One-dimensional inelastic Boltzmann equation: Stability and uniqueness of self-similar $L^{1}$-profiles for moderately hard potentials

TL;DR

This paper establishes the stability and uniqueness of self-similar $L^{1}$ profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, advancing understanding of long-time behavior in inelastic kinetic models.

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 case.

Findings

Proves stability of $L^{1}$ self-similar profiles under certain potential limits.

Establishes uniqueness of these profiles for small $oldsymbol{\gamma}$ in the collision kernel.

Introduces a perturbation method based on the Maxwell model and linearized operator analysis.

Abstract

We prove the stability of $L^{1}$ self-similar profiles under the hard-to-Maxwell potential limit for the one-dimensional inelastic Boltzmann equation with moderately hard potentials which, in turn, leads to the uniqueness of such profiles for hard potentials collision kernels of the form $|\cdot|^{\gamma}$ with $\gamma >0$ sufficiently small (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 and a careful study of the associated linearized operator recently derived in the companion paper \cite{maxwel}. The results can be seen as a first step towards a complete proof, in the one-dimensional setting, of a conjecture in \cite{ernst} regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation, at least, in a regime of moderately hard potentials.