THE KOLMOGOROV EQUATION ASSOCIATED TO THE STOCHASTIC NAVIER–STOKES EQUATIONS IN 2D

Abstract

We consider a 2D Navier–Stokes equation with Dirichlet boundary conditions perturbed by a stochastic term of the form [Formula: see text], where Q is a non-negative operator and Ẇ is a spacetime white noise. We solve the corresponding Kolmogorov equation in the space L2(H,ν) where ν is an invariant measure and prove the "carré du champs" identity. The key tool are some sharp estimates on the derivatives of the solution. Our main assumption is that the viscosity is large compared with the norm of Q. As a byproduct, we give a new simple proof of the uniqueness of the invariant measure and obtain an existence result for a Hamilton–Jacobi equation related to a control problem.