Phi-entropies for Fokker-Planck and kinetic Fokker-Planck equations

Abstract : This 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.
Type de document :
Article dans une revue
Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2018
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01672455
Contributeur : Jean Dolbeault <>
Soumis le : vendredi 27 juillet 2018 - 06:26:28
Dernière modification le : mardi 27 novembre 2018 - 06:56:46

Fichiers

DL2017-rev.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01672455, version 2
  • ARXIV : 1712.09897

Collections

Citation

Jean Dolbeault, Xingyu Li. Phi-entropies for Fokker-Planck and kinetic Fokker-Planck equations. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2018. 〈hal-01672455v2〉

Partager

Métriques

Consultations de la notice

76

Téléchargements de fichiers

52