Probability Theory/Partial Differential Equations
Malliavin calculus for highly degenerate 2D stochastic Navier–Stokes equations
[Calcul de Malliavin pour les équations de Navier–Stokes 2D stochastiques, hautement dégénérées.]

Présenté par : Paul Malliavin

Hairer, Martin1 ; Mattingly, Jonathan C.2 ; Pardoux, Étienne3
1 Math Department, The University of Warwick, Coventry CV4 7AL, UK
2 Math Department, Duke University, Box 90320, Durham, NC 27708 USA
3 LATP/CMI, université de Provence, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France
Comptes Rendus. Mathématique, Tome 339 (2004) no. 11, pp. 793-796.

Cette Note présente essentiellement les résultats de l'article “Malliavin calculus and the randomly forced Navier–Stokes equation”, de J.C. Mattingly et E. Pardoux. Elle contient aussi un résultat de l'article “Ergodicity of the degenerate stochastic 2D Navier–Stokes equation”, de M. Hairer et J.C. Mattingly. Nous étudions l'équation de Navier–Stokes sur le tore bidimensionel, excitée par un bruit blanc gaussien de dimension finie. Nous donnons des conditions sous lesquelles la loi de la projection sur tout sous-espace de dimension finie de la solution à un instant t>0 arbitraire a une densité régulière par rapport à la mesure de Lebesgue. Nos résultats sont en particulier vrais dans certains cas de bruit blanc gaussien de dimension quatre. Sous des hypothèses supplémentaires, nous montrons que la densité dont il est question ci-dessus est strictement positive partout. Les résultats de cette Note fournissent une part cruciale des arguments utilisés dans le second article cité ci-dessus, pour démontrer l'ergodicité de la solution.

This Note mainly presents the results from “Malliavin calculus and the randomly forced Navier–Stokes equation” by J.C. Mattingly and E. Pardoux. It also contains a result from “Ergodicity of the degenerate stochastic 2D Navier–Stokes equation” 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. This Note's results are a critical component in the ergodic results discussed in a future article.

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Accepté le :
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DOI : 10.1016/j.crma.2004.09.002
Hairer, Martin 1 ; Mattingly, Jonathan C. 2 ; Pardoux, Étienne 3

1 Math Department, The University of Warwick, Coventry CV4 7AL, UK
2 Math Department, Duke University, Box 90320, Durham, NC 27708 USA
3 LATP/CMI, université de Provence, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France 
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Hairer, Martin; Mattingly, Jonathan C.; Pardoux, Étienne. Malliavin calculus for highly degenerate 2D stochastic Navier–Stokes equations. Comptes Rendus. Mathématique, Tome 339 (2004) no. 11, pp. 793-796. doi : 10.1016/j.crma.2004.09.002. https://www.numdam.org/articles/10.1016/j.crma.2004.09.002/

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