Malliavin calculus for highly degenerate 2D stochastic Navier–Stokes equations

Abstract

This Note mainly presents the results from “Malliavin calculus and the randomly forced Navier–Stokes equation” by J.C. Mattingly and E. Pardoux. It also contains a result from “Ergodicity of the degenerate stochastic 2D Navier–Stokes equation” by M. Hairer and J.C. Mattingly. We study the Navier–Stokes equation on the two-dimensional torus when forced by a finite dimensional Gaussian white noise. We give conditions under which 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. In particular, our results hold for specific choices of four dimensional Gaussian white noise. Under additional assumptions, we show that the preceding density is everywhere strictly positive. This Note's results are a critical component in the ergodic results discussed in a future article. To cite this article: M. Hairer et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).