Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials.

Desvillettes, Laurent; Mouhot, Clément

Desvillettes, Laurent; Mouhot, Clément (2007), Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials., Asymptotic Analysis, 54, 3-4, p. 235-245

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00079949/en/

Date

2007

Nom de la revue

Asymptotic Analysis

Volume

Numéro

3-4

Éditeur

Amsterdam : IOS Press

Pages

235-245

Métadonnées

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Auteur(s)

Desvillettes, Laurent
Mouhot, Clément

Résumé (EN)

We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in $L^1 ((1 + |v|^s)dv)$ and $H^k$ norms, $s, k \ge 0$ hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in $L^1$ obtained by the so-called “entropy-entropy production” method in the context of dissipative systems with slowly growing a priori bounds (see reference [14]).

Mots-clés

uniform in time; regularity bounds; mo- ment bounds; soft potentials; spatially homogeneous; Boltzmann equation