Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the Torus
Evans, Josephine (2019-09), Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the Torus. https://basepub.dauphine.fr/handle/123456789/20113
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-02285252
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)This paper studies convergence to equilibrium for the spatially inhomogeneous linear relaxation Boltzmann equation in Boltzmann entropy and related entropy functionals, the p-entropies. Villani proved in  entropic hypocoercivity for a class of PDEs in a Hörmander sum of squares form. It was an open question to prove such a result for an operator which does not share this form. We prove a closed entropy-entropy production inequalityà la Villani which implies exponentially fast convergence to equilibrium for the linear Boltzmann equation with a quantitative rate. The key new idea appearing in our proof is the use of a total derivative of the entropy of a projection of our solution to compensate for an error term which appears when using non-linear entropies. We also extend the proofs for hypocoercivity for the linear relaxation Boltzmann to the case of Φ-entropy functionals.
Subjects / KeywordsConvergence to equilibrium; Hypocoercivity; Linear Boltzmann Equation; φ-entropy; Logarithmic Sobolev inequality; Beckner Inequality
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