3D Navier-Stokes Equation with Multivalued Friction

Abstract

We prove the existence of weak solutions and their continuous dependence on the data for a three-dimensional incompressible Navier-Stokes system subject to multivalued friction boundary conditions, within a Hilbert space framework. Under suitable assumptions, a fixed point argument yields the existence of a solution to the associated perturbed stationary problem. Using nonlinear semigroup theory, we then establish the existence of a mild solution to the non-stationary problem and prove a contraction principle. Finally, by combining the Crandall-Liggett implicit discretization scheme with a priori estimates for the approximating mild solutions, we obtain weak solutions that depend continuously on the initial data.