Abstract
This paper concerns the Cauchy problem in Rd for the stochastic Navier–Stokes equation $$∂_t\mathbf u=Δ\mathbf u−(\mathbf u,∇)\mathbf u−∇p+\mathbf f(\mathbf u)+[(σ,∇)\mathbf u−∇p̃+\mathbf g(\mathbf u)]○\dot W,\mathbf u(0)=\mathbf u_0, \mathrm{div} \mathbf u=0,$$ driven by white noise Ẇ. 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.
Citation
R. Mikulevicius. B. L. Rozovskii. "Global L2-solutions of stochastic Navier–Stokes equations." Ann. Probab. 33 (1) 137 - 176, January 2005. https://doi.org/10.1214/009117904000000630
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