Well-Posedness of Steady-State Bingham Type System by a Quasi Variational-Hemivariational Approach

Abstract

A stationary incompressible Navier-Stokes equation with a nonmonotone slip boundary condition in a bounded domain for a Bingham type fluid is studied. The weak formulation of the fluid problem leads to a new class of elliptic quasi variational-hemivariational inequality with constraints. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier-Fujita frictional slip condition. The analysis provides results on existence and uniqueness of solution to the quasi variational-hemivariational inequality, and on continuous dependence of the solution on the data. The proofs profit from results of nonsmooth analysis, the theory of pseudomonotone operators, and a fixed point argument.