A nonlinear stochastic diffusion-convection equation with reflection

Abstract

We study a nonlinear, pseudomonotone, stochastic diffusion-convection evolution problem on a bounded spatial domain, in any space dimension, with homogeneous boundary conditions and reflection. The additive noise term is given by a stochastic It\^{o} integral with respect to a Hilbert space valued $Q$-Wiener process. We show existence of a solution to the pseudomonotone stochastic diffusion-convection equation with non-negative initial value as well as the existence of a reflection measure which prevents the solution from taking negative values. In order to show a minimality condition of the measure, we study the properties of quasi everywhere defined representatives of the solution with respect to parabolic capacity.