H\"older regularity of the densities for the Navier--Stokes equations with noise

TL;DR

This paper proves that the finite-dimensional projections of weak solutions to the stochastic Navier-Stokes equations have densities that are both bounded and Hölder continuous, enhancing previous regularity results.

Contribution

It introduces new analytical estimates on a conditioned Fokker-Planck equation to establish regularity of densities for the Navier-Stokes equations with noise.

Findings

Densities of finite-dimensional projections are bounded.

Densities are Hölder continuous.

Improves upon previous regularity results.

Abstract

We prove that the densities of the finite dimensional projections of weak solutions of the Navier-Stokes equations driven by Gaussian noise are bounded and H\"older continuous, thus improving the results of Debussche and Romito [DebRom2014]. The proof is based on analytical estimates on a conditioned Fokker-Planck equation solved by the density, that has a "non-local" term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation.