Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres
type: Item
instance of: scholarly article
title: Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres
zbMATH Open document ID: 1178.82056
author: Stéphane Mischler
author: Clément Mouhot
published in: Communications in Mathematical Physics
publication date: 2009-09-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0701449
review text: The spatially homogeneous Boltzmann equation is considered for hard spheres undergoing inelastic collisions with a constant normal restitution coefficient in the range [0,1]. One needs to know rudiments of a physical theory of granular gases and a mathematical introduction to this theory, set by the present authors [part I with \textit{M. Rodriguez Ricard}, J. Stat. Phys. 124, No. 2--4, 655--702 (2006; Zbl 1135.82325), part II with \textit{C. Mouhot}, ibid., 703--746 (2006; Zbl 1135.82030)]. A brief review is given of why simplified Boltzmann models for inelastic Maxwell molecules or pseudo inealstic hard spheres do not capture some crucial physical features of the cooling process of granular gases. Basic mathematical works on spatially homogeneous inelastic hard spheres Boltzmann models are mentioned as a departure point of the analysis. In the present paper, the self-similarity properties of the Boltzmann equation for inelastic hard spheres are studied. The uniqueness and attractivity of self-similar solutions are established, thus giving a complete answer to the Ernst-Brito conjecture [\textit{M. H. Ernst} and \textit{R. Brito}, J. Stat. Phys. 109, No. 3--4, 407--432 (2002; Zbl 1015.82030)] for inelastic hard spheres with small inelasticity. The proofs are based on adopting a suitable elastic limit rescaling and the construction of a smooth path of self-similar profiles. Suitable tools from perturbative theory of linear operators are employed as well.
reviewed by: Piotr Garbaczewski
Mathematics Subject Classification ID: 82C22
Mathematics Subject Classification ID: 76T25
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 82C70
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 35F25
zbMATH DE Number: 5607340
zbMATH Keywords: homogeneous Boltzmann equation
zbMATH Keywords: granular gas
zbMATH Keywords: inelastic hard spheres gas
zbMATH Keywords: binary collisions
zbMATH Keywords: self-similar solutions
zbMATH Keywords: uniqueness
zbMATH Keywords: attractivity
zbMATH Keywords: elastic limit
zbMATH Keywords: Ernst-Brito conjecture
zbMATH Keywords: a posteriori estimates
zbMATH Keywords: self-similar profiles
zbMATH Keywords: convergence
zbMATH Keywords: semigroup operator
zbMATH Keywords: resolvent
zbMATH Keywords: spectral stability
MaRDI profile type: Publication
OpenAlex ID: W1973750289
arXiv ID: math/0701449
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DOI: 10.1007/S00220-009-0773-9
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