Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise

TL;DR

This paper proves that finite-dimensional projections of solutions to 3D stochastic Navier-Stokes equations driven by Gaussian noise have smooth densities, under mild noise assumptions, using a novel analytical approach.

Contribution

It establishes the existence and smoothness of densities for finite-dimensional functionals of 3D stochastic Navier-Stokes solutions with minimal noise assumptions, introducing a new proof technique.

Findings

Finite-dimensional projections have densities w.r.t. Lebesgue measure.

Densities exhibit smoothness in Besov spaces.

Results hold under very mild noise assumptions.

Abstract

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in Fournier and Printems (2010).