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
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.
|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|
|Publisher:||Elsevier Science B.V.|