Control and mixing for 2D Navier-Stokes equations with space-time localised noise
[Contrôle et mélange pour des équations de Navier-Stokes 2D avec un bruit localisé en espace-temps]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 2, pp. 253-280.

Nous considérons une perturbation aléatoire du système de Navier-Stokes 2D dans un domaine borné à bord régulier. On suppose que la force aléatoire est non dégénérée et que sa loi est périodique en temps et a un support localisé en espace et en temps. En ce qui concerne le problème non perturbé, on suppose qu'il est approximativement contrôlable en temps infini par une force extérieure dont le support est inclus dans celui de la force aléatoire. Sous ces hypothèses, on montre que le processus de Markov engendré par la restriction des solutions aux instants de temps proportionnels à la période possède une unique distribution stationnaire, qui est exponentiellement mélangeante. La démonstration est basée sur un argument de couplage, une propriété de contrôlabilité locale pour le système de Navier-Stokes, une estimation pour la distance en variation totale entre une mesure et son image par une application lisse et quelques résultats classiques de la théorie du transport optimal.

We consider randomly forced 2D Navier-Stokes equations in a bounded domain with smooth boundary. It is assumed that the random perturbation is non-degenerate, and its law is periodic in time and has a support localised with respect to space and time. Concerning the unperturbed problem, we assume that it is approximately controllable in infinite time by an external force whose support is included in that of the random force. Under these hypotheses, we prove that the Markov process generated by the restriction of solutions to the instants of time proportional to the period possesses a unique stationary distribution, which is exponentially mixing. The proof is based on a coupling argument, a local controllability property of the Navier-Stokes system, an estimate for the total variation distance between a measure and its image under a smooth mapping, and some classical results from the theory of optimal transport.

Publié le :
DOI : 10.24033/asens.2244
Classification : 35Q30, 60H15, 60J05, 93B05, 93C20.
Keywords: Navier-Stokes equations, stationary measures, exponential mixing.
Mot clés : Équations de Navier-Stokes, mesures stationnaires, mélange exponentiel. 
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     author = {Shirikyan, Armen},
     title = {Control and mixing  for {2D} {Navier-Stokes} equations  with space-time localised noise},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {253--280},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
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Shirikyan, Armen. Control and mixing  for 2D Navier-Stokes equations  with space-time localised noise. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 2, pp. 253-280. doi : 10.24033/asens.2244. http://www.numdam.org/articles/10.24033/asens.2244/

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