Exterior stokes problems and decay at infinity

Abstract

We study the exterior Dirichlet and Neumann problem for the 3-dimensional Stokes equations. By the use of the theory of hydrodynamical potentials, the decay at infinity can be completely characterized in the framework of weighted Sobolev spaces. The results are applied to the theory of weak solutions and to the Whitehead paradox.(1.3) T(u,p)where T ( u , p ) = (Tij(u,p))1Gi,j<3 denotes the stress tensor: 6, is the Kronecker symbol. The Stokes system is elliptic in the sense of Doughs-Nirenberg, both the boundary conditions fulfil the complementing condition (see [4], p. 27ff.). Odqvist ([26]) was the first to investigate exterior Stokes boundary value problems with potential thoeretic methods, he reduced the boundary value problems to Fredholm integral equations of the second kind on the boundary (see also [17], [21]). The more difficult two-dimensional case was treated by Hsiao and MacCamy ([18]). They reduced the exterior Stokes Dirichlet problem to a boundary integral equation of the first kind and explained the Stokes paradox. They gave a rigorous proof for their explanation in [19]. The analysis of the boundary integral equations in Sobolev spaces was completely investigated by Hsiao and Wendland (1201). In [14] and [15] Fischer, Hsiao and Wendland extended the methods of Hsiao and MacCamy to the three-dimensional exterior Dirichlet problem. They gave a rigorous justification for approximating the solutions of the nonlinear Navier-Stokes equations at small Reynolds numbers by solutions of suitable Stokes Problems and explained the three-dimensional Whitehead paradox. The formal proofs concerning the Whitehead paradox are due to Fischer ([14]). The main result in this paper is the following: Extending McOwen's methods from 1281, the solutions of the problems (l.l), (1.2) or (1.3) are completely characterized in the framework of the weighted Sobolev spaces Hp '(i2). These results are also essential for treating the instationary case of the Stokes equations which will be done in a forthcoming paper. The proceeding is now the following: In 11 the basic notations and essential auxiliary results are listed, in 111 the main theorem is presented, IV contents the proof of the main theorem. Since the Neumann problem can be treated in quite an analogous way, the proof is essentially restricted to the case of the Dirichlet problem, remarks about the differences between Dirichlet and Neumann problem are added at the end of sec. IV. In V we give some applications to the theory of "weak" solutions and the Whitehead paradox. I1 Notations, auxiliary resultsLet l2 C R3 be an exterior domain with compact boundary which we assume of class C" for simplicity, fi denotes the closure and i2-the complement of i2 in R3. As usual, Cr(i2) consists of all infinitely differentiable functions with compact support in 52. For 1 < r < 00, m E N, Hm*'(i2) {fe L'(R), D "~E L'(O), 0 < I a 1 < m} provided with the usual norm, and H"*'(sZ) stands

The Stokes Operator in Weighted $L^{q}$ -Spaces I: Weighted Estimates for the Stokes Resolvent Problem in a Half Space

“…Moreover, by Lemma 5.3 there is a solution (v, s) ∈ W 1, q ω (R n + ) n × L q ω (R n + ) of (17) corresponding to f = 0, g = 0 and γ(v) = φ − γ(W). Then u := v + W R n + and p := s + S R n + satisfy (17).…”

“…[9], [12], [17], [18]) in unbounded domains with weight functions vanishing or increasing for |x| → ∞ but being bounded from above and from below by positive constants near the boundary of the domain. We emphasize that our results hold for arbitrary Muckenhoupt weights, i.e., the weight function may become singular or vanish also at the boundary.…”

Weak solutions for the exterior Stokes problem in weighted Sobolev spaces

“…Our main result (Theorem 2.2) extends those of Specovius-Neugebauer in the following manner. On the one hand, the case of weak solutions has not been treated in [22] when n*3. On the other, the introduction of the logarithmic weight enables us to treat more cases.…”

“…For instance, Girault and Sequeira have studied in [11] the case where n"2 or 3, p"2 and "0, by variational methods. Specovius-Neugebauer has considered in [22] the problem of "nding strong solutions to the problem (S) when n*3 and n/p# , 9. The problem of weak solutions to the two-dimensional problem has also been treated by the same author in [23] (with again the restriction that 2/p# , 9).…”

Exterior Stokes problem in the half-space

“…They also provide some precise information on the behaviour of the functions at infinity, which is not obvious from the definition of H 1, p 0 ( c ω 0 ). For this approach, we refer to Girault and Sequeira [19] (when n = 2 or n = 3, p = 2 and α = 0), Specovius-Neugebauer ( [26] when n ≥ 3 and n p + α / ∈ Z for strong solutions and when n = 2 and 2 p + α / ∈ Z for weak solutions in [27]) and to Alliot and Amrouche [3].…”