Strong Convergence towards self-similarity for one-dimensional dissipative Maxwell models

TL;DR

This paper proves that solutions to one-dimensional dissipative Maxwell models become increasingly regular and converge strongly to a stationary state over time, regardless of initial regularity, extending previous results to inelastic Boltzmann equations.

Contribution

It establishes uniform propagation of regularity and strong convergence towards equilibrium for scaled solutions of dissipative Maxwell models, including inelastic Boltzmann equations, independent of inelasticity degree.

Findings

Strong convergence in Sobolev and L^1 norms towards stationary state

Propagation of regularity uniformly in time

Results hold regardless of inelasticity degree

Abstract

We prove the propagation of regularity, uniformly in time, for the scaled solutions of one-dimensional dissipative Maxwell models. This result together with the weak convergence towards the stationary state proven by Pareschi and Toscani in 2006 implies the strong convergence in Sobolev norms and in the L^1 norm towards it depending on the regularity of the initial data. In the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. This generalizes a recent result of Carlen, Carrillo and Carvalho (arXiv:0805.1051v1), in which, for weak inelasticity, propagation of regularity for the scaled inelastic Boltzmann equation was found by means of a precise control of the growth of the Fisher information.