Comptes Rendus
no. 1
Probability Theory/Partial Differential Equations
Markov selections and their regularity for the three-dimensional stochastic Navier–Stokes equations
[Sélections Markoviennes et leur régularité pour les équations stochastiques de Navier–Stokes tridimensionnelles]

Présenté par : Alain Bensoussan

Franco Flandoli1 ; Marco Romito2
1 Dipartimento di Matematica Applicata, Università di Pisa, via Bonanno Pisano, 25/b, 56126 Pisa, Italy
2 Dipartimento di Matematica, Università di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 47-50.

Il est établi que le problème de martingales associé aux équations de Navier–Stokes tridimensionnelles possède une famille de solutions qui satisfont la propriété de Markov. Ce résultat est obtenu par un principe abstrait de sélection. La propriété de Markov est fondamentale pour étendre la régularité du semi groupe de transition des petites échelles de temps à des échelles arbitraires, en établissant en particulier que chaque sélection de Markov dépend continûment des conditions initiales.

The martingale problem associated to the three-dimensional Navier–Stokes equations is shown to have a family of solutions satisfying the Markov property. The result is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times, thus showing that every Markov selection has a property of continuous dependence on initial conditions.

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DOI : 10.1016/j.crma.2006.04.025
Franco Flandoli 1 ; Marco Romito 2

1 Dipartimento di Matematica Applicata, Università di Pisa, via Bonanno Pisano, 25/b, 56126 Pisa, Italy
2 Dipartimento di Matematica, Università di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy 
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Franco Flandoli; Marco Romito. Markov selections and their regularity for the three-dimensional stochastic Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 47-50. doi : 10.1016/j.crma.2006.04.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.025/

[1] G. Da Prato; A. Debussche Ergodicity for the 3D stochastic Navier–Stokes equations, J. Math. Pures Appl. (9), Volume 82 (2003) no. 8, pp. 877-947

[2] A. Debussche; C. Odasso Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise (available on the arXiv preprint archive at the web address) | arXiv

[3] F. Flandoli; B. Maslowski Ergodicity of the 2-D Navier–Stokes equation under random perturbations, Comm. Math. Phys., Volume 172 (1995) no. 1, pp. 119-141

[4] F. Flandoli; M. Romito Markov selections for the 3D stochastic Navier–Stokes equations (available on the arXiv preprint archive at the web address) | arXiv

[5] N.V. Krylov The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat., Volume 37 (1973), pp. 691-708 (in Russian)

[6] D.W. Stroock; S.R.S. Varadhan Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, vol. 233, Springer-Verlag, Berlin, 1979

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