Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity

Abstract

In this paper we present a systematic study of a stochastic PDE with multiplicative noise modeling the motion of viscous and inviscid grade-two fluids on a bounded domain O of R2. We aim to identify the minimal conditions on the boundary smoothness of the domain for the well-posedness and time regularity of the solution. In particular, we found out that the existence of a H1(O) weak martingale solution holds for any bounded Lipschitz domain O. When O is a convex polygon the solution u lives in the Sobolev space W2,r(O) for some r>2 and rot(u−αΔu) is continuous in L2(O) with respect to the time variable. Moreover, pathwise uniqueness of solution holds. The existence result is new for the stochastic inviscid model and improves previous results for the viscous one. The time continuity result is new, even for the deterministic case when the domain O is a convex polygon.