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 gradient noise, which arises in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case B d/q−1 q,p 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.

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