Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains

Abstract

Stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for non-metric spaces. Moreover, some compactness and tightness criteria in non-metric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.