Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off
Hérau, Frédéric; Tonon, Daniela; Tristani, Isabelle (2019-07), Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off. https://basepub.dauphine.fr/handle/123456789/20151
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01599973
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
Laboratoire de Mathématiques Jean Leray [LMJL]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Département de Mathématiques et Applications - ENS Paris [DMA]
Abstract (EN)In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant framework for this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.
Subjects / Keywordsdissipativity; exponential rate of convergence; Boltzmann equation without cut-off; hard potentials; Cauchy theory; Spectral gap; long-time asymptotic; regularization estimates; pseudodifferential theory
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Cauchy theory and exponential stability for inhomogeneous boltzmann equation for hard potentials without cut-off Hérau, Frédéric; Tonon, Daniela; Tristani, Isabelle (2017) Document de travail / Working paper