We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogeneous cooling state present in the literature implies the strong convergence in Sobolev norms and in the norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.
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Eric A. Carlen, José A. Carrillo, Maria C. Carvalho, Strong convergence towards homogeneous cooling states for dissipative Maxwell models. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1675–1700DOI 10.1016/J.ANIHPC.2008.10.005