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

TL;DR

This paper proves the existence of martingale solutions for stochastic Navier-Stokes equations in unbounded 2D and 3D domains with multiplicative noise, using advanced approximation and compactness methods.

Contribution

It establishes the existence of solutions in unbounded domains with multiplicative noise, extending prior results to more general settings.

Findings

Existence of martingale solutions proved for stochastic Navier-Stokes equations.

Development of compactness and tightness criteria in non-metric spaces.

Generalization of Dubinsky's Theorem for unbounded domain analysis.

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.