Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces

In this paper we study the stochastic Navier–Stokes equations on the d-dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case B_q,p^d/q-1 for q∈[2,2d) and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2. Moreover, for d=2, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.