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We consider the spatially homogeneous Boltzmann equation for {\em inelastichard spheres}, in the framework of so-called {\em constant normal restitutioncoefficients} $\alpha \in [0,1]$. In the physical regime of a smallinelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive$\alpha_*>0$) we prove uniqueness of the self-similar profile for given valuesof the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and themomentum; therefore we deduce the uniqueness of the self-similar solution (upto a time translation). Moreover, if the initial datum lies in $L^1_3$, andunder some smallness condition on $(1-\alpha_*)$ depending on the mass, energyand $L^1_3$ norm of this initial datum, we prove time asymptotic convergence(with polynomial rate) of the solution towards the self-similar solution (theso-called {\em homogeneous cooling state}). These uniqueness, stability andconvergence results are expressed in the self-similar variables and thentranslate into corresponding results for the original Boltzmann equation. Theproofs are based on the identification of a suitable elastic limit rescaling,and the construction of a smooth path of self-similar profiles connecting to aparticular Maxwellian equilibrium in the elastic limit, together with toolsfrom perturbative theory of linear operators. Some universal quantities, suchas the "quasi-elastic self-similar temperature" and the rate of convergencetowards self-similarity at first order in terms of $(1-\alpha)$, are obtainedfrom our study. These results provide a positive answer and a mathematicalproof of the Ernst-Brito conjecture [16] in the case of inelastic hard sphereswith small inelasticity.Comment: 73 page

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