Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules

Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G.

doi:10.1016/j.anihpc.2009.11.005

Furioli, G. ; Pulvirenti, A. ; Terraneo, E. ; Toscani, G.

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 719-737.

Résumé

We prove propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for any value of the coefficient of restitution. The result follows from the uniform in time control of the tails of the Fourier transform of the solution, normalized in order to have constant energy. By standard arguments this implies the convergence of the scaled solution towards the stationary state in Sobolev and $L^{1}$ norms in the case of regular initial data as well as the convergence of the original solution to the corresponding self-similar cooling state. In the case of weak inelasticity, similar results have been established by Carlen, Carrillo and Carvalho (2009) in [11] via a precise control of the growth of the Fisher information.

MR Zbl

DOI : 10.1016/j.anihpc.2009.11.005

Mots clés : Granular gases, Kinetic models, Boltzmann equation

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     author = {Furioli, G. and Pulvirenti, A. and Terraneo, E. and Toscani, G.},
     title = {Convergence to self-similarity for the {Boltzmann} equation for strongly inelastic {Maxwell} molecules},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {719--737},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
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}

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Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 719-737. doi : 10.1016/j.anihpc.2009.11.005. http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.005/

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