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2) such that if "
1
T0
sup IVp(t,x)Idt
, rk
If 1
f T sup IVp(t,x)Idt > i. 0
xET"
then for any k > 2 we have: max I
r
j=1...n T 0
sup I
xET" axe
(t, x) I dt, > zek-
IT,
Inequality (5.3) is proved with the constant xk = min{rk, 2fk
;
n }.
References [1] Biryuk A. Spectral Properties of Solutions of Burgers Equation with Small Dissipation. Funct. Anal. Appl. 35, no. 1 (2001), 1-15. [2] Biryuk A. On Spatial Derivatives of Solutions of the Navier-Stokes Equation with Small Viscosity. Uspekhi Mat. Nauk 57 no 1. (2002), 147--148. [3] Dubrovin B. A.; Novikov S. P.; Fomenko A. T. Modern geometry-methods and applications. Part 1. The geometry of surfaces, transformation groups, and fields. Part II. The geometry and topology of manifolds. Graduate Texts in Mathematics 93, 104. Springer-Verlag 1984, 1985. [4] Friedman A. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [5] Frish U. Turbulence. The legacy of AN. Kolmogorov. Cambridge Univ. Press, 1995.
A. Biryuk
30
[6] Jefferson D. A numerical and analytical approach to turbulence in a special class of complex Ginzburg-Landau equations. Heriot-Watt University Thesis, 2002. (7] Halmos P. R. Finite-dimensional vector spaces. Springer-Verlag, New York - Heidelberg, 1974. 181 Hartman Ph. Ordinary Differential Equations. John Wiley & Sons Inc., New York, 1964.
191 Hormander L. Lectures on nonlinear hyperbolic differential equations. SpringerVerlag, Berlin, 1997. [101 Kolmogorov A. N. On inequalities for supremums of successive derivatives of a function on an infinite interval. Paper 40 in " Selected works of A.N. Kolmogorov, vol.1 " Moscow, Nauka 1985. Engl. translation: Kluwer, 1991. [11) Kukavica I., Grujic Z. Space Analyticity for the Navier-Stokes and Related Equations with Initial Data in L°. Journal of Functional Analysis 152, no. 2 (1998), 447-466.
[12] Kuksin S. Spectral Properties of Solutions for Nonlinear PDE's in the Turbulent Regime. GAFA, 9, no. 1 (1999), 141-184. [131 Pogorelov A. V. Extensions of the theorem of Gauss on spherical representation to the case of surfaces of bounded extrinsic curvature. Dokl. Akad. Nauk. SSSR (N.S.) 111, no.5 (1956), 945-947. (14) Spivak M. A comprehensive introduction to differential geometry. Vol. III, Publish or Perish, Boston, Mass., 1975.
Andrei Biryuk Independent University of Moscow 11, Bol'shoj Vlas'yevskij pyeryeulok Moscow 121002
Russia e-mail: biryuk®mccme.ru
Advances in Mathematical Fluid Mechanics, 31-51 © 2004 Birkhauser Verlag Basel/Switzerland
On the Global Well-posedness and Stability of the Navier-Stokes and the Related Equations Dongho Chae and Moon Lee Abstract. We study the problem of global well-posedness and stability in the scale invariant Besov spaces for the modified 3D Navier-Stokes equations with
the dissipation term, -Du replaced by (-A)"u, 0 < a < s. We prove the unique existence of a global-in-time solution in BZ 1
small BZ
for initial data having
.,in for all a E [0, ). We also obtain the global stability
of the solutions Bz 2,1
a
for a E [1, ). In the case 1 < a < 1, we prove y
+1-2n
the unique existence of a global-in-time solution in By . data, extending the previous results for the case a = 1.
for small initial
Mathematics Subject Classification (2000). 35Q30, 76D03. Keywords. Navier-Stokes equations, global well-posedness, stability.
1. Introduction In this paper, we are concerned with the sub-dissipative or hyper-dissipative Navier-Stokes equations.
(SNS)
div u = 0, u(0,x) = ue(x),
where it represents the velocity vector field, p is the scalar pressure. For simplicity, we assume that the external force vanishes, but it is easy to extend our results to the case of nonzero external force. J.L. Lions [22] proved the existence of a unique
regular solution provided a > 2. If a = 1, then the above system reduces to the usual Navier-Stokes equations. For the Navier-Stokes equations, Kato [20] proved the existence of global solution in C([0, oo); L3(R3)) if IIuo11 V is sufficiently small. After Kato's work [20], there were many important improvements using the scaling
Dongho Chae and Jihoon Lee
32
invariant function spaces. Especially, pioneered by Chemin [12], Cannon-MeyerPlanchon [7], initial value problems of the Navier-Stokes equations in some Besov spaces were extensively studied (see also [3], [4], [6] and [9]). For the Enter equations and compressible or incompressible Navier-Stokes equations, there are many recent
improvements using the notion of the Besov spaces and Triebel-Lizorkin spaces (see [10], [11], [13], [14], [16], [17], [21], [26] and references therein). Recently, Cannone-Karch [5] proved some existence and uniqueness theorems of global-intime solutions with external force and small initial conditions in some Besov type spaces in the case that -0+(-A)a replaces (-A)a. Considering scaling analysis, we find that if u(x, t) is a solution of (SNS)a, then ua(x, t) = \2a-1u(.\x, A2at) Bo9 +1-2a , I < pe Q - oo, are scaling invariant is also a solution of (SNS)a' PThus function spaces. Our first main result of this paper is the global existence and uniqueness result for the initial value problem (SNS)a with the initial data small A -2a in 82,1 norm. Precise statement is as follows.
Theorem 1. Let a E [0, be given. There exists a constant e > 0 such that 4) for any ua E B1 - 2a and IIuoII $-2n < e, the IVP (SNS)a has a global unique 1
82.1
solution u, which belongs to L°°(0, oo; B5 212a) ft L1(0, oo; B2 1) fl C([0, oo); B2 1)
with a =
f
-">
- 2a, if 2 < a - 4
-2a-b1, if0 0,
2°) u also belongs to L' (a, oo; B2,1) fl L1(a, oo; B2,i fl C((0, oo); B2,1), where -y = 1
- 62, if 0 < a < 2 , for any b2 > 0. Purthermore, the solution u satisfies the
1 2, if 2 - a < 4, following estimates
sup IIu(t)II
0 2) is contained in B2 1, our result in the case a = I can be regarded as improvement of the result of the corresponding stability result of the Navier- -Stokes equations in [23].
For the usual Navier-Stokes equations (SNS)1, Chemin [15] proved local in time existence in some critical Besov spaces and Cannone-Planchon [8] proved the global existence in some critical Besov spaces if the initial data has a small Besov norm. Cannone-Planchon [9] also derived various estimates for strong solutions in C([0, T]; L:1 (1R:1)) to the 3-dimensional incompressible Navier Stokes equations. Us-
ing the similar method originated from Fujita -Kato [19] and Kato [20], we can improve parts of Theorem 1 in the case z < a < i as follows. Theorem 3. Let a E
a ]bebe given. Suppose 1 < p < 2a . There exists a constant a}1-2° e > 0 such that for any un E Bpoo and uo a 11 _,, < e, the IVP (SNS),, has 11
II
+1-2° a global solution u E C([0, oo); B, ,0 ).
II
HP P,-
2. Littlewood-Paley decomposition We first set our notation, and recall definitions of the Besov spaces. We follow [24]. Let S be the Schwartz class of rapidly decreasing functions. Given f E S, its Fourier transform .F(f) = f is defined by
fef(x)dx.
f
1 {2,).,/z We consider V E S satisfying Supply C {l;' E W' z < IZ 15 2}, and ;p(t) > 0 if z < IC] < 2. Setting Oj = cp(2-- ) (in other words, 1pj(x) = 2j" p(2'x)), we can adjust the normalization constant in front of cp so that I
VV ER n \ {0}.
jEZ
Given k E Z, we define the function Sk E S by its Fourier transform Sk(i) = 1 -
j>k+1
Dongho Chae and Jihoon Lee
34 We observe
SuppcGflSuppop =0if Ij-j'I>2. Let s E R. p, q E [0, oo]. Given f E S', we denote Lj f = cpj * f . Then the homogeneous Besov semi-norm IIf1IBY ° is defined by IIf1IBy
[E 2j9aIIpi*fllL"]° ifgE[1,00) sup,
°
if q = oo.
f II LP ]
The homogeneous Besov space Bp.q is a quasi-normed space with the quasi-norm given by II II BY For s > 0 we define the inhomogeneous Besov space norm If II R;,° of f E S' as I I f I I s' for the Besov spaces.
IIf 111.P + I I f I I B' 4- Let us now state some basic properties
(i) Bernstein's Lemma : Assume that f E LP, 1 < p < oo, and supp f C {2J-2 < ICI < 2j}. Then there exists a constant Ck such that the
Proposition 1.
following inequality holds: Ck12jkIIfIILP < IIDkfIILP S Ck2'kIIfIILP-
(ii) We have the equivalence of norms DkfII $Pq N
II
IIfIIHp
(iii) Let s > 0, q E [1, oo], then there exists a constant C such that the inequality < C (IIIIILPL II9IIB;
IIf9IIB.
+ II9IIL'i IIf 11 B.
)
I
holds for homogeneous Besov spaces, where pl, r, E [1, oo] such that -
- + r2
+ -'
v
=
.
Let si, s2 < 'P such that s, + 32 > 0, f E Bp, and g E Bpj, B. Then f g E s, +a2- IV
Bp I
P and IIf9IIB,t,2-' p>.
We provide the proof of Proposition 2 in the appendix. Taking the divergence operation on the first equation of (SNS), we have the formula
-Op =
(9j (ok(u3uk).
j.k
Dongho Chae and Jihoon Lee
36
This enables us to define the general sub-dissipative Navier-Stokes type equations
J 8=u - A2°u = Q(u. u),
0 0. We provide only a where a > 0 and -y = l 2,if2]+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 0 similarly to the above: IISWIILT(BP)
_1
Since 1 has support in an annulus, we have
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
6a, gf,.(ba)
where C = C(I852I). We refer to the function behind the brace as bounding function, see Figure 4 for a sketch in a special situation.
FIGURE 4.
Bounding function of Ca (x), d = 2, 8S2 = B(0,1),
6=0.1,a=0.99,k=1,C=21r
Let C(t) = { (z, t) Id(z, 09D) < y, t = g6 (y), 6' < y < oo) be the cross section of the bounding function at the function value t and A(t) = IC(t) I the area of the cross section. Then 9a(6^)
J C6 (x) dx < C J Rd
U
A(t)dt.
Commutation Error of the Space Averaged Navier-Stokes Equations
65
From Lemma 4.1, we know A(t) < C(yd + y), with C depending only on Sl. Using ga(y) = t, changing variables and integrating by parts yield 9(6°)
f
r9(6)
A(t)dt
C
J
(yd + y)dt = C
0
J
d (yd + y) y (9a (y))dy
00
C C(bd° + 6°)9s (6°)
-d
yd-'9a (y)dy
- I 96 (y)dy
00
00
The integrals on the last line will be estimated using the change of variables y = 6/t and by monotonicity considerations of the arising integrand. For 6 sufficiently small, one obtains f C6 (x)dx < C 1 6d(°-k) + 6°-Rd) exp
\
Rd
(-
6k 62(1-o)
/
from what follows, since a < 1,
limJ Cs(x)dx=0.
6-0 Rd Now we will bound the second term in (19). The function B,kk (x) can be estimated from above in the following way Ba (x) < { Ian n B(x, 6°) I ° 9s (d(x, a11)) 1 0
if d(x, &I) < 61, if d(x, O fl) > 6°,
see Figure 5 for an illustration of the bounding function in a special situation. The bounding function is discontinuous, having a jump from the value 0 to the value Cg6(6°) at {x E !l I d(x,00) = 6°}.
FIGURE 5.
Bounding function of B6 (x), d = 2, asz = B(0,1),
6=0.1,a=0.99,k=1,C=6°
A. Dunca, V. John and W.J. Layton
66
Since 8i2 is smooth, we have I8 fl B(x, b")I < Cb(d-')" if 6 is small enough. It follows
f B6 (x)dx < C
b
A-1 ^k 4
g6 (d(x, 8 Z))dx.
{d(x.dsa)n/a}
+f
(1 + IIXII2)1/211 - 9a121VI2dx, IIxII2 0, IIxII2)-1/2 which does not depend on 6 and v, such that (1 + < Co for IIx112 > Tr/b. From (22) follows the pointwise estimate 11 -gd(x)I < 1 for any x E Rd. Thus, the first integral can be bounded by
f
IIxII2>*/a}
(1 + IIxII2)/211 - 9a121vI2dx
,r/d}
< CS f
(1 + IIxIl2)Ivl2dx.
(23)
{IIxII2>r/a}
A Taylor series expansion of (22) at IIxII2 = 0 and for fixed 6 gives
9a(x) = 1 -
a2a4II2
+ O(6'Ilxlli),
such that we have the pointwise bound 11
- ga(x)12 < C611X112
for any IIx112 < 7r/6 where C does not depend on S or x. In addition, IIxI12 (1 + IIx1I2)1/2 and consequently the second integral can be bounded as follows:
f
IIxII2:5
/h)
(1 + IIxIl)/211 - oI2II2dxCb f
(1+IIxl)II2dx. (24)
{IIxII2 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
1,
and C and a depend on a, k and IOill. Proof. Analogously to the begin of the proof of Proposition 4.3, one obtains
f v(x) Rd
k
g6(x - s),i(s)dsI dx in
C(k) [jd Iz(x)B6(x)Ik dx + f Iv(x)C6(x)Ik dx] II
IILP(as)),
Rd
where B6(x) and C6(x) are defined in the proof of Proposition 4.3. The terms on the right hand side are treated separately. In Proposition 4.3, it is proven that Ca E Ls (W') for every k E (0, oo). This implies
(Ch )p=C6°=Ca EL'(Rd),
A. Dunca, V. John and W.J. Layton
70
since k' E (0, oo). That means C6 E LP(Pd) for P E [1, oc). From the bounding function of C, it is obvious that C6 E L' (Rd), too. Using Holder's inequality for convolutions, see Adams [1, Theorem 4.30], and 11961ILI(R°) = 1, it follows IlVI1L9(Rd) 1 I1v"IIL9(R°) = IIvIILuk(Rd) < IIVIILvk(Rd)
By the regularity assumptions on v, it follows v E C°(Rd). This implies, together
with v = 0 outside 52, that v E LP(Rd) for every I < p < oo. Consequently, IIVIIL.k(Rd) < oo. Applying Holder's inequality, we obtain JR Rd
Iv(x)C6(x)Ikdx < IIvilLQk(R°)IIC6(x)IILr(Ra).
For the second factor, we can use the bound obtained in the proof of Proposition 4.3, replacing k by kp. Thus if 6 is small enough, we obtain
J
l626kQ)1 IIvIILak(Rd)
d
Jl
(25)
for every test function v which satisfies the regularity assumptions stated in Lemma 6.1.
The estimate of the second term starts by noting that the domain of integrar tion can// be restricted to a small neighbourhood of 85I
Iz(x)B6(x)lkdx =
J Rd
I7,(x)B6(x)Ikdx
J
IlzllL ({d(x,8S1) 0 such that lv(x) - v(y)I < CH llx
- yll2° for all x, y E S2.
By the Sobolev imbedding theorem, this constant can be estimated by CH < C(H)IIvilH2(sz) We fix an arbitrary x E {d(x,852) < 6°} and we take y E 852 with IIx - y112 = d(x, y). Since v vanishes on 852, we obtain Ilv(x)112 < CHd(x, On);'. It follows
IIvllL=({d(x.an) 0
if C is large enough. That means, also for the rational LES model, the kinetic energy of u can be estimated in form (34) and (35) if C is chosen sufficiently large and 6 sufficiently small.
Acknowledgment We thank Prof. G.P. Galdi for pointing out the result of Giga and Sohr in Remark 2.1.
References [1] R.A. Adams. Sobolev spaces. Academic Press, New York, 1975.
[2] A.A. Aldama. Filtering Techniques for Turbulent Flow Simulation, volume 56 of Springer Lecture Notes in Eng. Springer, Berlin, 1990. [3] L.G. Berselli, G.P. Galdi, W.J. Layton, and T. Iliescu. Mathematical analysis for the rational large eddy simulation model. Math. Models and Meth. in Appl. Sciences, 12:1131-1152, 2002.
[4] R.A. Clark, J. H. Ferziger, and W.C. Reynolds. Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mech., 91:1-16, 1979. [5] P. Coletti. Analytic and Numerical Results for k - e and Large Eddy Simulation Turbulence Models. PhD thesis, University of Trento, 1998. [6] A. Das and R.D. Moser. Filtering boundary conditions for LES and embedded boundary simulations. In C. Liu, L. Sakell, and T. Beutner, editors, DNS/LES - Progress and Challenges (Proceedings of Third AFOSR International Conference on DNS and LES), pages 389-396. Greyden Press, Columbus, 2001. [7] Q. Du and M.D. Gunzburger. Analysis of a Ladyzhenskaya model for incompressible viscous flow. J. Math. Anal. Appl., 155:21-45, 1991.
Commutation Error of the Space Averaged Navier--Stokes Equations
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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
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 2, 1 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) 3), Borchers and Miyakawa [5] developed such an approach and succeeded in the proof of IIVUIIL9(s:) 1, where supp u denotes the support of the function u. For a Banach space X we denote by B(X) the Banach space which consists of all bounded linear operators from X into itself. Given R > Ro, we take (and fix) two cut-off functions V)±,R satisfying
E Cx(Rn; [0, 1 ]),
V)±.R( x )
=
in H± \ BR+I, 1
01
in H:F U BR.
In some localization procedures with use of the cut-off functions above, the bounded domain of the form
D±,R={xEH±;R 0 independent off E Co (D±,R) (where Oj denotes all the j-th derivatives); and for all f E Co (D±.R) with f D* x f (x)dx = 0. By (2.2) the operator S±,R extends uniquely to a bounded operator from Wi'Q(D+,R) to W.j+"q(D+.R)°.
Navier-Stokes Flow Through an Aperture
87
For G = S2, H and a smooth bounded domain (n > 2), let Coo (G) be the set of all solenoidal (divergence free) vector fields whose components belong to Ca (G), and L7 (G) the completion of Co (G) in the norm II' Ilq.c If, in particular, G = 52,
then the space Lo(ft) is characterized as (1.2). The space Lq(G) of vector fields admits the Helmholtz decomposition Lq(G) = L9(G) ®Ln(G), 1 < q < oo, with L7 (G) _ {Vp E Lq(G); p E L (G)}; e [271, [60] for bounded domains, [4], [521 for G = H and [22], [53] for G = Q. Let Pq,c be the projection operator
,
from L9(G) onto Lq (G) associated with the decomposition above. Then the Stokes operator Aq,c is defined by the solenoidal part of the Laplace operator, that is,
D(Aq.c) = W2.q(G) n Wo q(G) n L9(G), Aq.G = -Pq.GO, for 1 < q < oo. The dual operator AQ,G of Aq,G coincides with Aq/(q-1),G on Lo (G)' = Lol(q-t)(G). We use, for simplicity, the abbreviations Pq for Pq,n and Aq for Aq,n, and the subscript q is also often omitted if there is no confusion. The Stokes operator enjoys the parabolic resolvent estimate II(. + AG)-' II B(Lo(G)) -< CE/IAI,
(2.3)
for [ arg Al < -7r - e (A i4 0), where e > 0 is arbitrarily small; see [29J, [61] for
bounded domains, [52], [4], [19], [20], [17] for G = H and [22] for G = Q. Estimate (2.3) implies that the operator -AG generates a bounded analytic semigroup {e-tAc; t > 01 of class (Co) in each Lo(G),1 < q < oo. We write E(t) = e-1Ay which is one of E±(t) = e-tA"*. The first theorem provides the Lq-Lr estimates of the Stokes semigroup a-tA for the aperture domain S2.
Theorem 2.1. Let n > 3. 1. Let 1 < q < r < oo (q # oo, r 0 1). There is a constant C = C(12, n, q, r) > 0
such that (1.3) holds for all t > 0 and f E L7 (l) unless q = 1; when q = 1, the assertion remains true if f is taken from V(S2) n Lo(Sl) for some s E (1,00).
2. Let 1 < q < r < n (r
1) or 1 < q < n < r < oo. There is a constant C = C(SZ, n, q, r) > 0 such that (1.4) holds for all t > 0 and f E Lo (1) unless q = 1; when q = 1, the assertion remains true if f is taken from L' (S2) n Ls. (11) for some s E (1, oc).
3. Let 1 < q < oo and f E Lg(Sl). Then as t -* 0 o(t_Q) lie-tA fllr = 1
Iloe- AfIIr = 0(t °-1/2)
if q < r < oo,
ast-oo ifq n even in Theorem 2.2, but we have asserted nothing about their decay rates since they do not seem to be optimal; see Remark 2.1 for the Stokes flow. On the other hand, in Theorem 2.3 the decay rates of Vu(t) in Lr(1l) for r > n are better than t-"/2 for exterior Navier Stokes flows shown by Wiegner [66]. Taking Theorem 5.1 of [17] for the Stokes flow in the half space into account, we would not expect u(t) E L'(9) in general. Thus the decay rates obtained in Theorem 2.3 seem to be optimal; that is, for example, IIu(t)II,o = 0(t-"/2) would not hold true. Concerning the exterior problem, Kozono [42], [43] made it clear that the Stokes and/or Navier-Stokes flows possess L'-summability and more rapid decay properties than (2.11) only in a special situation. Remark 2.7. In Theorem 2.2 one could not define a pressure drop (see Farwig [18, Remark 2.2]) since the solution does not always belong to Lr(S2) for r < n. Due to the additional summability assumption on the initial data, we obtain in Theorem 2.3 the pressure drop written in the form
[P(t)] = P+ (t) - P- (t) = if at+ u Vu - u)(t) wdx, t
where w E W2-4(S2), n/(n - 1) < q < oo, is a unique solution (given by [221) of the auxiliary problem
w-Ow+O7r=0. in 0 subject to wlasa = 0 and t¢(w) = 1. In fact, the formula above is derived from the relations
jw' Vp(t)dx = -[P(t)]O(w) = -[P(t)]. f u(t) Virdx = -[tr]¢(u(t)) = 0. 3. The Stokes resolvent for the half space The resolvent v = (A + AH)-' PH f together with the associated pressure Jr solves the system
Av - Ov+Vir= L in the half space H = H+ or H_ subject to vl8H = 0 for the external force f E L`+(H)l1 < q < oo, and A E C \ (-oo, 0]. In this section we are concerned with the analysis of v near A = 0. Our method is quite different from Abels [1]. One
Navier-Stokes Flow Through an Aperture
91
needs the following local energy decay estimate of the semigroup E(t) = e-tAH, which is a simple consequence of (1.3) for S2 = H.
Lemma 3.1. Let n > 2,1 < q < oo, d > 1 and R > 1. For any small c > 0 and integer k > 0 there is a constant C = C(n, q, d, R, e, k) > 0 such that IIo'ae E(t)PHf II q.)IR 0, f E L',j, (H) and j = 0, 1, 2. Proof. We make use of the estimate u E D(A;/H),
IIV'fhIr.H 1. This completes the proof.
Lemma 3.1 is sufficient for our analysis of the resolvent in this section, but the local energy decay estimate of the following form will be used in section 5.
Lemma 3.2. Let n > 2, 1 < q < oo and R > 1. Then there is a constant C = C(n, q, R) > 0 such that IIE(t)fII2.,I.HR + IIatE(t)fllq.Hn < C(1+t)-n/2giIfIID(A,,H),
(3.3)
fort > 0 and f E D(Aq,H). Proof. The left hand side of (3.3) is bounded from above by C(IIAIIE(t)fIIq.H + II E(t)fIIq,H) 1 it follows from (1.3) for S1 = H with
r = oo that IIE(t)f 11 ,11. < CII E(t)fII oo,H
.n/2g II f II q.H.
The other terms II o'E(t)f 11,11. < CII Aii2E(t)f II r.H S
Ct-j/2IIE(t/2)f IIr.H
(j=1,2)?
IIatE(t)fIIq.11R 0 there is a constant C= C(n, q, d, R, e) > 0 such that
m-1
IAI°Ilaa v(A)112.,,HR + Y, IWaav(A)112,q.H, < CIIfII q,H, k=O
for Re A > 0 (A # 0) and f E m
where
r (n-1)/2 =51 n/2 - 1
ifnisodd, if n is even,
n 1 1/2+e Q=A(e)=1+m-2+e= E
ifnisodd, ifniseven.
Furthermore, we have sup
IIv(A) - w112.q,H IIf IIq.H
f 34 0, f c- LIdi(H) -, 0,
(3.5)
asA-,0 with ReA>0, where
w=/
o
E(t)PH fdt.
Proof. We recall the formula
v(A) = (A + AH)-'PHf = f e-,\'E(t)P,jfdt,
(3.6)
0
which is valid in Lo(H) for Re A > 0 and f E L9(H). In the other region (A E C \ (-oo, 01; Re A < 0) we usually utilize the analytic extension of the semigroup {E(t); Re t > 0} to obtain the similar formula. For the case Re A = 0 (A 34 0) which is important for us, however, thanks to the local energy decay property (3.1), the formula (3.6) remains valid in the localized space Lq(HR) for f E (the function w in (3.5) is well-defined in Lq(HR) by the same reasoning). We thus obtain from (3.1) Il0'a,kv(A)11,,H,, 3, 1 < q < oo, d > Ro and R > R0. For any small e > 0 there are constants Ct = CI (0, n, q, d, R, e) > 0 and C2 = C2(Sl, n, q, d, e) > 0 such that rn- I
Ial°Ilaa T(A)fII2,q,S2R +:IIaaT(.)fII2,q,S:R < CIIIfIIq,
(4.5)
k=0
for Re A > 0 (A # 0) and f E Lq (S2); and m-I JAI" llaa Q(A)fllq+ E II0,k\Q(A)fllq S c'21Ifllq, k=0
for Re A > 0 with 0 < JAI < 2 and f E L' (Q), where m and Q = 0(e) are the same as in Lemma 3.3.
T. Hishida
98
Proof. In view of (4.2), we deduce (4.5) immediately from (3.4) together with (2.2). One can show (4.6) likewise, but it remains to estimate the pressures 7rf contained in (4.4). By (4.1) we have
aairf(x,A)dx=0,
1 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 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,
Isl21h1
for f E
(S2). Finally, to estimate 13, one does not need any localization. In fact,
since
,t(m)(s+h)f -4("_)(s)f = (-i)-+'(m+1)!
f
+h (ir+A)-(-+2)PfdT,
(2.3) gives
IIVm)(s + h) f - Vm)(s)f II2,q,s2R
< C114,(-)(s + h)f - $(m)(s)f IID(A,) < CIhI IsI-(m+1) IIf IIq,
for IsI > 2h° (> 2IhI) and f E Lq(S2). Therefore, we obtain 13 0 and f E
l,q.O
(ul).
T. Hishida
104
For the proof, the following lemma due to Shibata is crucial since we know the regularity of the Stokes resolvent given by Lemmas 4.2 and 4.4. Lemma 5.2. Let X be a Banach space with norm II ' II and g E L' (R; X). If there are constants 0 E (0,1) and M > 0 such that
fa
B llg(s) II ds + sup h#o IhI
then the Fourier inverse image
x
IIg(s + h) - g(s) Ilds < M,
G(t) = 2
e'stg(s)ds
of g enjoys
JIG(t)II < CM(1 + Itl)-B, with some C > 0 independent oft E R.
Remark 5.1. The assumption of Lemma 5.2 is equivalent to
g E (L' (R; X) W'.'(R; X))e.M , where )g denotes the real interpolation functor (the space to which g belongs is known as a Besov space).
Proof of Lemma 5.2. Although this lemma was already proved by Shibata [56], we give our different proof which seems to be simpler. Since IIG(t)II < M/2ir, it suffices
to consider the case Itl > 1. It is easily seen that if ht $ 2ja (j = 0,±1,±2,... then
etht
G(t)
etst(g(s + h) - g(s))ds,
J
27r(1 - etht) from which the assumption leads us to
a
IIG(t)II 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 5. When n = 3 or 4 (thus m = 1), as in Kobayashi and Shibata
[40), we have to introduce a cut-off function p E Co (R; [0, 1)) with p(s) = 1 near s = 0; then one can employ Lemma 5.2 with X = W2,q(S2) and g(s) = p(s)z/4(m)(s)f to obtain the desired result since a rapid decay of the remaining integral far from s = 0 is derived via integration by parts. We did not follow this procedure because Lemma 5.1 is sufficient for the proof of Theorem 2.1. The next step is to deduce the sharp local energy decay estimate (1.5) from Lemma 5.1.
T. Hishida
106
Lemma 5.3. Let n _> 3,1 < q < oo and R > R4. Then there is a constant C = C(12, n, q, R) > 0 such that Ie-tAf II1,q.t1n < Ct-nl2gII f
(5.4)
))q,
fort > 2 and f E Lo(ll); and f1ll.gslR II
+ I]ate-tAf Ilvsltt 5 C(1 + t)-"J24IIf IID(Aq),
(5.5)
fort > 0 and f E D(Aq). Proof. We employ a localization procedure which is similar to [38] and [40]. Given
f E Lo (1). we set g = e-'f E D(Ag) and intend to derive the decay estimate of u(t) = e-tAg = e-(t+1)A f in W1'g(SZR) for t > 1. We denote by p the pressure associated to u. We make use of the cut-off functions given by (2.1) and the Bogovskii operator introduced in section 2. Set
9t = V)t.R,+1 9 -
[9 '
and
v±(t) = E±(t)g 0 and that g± E D(Aq,H f) with .
Note that fD}
gV
R
I19±I) 5 C119±112.q.Ht 5 CI19112.q 5 CII9IID(A,) 5 CIIfII9,
(5.6)
by (2.2). We take the pressures zr± in H± associated to vt in such a way that D4.,,
n f (x, t)dx = 0,
(5.7)
for each t. In the course of the proof of this lemma, for simplicity, we abbreviate
'±to 0± and S±.R to S±. We now define {u±, p± I by ut(t) = V'±v±(t) - S±[vt(t)' VV't],
p±(t) = V)t'rt(t)
Then it follows from Lemma 3.2 together with (2.2) and (5.6) that IIu±(t)111.q.sttt 2 and f E L"(Q) on account of n < q < oo. Along the lines of the proof of Lemma 5.4, one can show
IIe`AfIIo.nt\n 0, U
u(t) =
(6.1)
J
by means of a standard contraction mapping principle, in exactly the same way as in Kato [39], provided that IIaIIn 5 b,,, where 6o = 50(fl, n) > 0 is a constant. The solution u(t) satisfies Ilu(t)IIr
Ct-1/2+n/2r IIaIIn
for n < r < oo,
(6.2)
(6.3)
]Iou(t)lln < Ct-112 IIalln,
for t > 0 together with the singular behavior IIU(t)IIr = o (t-1/2+n/2r)
for n < r < 00; IIVu(t)II.. = o (t-1/2)
,
(6.4)
as t 0. Furthermore, due to the Holder estimate (6.9) below which is implied by (6.2) and (6.3), the solution u(t) becomes actually a strong one of (1.1) with (2.6) (see [26], [32] and [64]). We now prove
lim IIu(t)IIn = 0,
(6.5)
for still smaller a E L'(Sl). To this end, we derive a certain decay property of u(t), which is weaker than (2.11) but sufficient for the proof of (6.5), assuming additionally a E L1(5l) fl L'(f) with small IIaIIn. Given ry E (0,1/2), we take q E (n/2, n) so that -y = n/2q - 1/2; then, t
Ilu(t)IIn 0, which together with (6.2) yields IIu(t)IIn < C(1 + t)-'(IIaIII + IIaIIn),
(6.6)
for t > 0 (this decay rate is not sharp and will be improved in Theorem 2.3). From now on we fix ry E (0,1/2) and set 6 = S.(1l,n,-t)/2. Given a E Lo (12) with IIaIIn 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) 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 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) 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
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
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
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 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 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)
1, then
Hence,
ft f
n
dx1
(3, then D 2V(12) and in the case q = 2 the theorem is proved. If lyo < (3, we continue the iteration process: (u, II) E D3?V(12)
Len=2.3(F, 1
G) E V1-1.2(l) X V1'2 (ft), 3l+1 = min ((1 +
(i)
(u, II)EV ' V(fl)
2 1,2 ... = (F, G) E V3i.;,!-1(12) x V3,+" A Lemma 2.3
Th,,,r< , 9.1 (i)
(u,II)EDs, ..,V(IZ), (3l+_=min ((1+m)'Yo,Q). 1,2
Taking m such that (l + m)-yo > /3, we obtain (u, II) E
Supplying mentioned above inclusions by the corresponding estimates we get for (u, II) the inequality (3.16) at q = 2. Let us consider the case q E (6/5, oo). If q E [6/5,2], /3 > I + 3/2 - 3/q, we have
r
j
f sldx
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) 0 such that if and let w E (E(i
l.t+z),6
IIw;
(S2) div(wz) E
72+1).a(Q)
Moreover, Oz E
(t-2,t-1).o(52
IIAz;
.(x) II
(4.9)
z+1).a(Q)II)
(4.10)
II + Ildiv (wz); A5(S) II < c IIh; (E(11+1)6ry
(Cryt,a2,t-1)'6(S2),
)II
y.4
div (wz) E Ay 2.l A
At-2,6 (p) I I + Odiv(wz); II ry
(E(ryt,i
< c (IIh;
IIz; (rt
IIw;
Proof. (i) Let h E
(4.8)
II 5 eo,
then problem (4.1), (4.2) has just one solution z with z E A7a(Sl) and IIz;
141
1ZT,(yt.2+1)'Q(S1),
i.e. h admits the representation
h(x) = r-255(0) + h(x). Wt+1,9(S)2 and h E Vy 9(52). For sufficiently small co in [10] with attributes b E is proved the existence of the unique solution z E7t;2t1)'9(St) of problem (4.1),
(4.2) which admits the asymptotic representation
z(x) = r- 2b(O) + i(x), z` E V'"(() (4.11) i.e. the "spherical" attributes of h and z coincide. Moreover, there holds the estimate 1, q > 1, -y E (1+2-3/q, 1+3-3/q). Then problem (4.16) has a unique solution cp which admits the asymptotic representation
p(x) = co(27rlxl)-1 +;i(x), where Cp E
(4.17)
There holds the estimate III; 1'1+' 9(Q)II + Icoi 2, b E (0, 1), ry E (l + 2 + d, l + 3 + b), ?'o > 1. Assume also that f satisfies compatibility condition (3.8). There exists a number e. > 0 such that if IIf;
7,3(fl)11 < C.,
then problem (1.7), (1.8) has exactly one solution ( v) E 1.2+1),601) x It 1; 1.t+2).bA satisfying the estimate (Eyr.2+i).d(f)II
II°;
+ IIv;
t.t+2).s(Q)II
Ayr
1
e:0-11011(1l)II.
< c IIf;
(6.12)
Remark 6.4. Note that constants in estimates (6.4), (6.12) depend on
Remark 6.5. F om the obtained results it follows, in particular, that in the case where f has a compact support the solution (v, v) admits the asymptotic representation v(x) = r2 E(O) + a(x), v(x) = rV (e) + v(x), with
IDQir(x)I = O(r-t-IaI-S),
ID'P(x)I =
O(r-2-IaI-7),
Jc J = 0,1, ... , l + 1,
]a] = 0, 1,...,1, V < 1.
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Asymptotic Behavior of Exterior 3D Steady Compressible Flow
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(8] S.A. Nazarov and K. Pileckas. Asymptotic conditions at infinity for the Stokes and Navier-Stokes problems in domains with cylindrical outlets to infinity. Advances in Fluid Dynamics, Quaderni di Matematica, P. Maremonti (Ed.) 4 (1999), 141-243. [9] S.A. Nazarov and B.A. Plamenevskii. Elliptic boundary value problems in domains unth piecewise smooth boundaries. Walter de Gruyter and Co, Berlin, 1994. [10] S.A. Nazarov, A. Sequeira and J.H. Videman. Asymptotic behaviour at infinity of three-dimensional steady viscoelastic flows. Pacific J. Math. 203 (2002), 461-488. [11] S.A. Nazarov, M. Specovius-Nengebaner and G. Thater. Quiet flows for Stokes and Navier--Stokes problems in domains with cylindrical outlets to infinity. Kyushu J. Math. 53 (1999), 369-394. (12] A. Novotny. On steady transport equation. Advanced Topics in Theoretical Fluid Mechanics, Pitman Research Notes in Mathematics, J. Malek, J. Netas, M. Rokyta (Eds.), 392 (1998), 118-146. [13] A. Novotny. About steady transport equation II, Shauder estimates in domains with smooth boundaries. Portugaliae Matheinatica, 54(3) (1997), 317-333. (14] A. Novotny and M. Padula. Physically reasonable solutions to steady compressible Navier-Stokes equations in 3D-exterior domains I (v- = 0). J. Math. Kyoto Univ. 36(2) (1996), 389-423. [15] A. Novotny and K. Pileckas. Steady compressible Navier-Stokes equations with large potential forces via a method of decomposition. Math. Meth. in Appl. Sci. 21 (1998), 665-684. [16] V.A. Solonnikov. On the solvability of boundary and initial-boundary value problems
for the Navier-Stokes system in domains with noncompact boundaries. Pacific J. Math. 93(2) (1981), 443--458.
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