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φ -Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorDolbeault, Jean
HAL ID: 87
ORCID: 0000-0003-4234-2298		*
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorLi, Xingyu*
dc.date.accessioned2018-03-19T11:17:23Z
dc.date.available2018-03-19T11:17:23Z
dc.date.issued2018
dc.identifier.issn0218-2025
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/17562
dc.language.isoenen
dc.subjectconfinement
dc.subjectspectral gap
dc.subjectHypocoercivity
dc.subjectlinear kinetic equations
dc.subjectFokker-Planck operator
dc.subjecttransport operator
dc.subjectdiffusion limit
dc.subjectPoincaré inequality
dc.subject.ddc515en
dc.titleφ -Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThis paper is devoted to ϕ-entropies applied to Fokker-Planck and kinetic Fokker-Planck equations in the whole space, with confinement. The so-called ϕ-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and L2 estimates. We review some of their properties in the case of diffusion equations of Fokker-Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker-Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operator plays an essential role in the relaxation process. Here we adopt the H1 point of view and establish a sharp decay rate. Rather than giving general but quantitatively vague estimates, our goal here is to consider simple cases, benchmark available methods and obtain sharp estimates on a key example. Some ϕ-entropies give rise to improved entropy – entropy production inequalities and, as a consequence, to faster decay rates for entropy estimates of solutions to non-degenerate diffusion equations. Our main result is to prove that faster entropy decay also holds at kinetic level and that optimal decay rates are achieved only in asymptotic regimes.
dc.relation.isversionofjnlnameMathematical Models and Methods in Applied Sciences
dc.relation.isversionofjnlvol28
dc.relation.isversionofjnlissue13
dc.relation.isversionofjnldate2018
dc.relation.isversionofjnlpages2637-2666
dc.relation.isversionofdoi10.1142/S0218202518500574
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-01672455
dc.relation.isversionofjnlpublisherWorld Scientific
dc.subject.ddclabelAnalyseen
dc.description.ssrncandidatenon
dc.description.halcandidatenon
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2020-04-14T10:09:34Z
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