Global L2-solutions of stochastic Navier–Stokes equations

Abstract

This paper concerns the Cauchy problem in Rd for the stochastic Navier–Stokes equation $$∂_t\mathbf u=Δ\mathbf u−(\mathbf u,∇)\mathbf u−∇p+\mathbf f(\mathbf u)+[(σ,∇)\mathbf u−∇p+\mathbf g(\mathbf u)]○\dot W,\mathbf u(0)=\mathbf u_0, \mathrm{div} \mathbf u=0,$$ driven by white noise W. Under minimal assumptions on regularity of the coefficients and random forces, the existence of a global weak (martingale) solution of the stochastic Navier–Stokes equation is proved. In the two-dimensional case, the existence and pathwise uniqueness of a global strong solution is shown. A Wiener chaos-based criterion for the existence and uniqueness of a strong global solution of the Navier–Stokes equations is established.