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

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. Bernoulli 16(2):343–360, 2010).