Nonlinear and oblique boundary value problems for the Stokes equations

Abstract

In this paper we consider the nonlinear boundary value problem governed by a stationary perturbed Stokes system with mixed boundary conditions (Dirichlet- maximal monotone graph), in a smooth domain. We first establish the existence result and some estimates for weak solutions of its approached problem. A specific regularity of the velocity and the pressure are obtained. The proof is based on the approach of maximal monotone graph by its Yosida regularization and the contraction method. 1. Introduction and formulation of the problem This paper concerns the study of the existence and regularity for the solution of the following problem. Let be a bounded open subset of R n (n = 2, 3) of class C 2 . The boundary = @ is assumed to be composed of two portions 1 and 2, with measure ( 1) > 0. The notationwill stand for a maximal monotone graph such that 0 2 �(0). For given body forces f 2 L 2 () n , we look for a solution (u, p) in H 2 () nH 1 () of the following problem: