Hairer, Martin; Mattingly, Jonathan C.; Pardoux, Étienne Malliavin calculus for highly degenerate 2D stochastic Navier-Stokes equations. (English) Zbl 1059.60074 C. R., Math., Acad. Sci. Paris 339, No. 11, 793-796 (2004). 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)]. Cited in 11 Documents 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 PDFBibTeX XMLCite \textit{M. Hairer} et al., C. R., Math., Acad. Sci. Paris 339, No. 11, 793--796 (2004; Zbl 1059.60074) Full Text: DOI arXiv References: [1] Aida, S.; Kusuoka, S.; Stroock, D., On the support of Wiener functionals, (Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990). Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser., vol. 284 (1993), Longman Sci. Tech.: Longman Sci. Tech. Harlow), 3-34 · Zbl 0790.60047 [2] Ben Arous, G.; Léandre, R., Décroissance exponentielle du noyau de la chaleur sur la diagonale. II, Probab. Theory Related Fields, 90, 3, 377-402 (1991) · Zbl 0734.60027 [3] Da Prato, G.; Zabczyk, J., Ergodicity for Infinite Dimensional Systems (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0849.60052 [4] Weinan, E.; Mattingly, J. C., Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation, Commun. Pure Appl. Math., 54, 11, 1386-1402 (2001) · Zbl 1024.76012 [5] Eckmann, J.-P.; Hairer, M., Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, Commun. Math. Phys., 219, 3, 523-565 (2001) · Zbl 0983.60058 [6] Flandoli, F.; Maslowski, B., Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171, 119-141 (1995) · Zbl 0845.35080 [7] M. Hairer, J.C. Mattingly, Ergodicity of the degenerate stochastic 2D Navier-Stokes equation, June 2004, submitted for publication; M. Hairer, J.C. Mattingly, Ergodicity of the degenerate stochastic 2D Navier-Stokes equation, June 2004, submitted for publication · Zbl 1059.60073 [8] M. Hairer, J.C. Mattingly, Ergodic properties of highly degenerate 2D Navier-Stokes equation, C. R. Acad. Sci. Paris, Ser. I, in press; M. Hairer, J.C. Mattingly, Ergodic properties of highly degenerate 2D Navier-Stokes equation, C. R. Acad. Sci. Paris, Ser. I, in press · Zbl 1059.60073 [9] Hörmander, L., The Analysis of Linear Partial Differential Operators I-IV (1985), Springer: Springer New York [10] Majda, A. J.; Bertozzi, A. L., Vorticity and Incompressible Flow, Cambridge Texts in Appl. Math., vol. 27 (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.76001 [11] J.C. Mattingly, É. Pardoux, Malliavin calculus and the randomly forced Navier Stokes equation, June, 2004, submitted for publication; J.C. Mattingly, É. Pardoux, Malliavin calculus and the randomly forced Navier Stokes equation, June, 2004, submitted for publication · Zbl 1059.60074 [12] Ocone, D., Stochastic calculus of variations for stochastic partial differential equations, J. Funct. Anal., 79, 2, 288-331 (1988) · Zbl 0653.60046 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.