| dc.contributor.author | Mouhot, Clément | |
| dc.contributor.author | Mischler, Stéphane | |
| dc.date.accessioned | 2010-02-09T12:50:36Z | |
| dc.date.available | 2010-02-09T12:50:36Z | |
| dc.date.issued | 2009 | |
| dc.identifier.uri | https://basepub.dauphine.fr/handle/123456789/3355 | |
| dc.language.iso | en | en |
| dc.subject | Spectrum | en |
| dc.subject | Degenerated perturbation | en |
| dc.subject | Elastic limit | en |
| dc.subject | Small inelasticity | en |
| dc.subject | Stability | en |
| dc.subject | Uniqueness | en |
| dc.subject | Self-similar profile | en |
| dc.subject | Self-similar solution | en |
| dc.subject | Hard spheres | en |
| dc.subject | Inelastic Boltzmann equation | en |
| dc.subject | Granular gases | en |
| dc.subject.ddc | 519 | en |
| dc.title | Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres | en |
| dc.type | Article accepté pour publication ou publié | |
| dc.description.abstracten | We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients αε[0,1]. In the physical regime of a small inelasticity (that is αε [α*,1) for some constructive α*ε[0,1)} we prove uniqueness of the self-similar profile for given values of the restitution coefficient αε [α*,1)} , the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation).
Moreover, if the initial datum lies in {L¹_3} , and under some smallness condition on (1-α*)depending on the mass, energy and {L¹ _3} norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called homogeneous cooling state).
These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the “quasi-elastic self-similar temperature” and the rate of convergence towards self-similarity at first order in terms of (1−α), are obtained from our study.
These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity. | en |
| dc.relation.isversionofjnlname | Communications in Mathematical Physics | |
| dc.relation.isversionofjnlvol | 288 | en |
| dc.relation.isversionofjnlissue | 2 | en |
| dc.relation.isversionofjnldate | 2009-06 | |
| dc.relation.isversionofjnlpages | 431-502 | en |
| dc.relation.isversionofdoi | http://dx.doi.org/10.1007/s00220-009-0773-9 | en |
| dc.identifier.urlsite | http://hal.archives-ouvertes.fr/hal-00124876/en/ | en |
| dc.description.sponsorshipprivate | oui | en |
| dc.relation.isversionofjnlpublisher | Springer | en |
| dc.subject.ddclabel | Probabilités et mathématiques appliquées | en |
