Global L₂-solutions of stochastic Navier-Stokes equations

2005/03/25 by R. Mikulevicius, B. L. Rozovskii
Mathematics · #math.PR #msc:60H15 #msc:35R60 #msc:76M35

paper · pdf · doi:10.1214/009117904000000630

published as Annals of Probability 2005, Vol. 33, No. 1, 137-176 · Published at http://dx.doi.org/10.1214/009117904000000630 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2005/03/25 · arxiv updated 2009/12/01

Abstract

This paper concerns the Cauchy problem in R^d for the stochastic Navier-Stokes equation ∂_tu=Δu-(u,∇)u-∇ p+f(u)+ [(σ,∇)u-∇ p+g(u)]∘ W, u(0)=u₀, divu=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.

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