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

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.