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Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : We consider the spatially homogeneous Boltzmann equation for {em inelastic hard spheres}, in the framework of socalled {em constant normal restitution coefficients} $alpha in [0,1]$. In the physical regime of a small inelasticity (that is $alpha in [alpha_*,1)$ for some constructive $alpha_*>0$) we prove uniqueness of the selfsimilar profile for given values of the restitution coefficient $alpha in [alpha_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the selfsimilar solution (up to a time translation). Moreover, if the initial datum lies in $L^1_3$, and under some smallness condition on $(1alpha_*)$ depending on the mass, energy and $L^1 _3$ norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the selfsimilar solution (the socalled {em homogeneous cooling state}). These uniqueness, stability and convergence results are expressed in the selfsimilar 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 selfsimilar 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 ``quasielastic selfsimilar temperature' and the rate of convergence towards selfsimilarity at first order in terms of $(1alpha)$, are obtained from our study. These results provide a positive answer and a mathematical proof of the ErnstBrito conjecture [16] in the case of inelastic hard spheres with small inelasticity. 





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