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January 2005 Global L2-solutions of stochastic Navier–Stokes equations
R. Mikulevicius, B. L. Rozovskii
Ann. Probab. 33(1): 137-176 (January 2005). DOI: 10.1214/009117904000000630

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 . 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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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

Information

Published: January 2005
First available in Project Euclid: 11 February 2005

zbMATH: 1098.60062
MathSciNet: MR2118862
Digital Object Identifier: 10.1214/009117904000000630

Subjects:
Primary: 35R60 , 60H15 , 76M35

Keywords: Kraichnan’s turbulence , Leray solution , Navier–Stokes , Pathwise uniqueness , Stochastic , strong solutions , Wiener Chaos

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 1 • January 2005
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