Martingale and stationary solutions for stochastic Navier-Stokes equations. (English) Zbl 0831.60072

Consider the stochastic Navier-Stokes equation in \(D\subset \mathbb{R}^d\), \[ \begin{split} {\partial u(t, x)\over \partial t}- \Delta u(t, x)+ (u(t, x)\cdot \nabla) u(t, x)\\ =- \nabla p(t, x)+ f(t, x)+ G(u, \xi)(t, x),\qquad t\in [0, T],\quad x\in D,\end{split} \] with the incompressibility, boundary and initial conditions. Here \(\xi(t, x)\) is a Gaussian random field, white noise in time, and \(G\) is an operator acting on noise and solution \(u\). The authors investigate three distinct cases of \(G\) such as (i) regular diffusion coefficient, (ii) coercive diffusion coefficient, (iii) cylindrical noise; and prove the existence of martingale solutions and of stationary solutions of the corresponding abstract stochastic evolution equation under different assumptions on \(G\). The proofs are due to a new method of compactness. The obtained result extends results of Z. Brzeźniak, M. Capiński and the first author [Math. Models Methods Appl. Sci. 1, No. 1, 41-59 (1991; Zbl 0741.60058), Stochastic Anal. Appl. 10, No. 5, 523-532 (1992; Zbl 0762.35083)] and M. Capiński and N. J. Cutland [Nonlinearity 6, No. 1, 71-78 (1993; Zbl 0765.76018)]. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well-defined. The related idea was presented by A. B. Cruzeiro [Expo. Math. 7, No. 1, 73-82 (1989; Zbl 0665.60066)].
Reviewer: I.Dôku


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
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