Vo'2()) X U°'2(12). From results of the paper [15] it follows that (u, II) E Wi 2(1I) x W,1? (12) and equations (3.11)o are satisfied almost everywhere in Q. We rewrite (3.11)o in the form

1-µ10u+VII=f+P°IVPon-(/11+ z2)V(Po1VPo'u), xE12, xE12,

tu=0,

(3.18)

xE812,

and consider (u, II) as a solution of Stokes problem (3.1) with the right-hand side (F, G, 0), where

F=If +fIO4P -(µl+p2)V(V4'u), G=g-V4, - u (remember that 04 = po 1Vp0). By Lemma 2.3 we get V Vy°2(12). Since f E V13 1'2(52) C

(3191

u E Vya2(1l), HVID E -t+l(H), g E VL2(12) C V'3_1+1(H), we conclude

F E V°'2(12),

GE

VI'T2(St),

T. Leonaviciene and K. Pileckas

136

with,31 = min(-yo,fl-1+1). For yo > 1,)3 E (1,1+1/2) we have 01-1 E (0,1/2) and in virtue of Theorem 3.1 (i) the solution (u, II) of Stokes problem (3.18) belongs to the space D3;2V(52) = Va;2(12) x 1 2(11). Applying again Lemma 2.3, we get ju E V.'+,.(1), 1104) E so that

0

FE

V12

(9),

2 2(n) G E Vwith

/32 = min(2yo, (3 - 1 + 2). We have (32 - 2 E (0,1/2) and Theorem 3.1 (i) implies (u, II) E

V(12). Repeating this argument 1 times we derive

(F G) E V

-1.2(12)

x VV;2(cl)

with (31 = min(lyo Q) E (1, 1 + 1/2) and conclude that (u, II) E Do2V(S2). If 1-yo > (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 <

(f

it

1/9

l1/q

(f IniQ 1x19 (-3+1-1) dx)

Iflglxlq(3-t+1) dx)

11

13

where 1/q + 1/q' = 1. Since q' E [2, 6], in view of Lemma 2.2 (i) there hold the embeddings DI'(12) = %, 2(S1) C V1jz 3/q'(S1) C Voy+1_1(St). Therefore,

f


11

Ifq>2, 3>1+3/2-3/q, then

1

1'U1711'dx)1/6UIxe dx)f2 f2


It

J
Asymptotic Behavior of Exterior 3D Steady Compressible Flow Moreover, by Lemma 2.2 (i) g E V,9(H) C

137

C L2(S2)

x )V2(S2) of

Therefore, there exists a unique weak solution (u, H) E V0 problem (3.11)0. As above we get the inclusion 04D u E V7o2(SI),

(3.20)

IIO4D E V°o2(S2).

In order to estimate the right-hand sides F and G, we use the inequality 1/a

C/

IhIslxls(,u+3/t 3/s-e)l{:L J-1J

I/t

r

< c (J IhjtIxI1 tydx)

(3.21)

sl

s1

which is true for arbitrary h E V,,0,1(0), 1 < s < t and `de > 0; can be easily proved by applying Holder's inequality.

Let us consider the case q E [6/5,21. We take eo = -to - 1. Then (3.20) and (3.21) with s = q, t = 2, µ = yo imply V4P u E V,+3/2-3/q(1Z), HV4 E +3/2-3/q(H) Since 1 + 3/2 - 3/q < /3 - I + 1, we conclude F E V+ 3/2-3/q(1),

G E VI+3/2-3/9A.

(3.22)

Let q E (2, 6]. From (3.21) with t = q, s = 2, µ = Q -1 + 1 follows that (f,g) E V3-1+1+3/9-3/2-e1(Q) X VQ-1+1+3/9-3/2-e1A C

V,0,2

2

(f1) X V11_ E1(9),

b'e1 > 0.

Since -yo > 1, inclusions (3.20) imply F E V°' E, (f1),

G E Vil_21(S2),

and, if eI < 1/2, by Theorem 3.1 (i) we conclude u E Vj_f1(0),

n E Vli'i,(S1).

In virtue of Lemma 2.2 (i) we obtain u E V1E9 _3/q+3/2(S2), II E V°'9 -3/q+3/2(f1)

and by Lemma 2.3 VID u E V70-C1-3/q+3/2(n)' HV1 E Vry0.9-3/9+3/2(0)' Taking ei so that el < -yo - 1, e1 < 1/2, we get VIb U E V11 3/q+3/2(f),

HV4' E VO'3/q+3/2(x)

and, therefore, (F, G) satisfy inclusions (3.22). Finally, let q > 6. Then inequality (3.21) gives f E V°'2 +1+3/9-3/2-e1(n) C V0_21 (12),

9 E V,1 e1(H)

and as in the previous case we derive the inclusions

(u, H) E Vl-et (Q) X V11_2 1 te1n1

2.30

u E V16

1,.= 2.2(u, (cl

II) E V11-e1 (l) x V0-e (f 1)

IIO4PE V0'`'

/4

T. Leonavi5iene and K. Pileckas

138

By inequality (3.21) with u =)3 - l + 1, s = 6, t = q we have f c V0Os+1/2+3/q-E,(Sl) C VZ_e,(S2),

g E Vz=6,(S2).

Hence,

F E V°'E, (0),

GE

V2'-'6,,

(f2).

Since sI < 1/2, we have (2 - el) - 1 E (1/2, 3/2) and by Theorem 3.1 (i) we get u E v22-6 , (9), II E V2'-'6,, (n). Lemma 2.2 implies u E AZ'le2(S2), II E A?'EZ(f ). It

is easy to compute that the last relations imply the inclusions V(D u E V7o+3/2-3/q-el (Il) C V1+3/2-3/q(")'

IIVIF E V0+13/2-3/q(4

Thus, we have proved the inclusions (3.22) for arbitrary q E [6/5, oo). Now, arguing

as in the case q = 2, by repeated application of Theorem 3.1 (i) and Lemma 2.3 we derive the sequence of inclusions (F, G) E

X V"' g(f2) TI'"°

Lemn,g 2.3... Then Lem

23

3.1 (i)(u, II) E DPo V(S2)

3.1 (i)(u H) E D!'gV(1)

I'heo ==> 3.1 (i)(u,

H) E

DID,+m V(52) C

where ,30 = 1+3/2-3/q,...,/3i = min(yol + $oi Q), ..., ,QI+,n = min ((l + m)-yo + /3o, /3), (1+m)ryo+$o > 0. Supplying the obtained inclusion with the corresponding estimates we obtain estimate (3.16) for arbitrary q E (6/5, oo). (ii) Let (f,g) E Aa 1'6(SZ) xA"a(f2), l > 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)
(3.23)

(ii) Let (f, g, h) E Rl6A(Sl), I > 1, 6 E (0,1), Q E (1 + 6 + 3/2,1 + 6 + 2). and

Then problem (3.11) has a unique solution (u, H) E

II(u,H);D6A(1)II
(3.24)

it can be extended as a Proof. Since h E W1+1-1/4,4(on) (h E (Cl "1,6(52)) and having a compact support. function H belonging to Hence, H E V01}1,4(52) (h E Aa 1'6(Sl)). For U = u - H and II we get problem (3.11)o with the new right-hand side (fl, g), 0). Theorem 3.6 follows now from Theorem 3.5.

Now we study problem (3.11) in weighted functions spaces with detached asymptotics. Theorem 3.9. (i) Let (f, g, h) E Rti421(cl), l > 1, q E [6/5, oc), y E (1+2-3/q, 1+ 3-3/q) and let ,3(9) satisfies the compatibility condition (3.8). Then problem (3.11) has a unique solution (u, II) E D 4'XT(S2). There holds the estimate II (u, H); DY,°tZ(II) II

c 11f, g, h);

II.

(3.25)

(ii) Let (f, g, h) E R;6e:(S), 1 > 1, 6 E (0,1), y E (1 + 2 + 6, 1 + 3 + 6) and let the compatibility condition (3.8) be valid. Then problem (3.11) has a unique and solution (u, II) E jj (u, H); Vi

(II) jI < c II (f, g, h); R764t(II)

II.

(3.26)

T. Leonaviciene and K. Pileckas

140

R1,g21(1) C IZI-q V

(Q) with arbitrary Q E (1+3/2-3/q, 1+2-3/q), by Theorem 3.5 there exists a unique solution (u, II) E DtigV (51), ,3. = 1 + 2 3/q - e, 0 < e < inin(1/2, -y - 1). From Lemma 2.3 it follows that Proof.

(i) Since

v. u E

C

V,'+'.y(c ).

HV$ E V13.+ro(n) C Vy'Q(Il).

Therefore, (u, H) can be considered as a solution of Stokes problem (3.1) with the right-hand side (F, G, h) E R 1SI7(fl). In virtue of Theorem 3.3 we conclude that (u, 11) E D;g21(1l) and the estimate (3.25) holds true. (ii) The proof of the second part of the theorem is completely analogous to the that of the first part. One have just to change weighted Sobolev spaces into weighted Holder spaces and to use Theorem 3.4 instead of Theorem 3.3.

4. Transport equation and Poisson-type equation 4.1. Transport equation Let us consider the transport equation

z +div (wz) = h,

(4.1)

where w satisfies the condition (4.2) xE851 Problem (4.1), (4.2) was studied in many papers using various functional

settings (e.g [12, 13] and references cited there). Here we need results concerning the solvability of (4.1), (4.2) in weighted Sobolev and Holder spaces with detached asymptotics.

Theorem 4.1. (i) Let h E

%1(1.1+1).q(Q)

with 1 > 2, q > 3/2,

y E (I + 2- 3/q, 1+ 3- 3/q)

and let w E 21;! i

1.1+z).q(51)

(4.3)

There exists a number co > 0 such that if 11w;Zyi + 1.1+2).90)11

(4.4)

< Col

then problem (4.1), (4.2) has just one solution z with z E 27yt;2+1)'1(S ), div(wz) E

(4.5)

There holds the estimate 11z;2J(1;2+1)"(1)11+Ildiv(wz);VV'`f(1l)11

Furthermore, AE IIAz ,D(2

I Odiv (wz) E + 11Adiv (wz); (11h;21(1.2+1).q(S1)11

c

+


(4.6)

and

V1-2.9(1l)ll

I1w;,Ury1+1.1+2).q(n)llll=;21(1.2+1).v(Q)1i)

(4.7)

Asymptotic Behavior of Exterior 3D Steady Compressible Flow y.2+1)'a(11)

(ii) Let h E

I > 2,

b E (0, 1),

yyE(1+2+b,1+3+5) (t 1,t+2)'6(-) There exists a number co > 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
(4.12)

Since z' and h have the same "spherical" attributes, from equation (4.1) we get div (wz) = h - a' E V7,9(5l) and estimate (4.6) follows from (4.12). Applying the operator 0 to equation (4.1) we obtain

Oz + div (wOz) = Ah - div (iwz) - 2div (Vw Vz).

(4.13)

From Lemmata 2.1 and 2.5 we conclude Ah

E'Ihl,;2.1-1)(SZ),

div (Owz) + 2div (Vw Vz) E VI-2,9(n)

and

Ildiv (Owz) + 2div (Vw . Vz); (1+1,1+2),9(Q)II


(1,1+1)y (Q)II.

(4 . 14)

27ryt'42'1-1)

Therefore, Az E 9(52) may be interpreted as a unique solution of (4.1) with the right-hand side h1 = Ah - div (Awz) - 2div (Vw Vz) E `D,y,a2't-1)'0(52) Now, from results of the paper [10] (see Section 4) follows the estimate IIAz;%;7t.42,t-1).9(1)II
(4.15)

T. Leonavitiene and K. Pileckas

142

Since div (wLz) = h1 - Az, from (4.14), (4.15) we derive estimate (4.7). (ii) The proof is completely analogous to that of the case (i). We only note that the existence of the unique solution z E to (4.1) is proved in (rli,2+1>.6(!)

[5].

Remark 4.2. Representing the solution z in the form (4.11) we get for the remain-

der a the transport equation (4.1) with the right-hand side h - div (wr-215(8)). That oblige us to require more regularity for the "spherical" attributes of z and h. This choise is possible only because they coinside.

4.2. Boundary value problems for the Poisson equation Let us consider Neumann problem for the operator &a, = div (po(x)V), where Po(x)=p.exp $x), 4' 'Yo > 1, i.e. we consider the problem ( EA1+1.6

-AN0c° = V),

= 0,

xE12, xE812,

O(x)-+0, lxl -oo.

(4.16)

Theorem 4.3. (i) Let 7P E VI,-',9(12), l > 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 <- c 11 0;

Vry_'.9(Q)II

(4.18)

(ii) Let ib E Ay 1.6(12), l > 1, 6 E (0,1), ry E (l + 2 + 6,1 + 3 + 6). Then problem (4.16) has a unique solution cp which admits the asymptotic representation (4.17) with ip E A7+1.6(12) and there holds the estimate

IP;A'ry "(Q)II+Icoi
(4.19)

The proof of Theorem 4.3 can be found in [9]. Let us consider now the Dirichlet problem for the Laplace operator xE11,

1;=0, xE812.

(4.20)

Theorem 4.4. [9]. Let S E VI,-',9(12), l > 1, q > 1, y > 1 + 1 - 3/q and let E V°°(12) be a solution of problem (4.20). Then l;' E and lIe;

The solution

vy+1.a(O)II

E V°_i_j(I) is unique.

<

c11<;

VI-1,q(Q)II.

(4.21)

Asymptotic Behavior of Exterior 3D Steady Compressible Flow

143

5. Linearized problem Let us consider problem (1.9'). Theorem 5.1. Let S2 E 1R'; be an exterior domain with the smooth boundary aft, (b E

A'+''6 t+t+svo (SI) 7o > 1, 1> 2, b E (0, 1 ) Po(x) = p,exp4P (x). Suppose that wE c(l; t.t+2),9(52), %I'-t,t).9(12), FE 9 E V4(1), q > 3/2, -y E (1 + 2 - 3/q, I + 3 3/q), be given functions with

w0, div w = 0,

(5.1)

x E 49Q,

and F satisfies the compatibility condition (3.8). There exists a number eo > 0 such that if IIw;fIyl;

1.t+2).9 All < Eo,

(5.2)

then problem (1.9') has a unique solution (o,v) E 277t2 Moreover, div (wa) E V,+1'Q(0) and there hold the estimates

All

'q(mll + 11v;

(IIF+T;`a Ildiv (wa); Vy 9(S2)II

x VU t.t+2),q(D)

.,.t),

c (IIF+Z,r,3 1

q(S2)II + Il9 V7 9(n)II),

.t).9(11)II

+ IIg; V,9(n) II).

(5.3) (5.4)

Proof. As it is mentioned in Introduction, problem (1.9') is equivalent to three problems (1.13)-(1.15) and the solution (a,v) of (1.9') can be found as a fixed point of the linear mapping G: r -b a defined by (1.13)-(1.15). Let us give exact meaning to the formal decomposition scheme described in Introduction. Define the Banach space 93"q(Q) = {a: a E 93("'+ )'9(D), Mdiv (wa) E V1-2,9(II)}

with the norm IIA(div (wa));

IIa

vY-29(f1)II.

Let r E'B79(52) be given. First, we find p as a solution of the Neumann problem Dao

-div (rw) + g, x E SZ;

= 0, x E 60.

(5.5)

Second, we find the solution (u, II) of the Stoke-type problem (1.14) taking in (1.16) a = r, and, finally, we find a as the solution of transport equation (1.15). In such a way we define the operator G: r - a. We show that L is a contraction in the space Byq(fl). The fixed point a of G and the corresponding to a solutions O (solution of (1.13)) and (u, II) (solution of (1.14)) solve (1.13)-(1.15). The solution of the original linearized problem (1.9) has the form (a, v) = (a, u + VW). Consider problem (5.5). First, we realize that

lldiv (wr); V7-; 'All

(11+1)

C Il r Q

9(Q) II IIw;1u

+ t.1+2).9(II) II

(5.6)

T. Leonaviciene and K. Pileckas

144

(see Lemma 2.5. Further, since (divw)Iau = 0, wlau = 0, we get

div (wr) = 0,

x E as1.

Therefore, div (wr) can be considered as the solution of the Dirichlet problem

A(div (wr)) = 0(div (wr)), div (wr) = 0,

x E S1,

x E 811.

By Theorem 4.4 the solution div (wr) of the problem (5.7) satisfies the estimate Ildiv (wr); Vy,9(11)II < c IIO(div (wr));

(5.8)

Vy-2,q(Q) 11

Now, in virtue of Theorem 4.3 there exists a unique solution Ip of problem (5.5) admiting the asymptotic representation (4.18) with < p E V +2. (fl). Moreover, estimates (4.19), (5.6), (5.8) give

III;

ICI

(I)II+llg,v'1.9(Q)II), VI+2.9(S1)II

IIi;

(5.9)

+ IcoI

< c (Ik div (wr); V1-2 qp)II + Ilg; Vi

(5.10)

4(H)II).

Next, we consider problem (1.14) with G defined by (1.16) at o, = r. Using Lemmata 2.1 -2.5 and the asymptotic representation (4.18) for