On stochastic modified 3D Navier-Stokes equations with anisotropic viscosity

Navier-Stokes equations in the whole space R3 subject to an anisotropic viscosity and a random perturbation of multiplicative type is described. By adding a term of Brinkman-Forchheimer type to the model, existence and uniqueness of global weak solutions in the PDE sense are proved. These are strong solutions in the probability sense. The Brinkman-Forchheirmer term provides some extra regularity in the space L2α+2(R3), with α>1. As a consequence, the nonlinear term has better properties which allow to prove uniqueness. The proof of existence is performed through a control method. A Large Deviations Principle is given and proven at the end of the paper.