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0;
(6)
see 7 , Theorem 4.15, and 5 , loc. cit. Referring to Dahlberg, Kenig, Verchota 2 , Theorem 4.6 and t o 5 , loc. cit., we further see there exists a solution (u,7r) of (1), (3) fulfilling (6) with p = 2, provided / is given in L2(Q)3 and h in L2{dQ)3 with
/ h -^ dtt = 0 Jan IdCL
for ^ 6 Z{0Q),
(7)
where Z( • d£l »-)■ M3 such that tp(x)
= a + b x x (x G 0 ;
see Dahlberg, Kenig 1 , Theorem 4.18. Moreover, for p G (1,2], F eLp(tt), Ge W1,p(dQ), boundary value problem (8), (9) may be solved by a function U with
u G wfc*(Q) n w1 + 1/p-€>p{ty This is implied by Verchota L*(n), if GLP( 0 .
(11)
, Theorem 5.1. Finally, let p G (1, 2] ,
I H d£l Jen
F G
I F dx = Q. Jo,
Then it may be deduced from Theorem 4.18 in 1 that problem (8), (10) has a solution U which fulfills (11). These results are also valid for arbitrary Lipschitz domains. We see that for solutions of the Poisson equation on Lipschitz domains, a rather complete Lp-theory is available, whereas for the Stokes system, only a L 2 -theory could be developed. This, admittedly, was difficult enough, but this still raises the question what to expect if p ^ 2 . Here we shall give a partial answer to this question: Restricting ourselves to the case of our special Lipschitz domain Q,, we shall prove that among the preceding three results on solutions of the Laplace equation, two are valid for solutions of the Stokes system as well. In order to be able to state our results, we fix a non-tangential direction field m : dCt »-> M 3 . This function should be smooth and satisfy the ensuing conditions: |ra(x)| = 1,
x — K, • m(x) G ^2,
x + K • m(x) G M 3 \Q
for x G 9fi, KG (0,X>i), and | x + K - m(x) — x' — K1 - m(x/) | >
V2 • (\x-x'\
+
|«-«;|)
for x, x' G dSl, K, K! G (— V\, V\), with certain constants Vi, T>2 > 0. Some indications on how to construct such a function are given in 1 2 . Now our results may be stated as follows:
4 3
T h e o r e m 1 Let p G [2, o o ) , / G LP(Q)3»^ , g G Lp(d£l)^3 ■with (5). Then there is a solution (u, re) of (1), (2) satisfying (4), with u taking boundary values in this sense: /
\9{x)') ~ u(x — K • m(x))
3
\ dQ(x) —> 0
Moreover, for any e > 0 , there is a constant and e such that IMII/P-
C | P
0 on/y depending
(II/IIP +
on Q, p
IUIIP)-
T h e o r e m 2 Let p G ( 1 , 2 ] , / G L p ( f ti)) 33 ,, ft G L p ((/, *) := (S(f)
+ V(*)) | fi ,
+ Q($)) | Q .
Collecting our previous results, we find that the pair of functions («,*) =
(«(/,*),*(/,*))
satisfies the regularity conditions stated in (4), and (U.TT) =
(«(/,*), e(/,*))
those in (6), provided / G L p (ft) 3 , $ G Lp( 0, there is a constant C = C(fi, p, e) > 0 with
M / . * ) | n | | 1 + 1 / , _ e , , + ll«(/.*)in|li /p _«, P + I M / , * ) N 1 / p _ £ , p < c-di/H,, + Hsu,), as follows from (12) and (14). We finally observe that /
I A ( l , p , f i )($)(*) + « ( / ) ( * ) - u{f,$)(x-K-m{x))\Pi ( * ) ) | P dQ(x) dn{x)^0,
JdCL '
/
I A* ( - l . p . f l )(*)(*) + T(R(f), T(R(f),S(f))(x).nM(x) - T ( « ( / , * ) , $ ( / , * ) ) ( x - « . m ( a j ) ) • n ^ ( x ) T df2(x)
—> 0
for K 4- 0 . Thus, referring to (13), we see that Theorem 1 and 2 are now reduced to the ensuing claims pertaining to certain integral operators on d£l:
8 3 3 Lpn(d£i) := { g G Lpp{dti) T h e o r e m 3 Set L£(),
is bounded invertible. T h e o r e m 4 Set L%(dQ) := {he Lp(dQ) {dQ)33 : h fullfils (7) } . Then, t/pifp 2 . It is well known t h a t kernel A* ( 1l ,, 2, ft) = s p a n { n ^ ) } ,
dimkernel dim kernel A* ( - 11 ,, 2, ft) =
0ft), k e r n e l A ( - l , 2, ft) = Z (Z{dQ),
dim d i m kkernel e r n e l A (( l1,, 2, ft) =
6,
1;
see L e m m a 13.8 i n 4 and Theorem 4.6 i n 2 . From this we deduce the relations kernel A* ( 1, g, q, ft) D s p a n { n ( n ) } ,
dim kernel A* ( -—1 ,1, q,g, ft ft) dimkernel ) >> 66, ,
for gq < < 2 , and kkernel e r n e lA( A (--1l , g, q, ft) = ZZ((5dfftt)),,
dim kernel A ( 1, g, ftft)) < dimkernel
1;
for gq > > 2 . Since the operators in (15) have index 0 for g G ( 1 , 2 ] , and because index A ( l , g, ft) = = dimkernel dimkernel A A (( ll ,, gg ,,
ft) ft)
dimkernel 1/g)" 11 ,, ft) dimkernel A* A* (( 1, 1, (1 (1 -- 1/g)" ft)
--
for for gg > > 22 ,, and and i n d e x A * ( - l , g, ft) = dimkernel A* ( - l , g ,
ft)
-
dimkernel A ( - l , (1 - 1 / g ) " 1 , ft)
10
for q < 2 , we may now conclude kernel A* ( 1, q, ft) = = span{n( n )},
dim kernel A* ( - 1 , q, g, ftft)) = 6,(16)
for q < 2 , and kernel A( A ( -- 11,, g, ft) = Z(dQ) ,
dim kernel A ( 1, g, ft ) = 1;
(17)
for q > 2 . Now Theorem 3 and 4 follow by the theory of Fredholm operators. We mention that due to (16) and (17), boundary value problems (1), (2) and (1), (3) may* be solved in the exterior domain — IR\ft as well. This mayJ ~—~* v / ' v / — — — \ — be shown by the same arguments as used in the case of a smoothly be shown by the same arguments as used in the case of11 a smoothly bounded bounded 6 6 , p. 187-196, where results from 11 were worked out in domain. We refer t o domain. We refer t o , p. 187-196, where results from were worked out in detail. detail. References 1. D. E. J. Dahlberg and C. E. Kenig, Ann. of Math. 125, 437 (1987). 2. B. E. J. Dahlberg, C. E. Kenig and G. C. Verchota, Duke Math. J. 57, 795 (1988). 3. M. Dauge, SIAM J. Math. Anal. 20, 74 (1989). 4. P. Deuring, The Stokes system in an infinite cone (Akademie Verlag, Berlin, 1994). 5. P. Deuring and W. von Wahl, Math. Nachr. 171, 111 (1995). 6. P. Deuring, W. von Wahl and P. Weidemaier, Bayreuth. Mathematische Schriften 27, 1 (1988). 7. E. B. Fabes, C. E. Kenig and G. C. Verchota, Duke Math. J. 57, 769 (1988). 8. V. A. Kozlov, V. G. Maz'ya and C. Schwab, J. Reine Angew. Math. 465, 65 (1994). 9. V. G. Maz'ya and B. A. Plamenevskii, Z. Anal. Anw. 2, 335 (1983). 10. V. G. Maz'ya and B. A. Plamenevskii, AMS Translations 123, 109 (1984). 11. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow (Gordon and Breach, New York e.a., 1969). 12. J. Necas, Les methodes directes en theorie des equations elliptiques. (Masson, Paris, 1967). 13. G. C. Verchota, J. Funct. Anal. 59, 572 (1984).
11
W E I G H T E D ESTIMATES FOR T H E OSEEN EQUATIONS A N D T H E NAVIER-STOKES EQUATIONS IN E X T E R I O R DOMAINS*
REINHARD FARWIG Fachbereich Mathematik, Technische Hochschule Darmstadt 64289 Darmstadt, Germany HERMANN SOHR Fachbereich Mathematik-Informatik, Universitat-GH Paderborn, 33095 Paderborn, Germany In this paper we develop weighted L^-estimates for the linear Oseen equations in I] R n , n > 2, and extend them by perturbation to the nonlinear case. In case n = 3 these estimates are used to prove decay properties of solutions with finite Dirichlet integral and nonzero velocity at infinity of the stationary Navier-Stokes equations in exterior domains. It follows that these solutions are P-R-solutions in the sense of Finn. This yields a short new proof of Babenko's result 1 (for another approach see Galdi 8 , 9 ' 1 0 ) and extends it to a larger class of forces with unbounded support. Furthermore this method avoids the use of the explicit integral representation of the solution.
1
Introduction
In an exterior domain Q C IRn with boundary T = dQ of class C2,n > 2, consider the stationary Navier-Stokes system —vAu + u • Vw -f Vp — / , div u = 0 in Q, —isAu u\r = up, uu —> ^oo Uoo as \x\ —»> oo; oo; = wr,
(1-1) (1-1)
here v > > 0 is a constant, / = ( / i , . . . , / n ) • ^& -> El Hln , ur : T —> IR n , and n Uoo £ G IR are the prescribed data while the velocity field u = (t/i,.. ., un) and the pressure p are the desired solutions representing a flow within Q. It is well known 15 ' 16 that for a suitable right-hand side (1.1) has a weak solution u with * Research supported by Sonderforschungsbereich 256 "Nichtlineare " Nichtlineare partielle Differentialgleichungen", Bonn, and by the DFG research group "Gleichungen der Hydrodynamik", Bayreuth/Paderborn. Mathematics Subject Classification: 35 Q 30, 76 D 05 Key Words: D-solutions, PK-solutions, Oseen equations, stationary Navier-Stokes equations, weighted estimates.
12 finite Dirichlet integral | | V i i | | ! = /f \Vu\2dx ||Vii||2
< oo,
(1.2)
called a D-solution; for a precise definition see Section 4. For the construction of strong solutions with finite Dirichlet integral in weighted function spaces we refer to F i n n 7 and Farwig 3 , 4 . A classical problem is to derive regularity properties at infinity of a given Dsolution. T h e pointwise decay \u(x) - Uoo| -» 0 as | z | -»■ oo was proved by Uoo = 0 and much later 5 ' 1 5 for u^ Uoo ^ 0 . To establish the rate of Leray 1 6 for u^ decay of \u(x) - Uoo| - » 0 let u^ ^ 0. Replacing w u ^ and using a u by u - Uoo "cut-off" near oo the linearization of (1.1) leads to the system —vAu -f 4- UOQ • Vw 4+ V p = / , div u — = g
(1.3) (1.3)
in the whole space M n . These Oseen equations play a fundamental role where due to the cut-off procedure it is necessary to a d m i t g ^ 0. In a series of papers G a l d i 8 , 9 ' 1 0 ' 1 2 developed a new method to investigate the equations (1.1), (1.3). Using Lizorkin's multiplier theorem in the analysis of (1.3) he found a new proof of Babenko's famous result 1 : if T, / are smooth enough and supp / is bounded, then a D-solution is a PR-solution in the sense of F i n n 6 ' 7 , i.e. K*)-t*oo| \. Our purpose is to extend Galdi's theory in two directions. First we develop new a priori estimates of (1.3) in weighted Sobolev spaces using weights of the form M(x) = (l + | z | r , 0 < a < 1, see Theorem 3.1. To prepare these estimates we need results without weights, see Theorems 2.3 and 2.4, which rest on Lizorkin's multiplier theorem too, b u t extend Galdi's results concerning uniqueness and embedding properties. Next we introduce a perturbation argument to extend these weighted results to the nonlinear equations Moo • V u + —vAu 4- Uoo 4- u • Vw 4- V p = / , div u = g
(1.4) (1.4)
in IR n , see Theorem 4 . 1 . Writing u • Vw as u • Vw where w = u is considered to be fixed, we treat u • Vw as a linear perturbation of the first two terms in (1.4) and apply K a t o ' s perturbation criterion 1 4 . Here we need a smallness
13
assumption on Vtu which is satisfied in our application to the original equations (1.1) after using an appropriate cut-off. Up to this point the results are valid for all dimensions n > 2. If n — 3, let u be a D-solution of (1.1) under the assumption ||(1 ++ | I••| )iar//| l| ,l ,