zbMATH — the first resource for mathematics

Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D domains. (English) Zbl 1259.35230
Summary: Stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Moreover, some compactness and tightness criteria in nonmetric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.

35R60 PDEs with randomness, stochastic partial differential equations
35Q30 Navier-Stokes equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76M35 Stochastic analysis applied to problems in fluid mechanics
Full Text: DOI