Malliavin calculus and ergodic properties of highly degenerate 2D stochastic Navier--Stokes equation

The objective of this note is to present the results from the two recent papers. We study the Navier--Stokes equation on the two--dimensional torus when forced by a finite dimensional white Gaussian noise. We give conditions under which both the law of the solution at any time t>0, projected on a finite dimensional subspace, has a smooth density with respect to Lebesgue measure and the solution itself is ergodic. In particular, our results hold for specific choices of four dimensional white Gaussian noise. Under additional assumptions, we show that the preceding density is everywhere strictly positive.