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Malliavin calculus for highly degenerate 2D stochastic Navier-Stokes equations. (English) Zbl 1059.60074

Summary: This note mainly presents the results from “Malliavin calculus and the randomly forced Navier-Stokes equation” (to appear) by J. C. Mattingly and E. Pardoux. It also contains a result from “Ergodicity of the degenerate stochastic 2D Navier-Stokes equation” (to appear) by M. Hairer and J. C. Mattingly. We study the Navier-Stokes equation on the two-dimensional torus when forced by a finite-dimensional Gaussian white noise. We give conditions under which the law of the solution at any time \(t>0\), projected on a finite-dimensional subspace, has a smooth density with respect to Lebesgue measure. In particular, our results hold for specific choices of four-dimensional Gaussian white noise. Under additional assumptions, we show that the preceding density is everywhere strictly positive. The results are a critical component in the ergodic results discussed in the paper announced above [M. Hairer and J. C. Mattingly, C. R., Math., Acad. Sci. Paris 339, No. 12, 879–882 (2004; Zbl 1059.60073)].

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
35Q30 Navier-Stokes equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76D05 Navier-Stokes equations for incompressible viscous fluids
76M35 Stochastic analysis applied to problems in fluid mechanics

Citations:

Zbl 1059.60073
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References:

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