Stochastic 3D Navier-Stokes equations with nonlinear damping: martingale solution, strong solution and small time large deviation principles

Abstract

In this paper, by using classical Faedo-Galerkin approximation and compactness method, the existence of martingale solutions for the stochastic 3D Navier-Stokes equations with nonlinear damping is obtained. The existence and uniqueness of strong solution are proved for β > 3 with any α > 0 and α ≥ 1 2 as β = 3. Meanwhile a small time large deviation principle for the stochastic 3D Navier-Stokes equation with damping is proved for β > 3 with any α > 0 and α ≥ 1 2 as β = 3.