A Stochastic View over the Open Problem of Well-posedness for the 3D Navier-Stokes Equations

This series of lectures discusses the open problem of well-posedness of 3D Navier-Stokes equations from the viewpoint of stochastic analysis, namely attempting to understand whether noise may improve the well-posedness. Results and obstructions of the deterministic theory are first recalled. Then the difficulties met to prove weak well-posedness by additive noise are discussed, in relation with Girsanov transform and Kolmogorov equations. Finally, the vorticity equation is considered and it is shown that a linearized version of it, under a suitable multipicative noise, has better properties than the deterministic one, in particular the blow-up due to stretching is prevented.