Stochastic Navier-Stokes equations for turbulent flows in critical spaces

Abstract

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

53 pages. Accepted for publication in CIMP

References in corpus (4)

• Global L_2-solutions of stochastic Navier-Stokes equations
• Critical spaces for quasilinear parabolic evolution equations and applications
• Eddy heat exchange at the boundary under white noise turbulence
• Nonlinear parabolic stochastic evolution equations in critical spaces Part II. Blow-up criteria and instantaneous regularization