Global L_2-solutions of stochastic Navier-Stokes equations

Details Global L_2-solutions of stochastic Navier-Stokes equations Global L_2-solutions of stochastic Navier-Stokes equations

2005-03-25

arxiv.org/abs/math/0503597v1

Annals of Probability 2005, Vol. 33, No. 1, 137-176

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000630 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000630

This paper concerns the Cauchy problem in R^d for the stochastic Navier-Stokes equation \partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+ [(\sigma,\nabla)u-\nabla \tilde p+g(u)]\circ \dot W, u(0)=u_0,\qquad divu=0, driven by white noise \dot 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.

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