# Absolute continuity of the law for the two dimensional stochastic Navier-Stokes equations

## TL;DR

This paper proves the existence, uniqueness, and regularity of solutions to the 2D stochastic Navier-Stokes equations with Gaussian noise, demonstrating the absolute continuity of the solution's law under certain conditions.

## Contribution

It establishes the absolute continuity of the law for solutions to the 2D stochastic Navier-Stokes equations with Gaussian noise, including regularity and differentiability results.

## Key findings

- Existence and uniqueness of weak solutions.
- Space-time continuity of solutions with continuous initial vorticity.
- Absolute continuity of the solution's law with respect to Lebesgue measure.

## Abstract

We consider the two dimensional Navier-Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and coloured in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $\xi$ and we show that, if the initial vorticity $\xi_0$ is continuous in space, then there exists a space-time continuous version of the solution. In addition we show that the solution $\xi(t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.