One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials

TL;DR

This paper establishes the uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, extending understanding beyond Maxwell interactions and providing a foundation for analyzing long-term behavior.

Contribution

It provides the first uniqueness result for self-similar profiles in inelastic Boltzmann models with strong inelasticity, using a perturbation approach from the Maxwell model.

Findings

Uniqueness of self-similar profiles for certain inelastic Boltzmann equations.

Extension of trend to equilibrium results in weighted Sobolev spaces.

Perturbation method from Maxwell model to inelastic case.

Abstract

We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form | $\bullet$ | $\gamma$ for $\gamma$ > 0 small enough (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $\gamma$ = 0). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated linearised operator. In particular, a part of the paper is devoted to the trend to equilibrium for the Maxwell model in suitable weighted Sobolev spaces, an extension of results which are known to hold in weaker topologies. Our results can be seen as a first step towards a full proof, in the one-dimensional setting, of a conjecture in Ernst \& Brito (2002) regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation.