Stability, convergence to selfsimilarity and elastic limit for the Boltzmann equation for inelastic hard spheres
Mouhot, Clément; Mischler, Stéphane (2009), Stability, convergence to selfsimilarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Communications in Mathematical Physics, 288, 2, p. 431502. http://dx.doi.org/10.1007/s0022000907739
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Article accepté pour publication ou publiéExternal document link
http://hal.archivesouvertes.fr/hal00124876/en/Date
2009Journal name
Communications in Mathematical PhysicsVolume
288Number
2Publisher
Springer
Pages
431502
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Show full item recordAbstract (EN)
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of socalled 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 selfsimilar profile for given values of the restitution coefficient αε [α*,1)} , the mass and the momentum; therefore we deduce the uniqueness of the selfsimilar solution (up to a time translation). Moreover, if the initial datum lies in {L¹_3} , and under some smallness condition on (1α*)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 selfsimilar solution (the socalled homogeneous cooling state). These uniqueness, stability and convergence results are expressed in the selfsimilar 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 selfsimilar 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 “quasielastic selfsimilar temperature” and the rate of convergence towards selfsimilarity 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 ErnstBrito conjecture [16] in the case of inelastic hard spheres with small inelasticity.Subjects / Keywords
Spectrum; Degenerated perturbation; Elastic limit; Small inelasticity; Stability; Uniqueness; Selfsimilar profile; Selfsimilar solution; Hard spheres; Inelastic Boltzmann equation; Granular gasesRelated items
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