Hemivariational inequalities for stationary Navier–Stokes equations

Abstract

In this paper we study a class of inequality problems for the stationary Navier–Stokes type operators related to the model of motion of a viscous incompressible fluid in a bounded domain. The equations are nonlinear Navier–Stokes ones for the velocity and pressure with nonstandard boundary conditions. We assume the nonslip boundary condition together with a Clarke subdifferential relation between the pressure and the normal components of the velocity. The existence and uniqueness of weak solutions to the model are proved by using a surjectivity result for pseudomonotone maps. We also establish a result on the dependence of the solution set with respect to a locally Lipschitz superpotential appearing in the boundary condition.