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Malliavin calculus for the stochastic 2D Navier-Stokes equation. (English) Zbl 1113.60058
The authors consider a two-dimensional incompressible fluid satisfying the stochastic Navier-Stokes equation \[ {\partial w\over\partial t}(t,x)+B(w,w)(t,x)=\nu\Delta w(t,x)+{\partial W\over\partial t}(t,x) \] on the two-dimensional torus. Here \(\nu\) is the viscosity constant, \(B(w,w)\) is some nonlinear functional, and \(W(t,x)=\sum W_k(t)e_k(x)\) is some force which can be written in terms of a finite number of standard one-dimensional Brownian motions \(W_k(t)\). It is proved that the law of any finite-dimensional projection of \(w(t,.)\) has a smooth strictly positive density with respect to the Lebesgue measure. Notice that the stochastic force is finite-dimensional, but is propagated to all the components by means of the nonlinearity of the equation. The main tool is Malliavin’s calculus, and the study of the Malliavin covariance matrix uses some technical results about stochastic partial differential equations. The results of this paper are an essential tool for studying the ergodicity of the equation.

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76D05 Navier-Stokes equations for incompressible viscous fluids
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