On self-similar solutions to the homogeneous Boltzmann equation

Marco Cannone; Grzegorz Karch

doi:10.3934/krm.2013.6.801

Article Contents

2013, Volume 6, Issue 4: 801-808. Doi: 10.3934/krm.2013.6.801

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On self-similar solutions to the homogeneous Boltzmann equation

Marco Cannone^1, and
Grzegorz Karch^2,

1.
Université Paris-Est-Marne-la-Vallée, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050 CNRS, 5 boulevard Descartes, Cité Descartes Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2
2.
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław

Received: June 30, 2013

Revised: July 31, 2013

Published: November 2013

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Abstract

In this note, our results from [ Comm. Pure Appl. Math. 63 (2010), 747--778] on infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules are discussed, presented in a different context, and improved by using recent observations by Morimoto and Yang. In particular, similarities between the homogeneous Boltzmann equation and the fractional heat equation are emphasized. Moreover, we show that a certain conjecture by Bobylev and Cercignani on regularity of self-similar solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules has a positive answer.

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Mathematics Subject Classification: Primary: 82C40; Secondary: 76P05.

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