From Poincaré to logarithmic Sobolev inequalities: a gradient flow approach

Savaré, Giuseppe; Nazaret, Bruno; Dolbeault, Jean

Savaré, Giuseppe; Nazaret, Bruno; Dolbeault, Jean (2012), From Poincaré to logarithmic Sobolev inequalities: a gradient flow approach, SIAM Journal on Mathematical Analysis, 44, 5, p. 3186-3216. http://dx.doi.org/10.1137/110835190

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00595042/fr/

Date

2012

Journal name

SIAM Journal on Mathematical Analysis

Volume

Number

Publisher

SIAM

Pages

3186-3216

Abstract (EN)

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogorov-Fokker-Planck (KFP) equation.

Subjects / Keywords

Kolmogorov-Fokker-Planck equation; Gradient flows; Action functional; Continuity equation; Generalized Poincaré inequality; Kantorovich-Rubinstein-Wasserstein distance; Optimal transport