# On stochastic modified 3d navier-stokes equations with anisotropic viscosity

## TL;DR

This paper investigates stochastic 3D Navier-Stokes equations with anisotropic viscosity, establishing existence, uniqueness, and large deviations principles for solutions with added Brinkman-Forchheimer terms.

## Contribution

It introduces a novel approach combining anisotropic viscosity, stochastic perturbations, and Brinkman-Forchheimer terms to prove existence, uniqueness, and large deviations for the model.

## Key findings

- Existence of global weak solutions is proven.
- Uniqueness of solutions is established under new regularity conditions.
- A Large Deviations Principle for the solutions is demonstrated.

## Abstract

Navier-Stokes equations in the whole space R^3 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 convective term given in terms of the Brinkman-Forchheirmer provides some extra regularity in the space L^{2$\alpha$+2} (R^3), with $\alpha$ > 1. As a consequence, the nonlinear term has better properties which allows 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.