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

Analysis of PDEs;Functional Analysis;Probability