In this talk I will talk about our recent work on a class of singular SPDEs via convex integration method. In particular, we establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier–Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition in L2 ∪ B∞,∞−1+κ, κ > 0, and implies also non-uniqueness in law. Finally I will show the existence, non-uniqueness, non-Gaussianity and non-unique ergodicity for singular quasi geostrophic equation in the critical and supercritical regime.
This video was produced by the University of Münster, as part of the workshop Stochastic Analysis meets QFT – critical theory.
