Stability,Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients . In the physical regime of a small inelasticity (that is for some constructive ) we prove uniqueness of the self-similar profile for given values of the restitution coefficient , the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in , and under some smallness condition on depending on the mass, energy and norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called homogeneous cooling state). These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the “quasi-elastic self-similar temperature” and the rate of convergence towards self-similarity at first order in terms of (1−α), are obtained from our study. These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.