On the Stokes Equation with the Leak and Slip Boundary Conditions of Friction Type: Regularity of Solutions

Abstract

We consider the Stokes equations under some nonlinear boundary conditions, which are described in terms of subdifferentials of maximal monotone graphs and are called leak and slip boundary conditions of friction type. The main objective is to show the existence of strong solutions, say $u\in H^2$ and $p\in H^1$, to these problems. We start with weak solutions to variational inequalities, and then study the regularity of weak solutions. Our main theorems imply the maximality of Stokes operators with such nonlinear boundary conditions in a suitable Hilbert space and they are of use in analysis of time-dependent problems. Linear boundary conditions of Neumann type, such as slip and penetration conditions, are also discussed.

A correction to this paper is available.