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 \(\mathbb {B}^{d/q-1}_{q,p}\) ${\mathbb{B}}_{q,p}^{d/q-1}$ for \(q\in [2,2d)\) $q\in [2,2d)$ 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 \(d\geqslant 2\) $d⩾2$ . Moreover, for \(d=2\) $d=2$ , we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.