Time regularity of the densities for the Navier--Stokes equations with noise

TL;DR

This paper proves that the probability densities of finite-dimensional projections of 3D stochastic Navier--Stokes solutions are H"older continuous in time within certain function spaces, aligning with diffusive scaling expectations.

Contribution

It establishes the time regularity of densities for stochastic Navier--Stokes equations in three dimensions, extending understanding of their probabilistic properties.

Findings

Density is H"older continuous in time in L^1 space.

H"older continuity persists in Besov spaces.

Exponents match diffusive scaling expectations.

Abstract

We prove that the density of the law of any finite dimensional projection of solutions of the Navier--Stokes equations with noise in dimension $3$ is H\"older continuous in time with values in the natural space $L^1$. When considered with values in Besov spaces, H\"older continuity still holds. The H\"older exponents correspond, up to arbitrarily small corrections, to the expected diffusive scaling.