The Annals of Probability

Global L2-solutions of stochastic Navier–Stokes equations

R. Mikulevicius and B. L. Rozovskii

Full-text: Open access

Abstract

This paper concerns the Cauchy problem in Rd for the stochastic Navier–Stokes equation

tuu−(u,∇)u−∇p+f(u)+[(σ,∇)u−∇+g(u)]○,u(0)=u0,  div 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.

Article information

Source
Ann. Probab., Volume 33, Number 1 (2005), 137-176.

Dates
First available in Project Euclid: 11 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1108141723

Digital Object Identifier
doi:10.1214/009117904000000630

Mathematical Reviews number (MathSciNet)
MR2118862

Zentralblatt MATH identifier
1098.60062

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 76M35: Stochastic analysis

Keywords
Stochastic Navier–Stokes Leray solution Kraichnan’s turbulence Wiener chaos strong solutions pathwise uniqueness

Citation

Mikulevicius, R.; Rozovskii, B. L. Global L 2 -solutions of stochastic Navier–Stokes equations. Ann. Probab. 33 (2005), no. 1, 137--176. doi:10.1214/009117904000000630. https://projecteuclid.org/euclid.aop/1108141723


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