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Stochastic nonhomogeneous incompressible Navier-Stokes equations

Cutland, N.J. and Enright, B. (2006) Stochastic nonhomogeneous incompressible Navier-Stokes equations. Journal of Differential Equations, 228 (1). pp. 140-170. ISSN 0022-0396

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Abstract

We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier–Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331–332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240–1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier–Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992].

The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier–Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier–Stokes equations, Applicandae Math. 25 (1991) 59–85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 23 Apr 2009 15:54
Last Modified: 23 Apr 2009 15:54
Published Version: http://dx.doi.org/10.1016/j.jde.2006.04.009
Status: Published
Publisher: Elsevier Science B.V.
Identification Number: 10.1016/j.jde.2006.04.009
URI: http://eprints.whiterose.ac.uk/id/eprint/6560

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