In this talk I will talk about our recent work on the three dimensional stochastic Navier-Stokes equations via convex integration method.
First we establish non-uniqueness in law, existence and non-uniqueness of probabilistically strong solutions and non-uniqueness of the associated Markov processes. Second we prove existence of infinitely many stationary solutions as well as ergodic stationary solutions to the stochastic Navier-Stokes and Euler equations. Moreover, we are able to make conclusions regarding the vanishing viscosity limit and the anomalous dissipation.
Finally I will show 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, the convective term is ill-defined in the classical sense and probabilistic renormalization is required.