Densities for the Navier-Stokes equations with noise

We present a proof of existence of the density with respect to the Lebesgue measure for the finite dimensional marginal distributions of the law of solutions of the 3D Navier-Stokes equations forced by Gaussian noise, as well as of regularity of the density in Besov spaces.

Classical methods, such as the Malliavin calculus, do not work in this setting for reasons which are strongly related to the three dimensional case. Existence of a density is then ensured by an ad-hoc probabilistic method.

The same method provides also regularity in time of the densities, as well as absolute continuity of the laws of some quantities (the energy and dissipation) that depend on a infinite number of components.

When the random forcing has no full support in Fourier space, we prove existence of a density, although without any regularity, by using the backward local smoothness of trajectories, and weak--strong uniqueness.