Regularity of Solution to Generalized Stokes Problem

Presented work deals with the interior regularity of solution to generalized Stokes problem. For this purpose the method of difference quotients is used. Some results from my diploma thesis are needed and will be sumarized in the first section. The second section contains the proof of the main theorem. Introduction For given datas Ω ⊂ Rd, A : Ω 7→ Rd2×d2 , B : Ω 7→ Rd2 , f : Ω 7→ Rd and g : Ω 7→ R the following problem is considered −divADu+B∇p = f on Ω div u = g on Ω u = 0 on ∂Ω (1) Here Du is symmetric part of gradient, it means (Du)lj = 12 ( ∂ul ∂xj + ∂uj ∂xl ) . The study of this problem is motivated by the research concerning partial regularity of weak solutions of systems of equations describing the flow of incompressible fluid whose viscosity depends on the pressure and the shear rate. More about such models can be read in papers of Málek, Buĺıček, Franta and Rajagopal (see e.g. [2], [3]). Generalized Stokes problem can be obtained as linearization of equations published there. The special case of constant matrix B was investigated in [4]. Existence and uniqueness of weak solution of equation (1) was studied in my thesis. Results, which are important for this article, will be reminded here. We assume that Ω is always an open Lipschitz domain in Rd. For spaces of functions defined on Ω, we omit it in notation. Norms will be denoted in following way: ‖.‖k,p,V is norm in space W k,p(V ), ‖.‖p,V is norm in space Lp(V ) resp. ‖.‖k,p is norm in W k,p and ‖.‖p is norm in Lp. As before on subscript V is omitted for V = Ω. Letter c will denote a constant that can vary from line to line but it is always independent on solutions and datas f and g. Shifted function fv is defined as fv(x) def = f(x− v). A d2 × d2 matrix A is symmetric, i.e. Akl ij = Ail kj = A ij kl ∀i, j, k, l = 1, . . . , d For A ∈ L∞(Ω, Rd2×d2), B ∈ W 1,∞(Ω, Rd2), f ∈ W−1,2(Ω, Rd) and g ∈ L2(Ω, R) a weak solution of (1) is defined as a couple (u, p) ∈W 1,2 0 (Ω, Rd)× L2, div u = g, fulfilling equation1 ∫ Ω A ij (Du)lj(Dφ)ki + ∫ Ω p ∂ ∂xj (Bkjφk) = [f, φ]W 1,2 0 (2) for all φ ∈W 1,2 0 . Notation [f, φ]W 1,2 0 is duality between spaces W −1,2 and W 1,2 0 . Definition 1. We say, that condition (p1) is fulfilled, if matrix A ∈ L∞(Ω, Rd2×d2) is symmetric and elliptic with constant α > 0 (it is ∫ ΩADφDφ ≥ α‖φ‖ 2 2 ∀φ ∈ W 1,2 0 ), matrix B can be Summation convention is used throughout the paper 80 WDS'09 Proceedings of Contributed Papers, Part I, 80–83, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS