Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres

Abstract

We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$. In the physical regime of a small inelasticity (that is $\alpha \in [{\alpha }_{\ast },1)$ for some constructive ${\alpha }_{\ast }>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $\alpha \in [{\alpha }_{\ast },1)$, 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 $L_3^1$, and under some smallness condition on $(1-{\alpha }_{\ast })$ depending on the mass, energy and $L_3^1$ norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called {\em 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 $(1-\alpha )$, 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.

73 pages

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