Umeå universitets logga

Abstract.: We prove the stability of L1 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 potential collision kernels of the form (Formula presented.) with γ > 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 γ = 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 [1]. The results can be seen as a first step towards a complete proof, in the one-dimensional setting, of a conjecture in [2] regarding the determination of the long-time behavior of solutions to the inelastic Boltzmann equation, at least in a regime of moderately hard potentials.