The Asymptotic Properties of the Solution to the Stokes Problem in Domains That Are Layer-Like at Infinity

Abstract

Asymptotic formulae are derived for solutions to the Stokes problem in domains which, outside a ball, coincide with the three-dimensional layer R 2 × (0, 1). The properties of detached asymptotic terms differ in the transversal and longitudinal directions. In order to justify the asymptotic expansions the procedure of dimension reduction is employed together with estimates for miscellaneous weighted norms of the solutions.Mathematics Subject Classification (1991). 35Q30, 76D07.

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

“…If the right-hand side f of problem (1.2) decays sufficiently fast, the condition β < −1 in Lemma 2.1 can be replaced by −1 < β < 0, and then also p ∞ is uniquely determined. The solution (v ∞ , p ∞ ) gets a special asymptotic form, as it was shown in [20]. We distinguish between the longitudinal components, v ∞ , and the transversal component v ∞ z of the vector v ∞ , then…”

“…As shown in [16,[18][19][20], the following anisotropic weighted Sobolev norms (2.1) are especially adapted to a wide class of elliptic boundary value problems in layer-like domains. We recall that x = (y, z) and r = |y|, similarly derivatives…”

“…To justify formula (2.4), we present simplified results of [20] which are sufficient for the further use in this paper: In the following theorem, the estimates of the remainders in (2.4) are not optimal with respect to the smoothness properties of the data and the solutions, in particular, the assumptions on the right-hand side f are too restrictive. Nevertheless, in the case f ∈ C ∞ 0 (Ω) 3 the indices l and N can be taken arbitrary and then Theorem 2.2 provides an explicit information on the power-law asymptotic behavior of the solution.…”

“…These asymptotic expansions (see formulae (2.2)-(2.5)) were obtained in [20] with the help of a method developed in [15][16][17][18], they contain the plane harmonics P N . The approximation problem in the bounded domain Ω R is composed from the Stokes (or Navier-Stokes) equations, the Dirichlet conditions restricted to Σ R = {x ∈ ∂Ω : r < R}, and the ABC on the truncation boundary Γ R = {x : r = R, |z| < 1/2}, in the linear case this means 4) where the operator M R has to be chosen properly.…”

“…Under suitable restrictions for the data it is possible to obtain solutions to the nonlinear problem with the same ABC as for the linear problem together with error estimates of type (1.6) (see Theorem 6.2). However, by using existence results of [21] and the results on the asymptotic behavior of the solutions to (1.2) (see [20,27]) and (1.3) it becomes clear how the nonlinearity influences the asymptotics at infinity of suitable strong solutions to (1.3) -these results are explained in Section 5. Thus for the nonlinear problem the order of convergence is limited by N ≤ 3 in (1.6), even if the right hand side f is infinitely smooth with compact support.…”