On the convergence of smooth solutions from Boltzmann to Navier–Stokes

Gallagher, Isabelle; Tristani, Isabelle

doi:10.5802/ahl.40

On the convergence of smooth solutions from Boltzmann to Navier–Stokes
[Sur la convergence de solutions régulières de Boltzmann vers Navier–Stokes]

Gallagher, Isabelle ¹ ; Tristani, Isabelle ²

¹ DMA, École normale supérieure, CNRS, PSL University, 75005 Paris, France and UFR de mathématiques, Université Paris-Diderot, 75013 Paris, France
² Département de mathématiques et applications, École normale supérieure, CNRS, PSL University, 45 rue d’Ulm, 75005 Paris, France

Annales Henri Lebesgue, Tome 3 (2020), pp. 561-614.

Résumé
Abstract

On s’intéresse dans ce travail au lien entre les solutions fortes des équations de Boltzmann et de Navier–Stokes. Pour justifier cette relation, notre idée principale est d’utiliser des informations sur le système limite (par exemple le fait que les équations de Navier–Stokes ont une solution globale unique en deux dimensions d’espace, ou quand la donnée initiale est petite). En particulier on démontre que le temps d’existence de la solution de l’équation de Boltzmann remise à l’échelle est toujours plus grand que celui du système de Navier–Stokes. On considère des données initiales générales dans l’espace entier en dimensions 2 et 3, et nous traitons également le cas de données bien préparées dans le cas de conditions aux limites périodiques.

In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier–Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier–Stokes equations are globally wellposed in two space dimensions or when the initial data is small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier–Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions.

Reçu le : 2019-03-05
Révisé le : 2019-09-09
Accepté le : 2019-11-06
Publié le : 2020-07-10

DOI : 10.5802/ahl.40

Classification : 35Q35, 35Q30, 76D05, 82B40, 82C40, 82D05
Mots-clés : équation de Navier–Stokes, équation de Boltzmann

Affiliations des auteurs :

Gallagher, Isabelle ¹ ; Tristani, Isabelle ²

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     title = {On the convergence of smooth solutions from {Boltzmann} to {Navier{\textendash}Stokes}},
     journal = {Annales Henri Lebesgue},
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     year = {2020},
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     url = {http://www.numdam.org/articles/10.5802/ahl.40/}
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Gallagher, Isabelle; Tristani, Isabelle. On the convergence of smooth solutions from Boltzmann to Navier–Stokes. Annales Henri Lebesgue, Tome 3 (2020), pp. 561-614. doi : 10.5802/ahl.40. http://www.numdam.org/articles/10.5802/ahl.40/

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