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

Dolbeault, Jean; Li, Xingyu (2018), φ -Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations, Mathematical Models and Methods in Applied Sciences, 28, 13, p. 2637-2666. 10.1142/S0218202518500574

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01672455
Date
2018
Journal name
Mathematical Models and Methods in Applied Sciences
Volume
28
Number
13
Publisher
World Scientific
Pages
2637-2666
Publication identifier
10.1142/S0218202518500574
Metadata
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Author(s)
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Li, Xingyu
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.
Subjects / Keywords
confinement; spectral gap; Hypocoercivity; linear kinetic equations; Fokker-Planck operator; transport operator; diffusion limit; Poincaré inequality

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