Decay Rates in Probability Metrics Towards Homogeneous Cooling States for the Inelastic Maxwell Model |
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Authors: | M Bisi J A Carrillo G Toscani |
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Institution: | (1) Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53/A, I-43100 Parma, Italy;(2) ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain;(3) Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy |
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Abstract: | We quantify the long-time behavior of solutions to the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures. Existence, uniqueness, and precise estimates of overpopulated high energy tails of the self-similar profile proved in ref. 9 are revisited and derived from this new Liapunov functional. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as t → ∞ to the self-similar solution in this distance, which metrizes the weak convergence of measures. Moreover, we can relate this Fourier distance to the Euclidean Wasserstein distance or Tanaka functional proving also its exponential convergence towards the homogeneous cooling states. The findings are relevant in the understanding of the conjecture formulated by Ernst and Brito in refs. 15, 16, and complement and improve recent studies on the same problem of Bobylev and Cercignani(9) and Bobylev, Cercignani and one of the authors.(11) |
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Keywords: | inelastic collisions Maxwell models asymptotic behavior Fourier metrics Ernst-Brito conjecture |
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