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

Article type: Research Article

Authors: Desvillettes, Laurent | Mouhot, Clément

Affiliations: CMLA, École Normale Supérieure de Cachan, 61, avenue du Président Wilson, 94235 Cachan Cedex, France. E-mail: desville@cmla.ens-cachan.fr | CNRS and CEREMADE, Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny 75775, Paris Cedex 16, France. E-mail: clement.mouhot@ceremade.dauphine.fr

Abstract: We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in L1((1+|v|s) dv) and Hk norms, s, k≥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 L1 obtained by the so-called “entropy–entropy production” method in the context of dissipative systems with slowly growing a priori bounds (G. Toscani and C. Villani, J. Statist. Phys. 98 (2000), 1279–1309).

Keywords: Boltzmann equation, spatially homogeneous, soft potentials, moment bounds, regularity bounds, uniform in time

Journal: Asymptotic Analysis, vol. 54, no. 3-4, pp. 235-245, 2007