Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability

Abstract

UDC 519.21 We initiate the investigation of stochastic Navier–Stokes variational inequalities involving unilateral boundary conditions and nonlinear forcings driven by Wiener processes for which we establish the existence of a probabilistic weak (or martingale) solution.  Our approach involves an intermediate penalized problem whose weak solution is obtained by means of Galerkin's method in combination with some analytic and probabilistic compactness results.  The required probabilistic weak solution of the stochastic Navier–Stokes variational inequality is consecutively obtained through the limit transition in the penalized problem. The main result is new for stochastic Navier–Stokes variational inequalities. It is a stochastic counterpart of the work of Brezis on deterministic Navier–Stokes variational inequalities and generalizes several previous results on stochastic Navier-Stokes equations to stochastic Navier–Stokes variational inequalities with unilateral boundary conditions.