Comptes Rendus
Functional Analysis
The optimal evolution of the free energy of interacting gases and its applications
[L'évolution de l'énergie totale d'un gaz le long d'un transport optimal et applications]
Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 173-178.

Nous établissons une inégalité reliant l'énergie totale – interne, potentielle et interactive – de deux densités de probabilité, leur distance de Wasserstein, leurs barycentres ainsi que leur entropie relative généralisée. Cette inégalité implique plusieurs des inégalités géométriques classiques, ainsi qu'une correspondence remarquable entre les solutions de certaines équations quasilinéaires (ou semi-linéaires) et les solutions stationnaires d'équations du type Fokker–Planck. On établit aussi des inégalités HWBI – généralisant les inégalités HWI de Otto et Villani [J. Funct. Anal. 173 (2) (2000) 361–400] et de Carrillo et al. [Rev. Math. Iberoamericana (2003)], où le « B » refère au nouveau terme barycentrique – dont découlent plusieurs inégalités gaussiennes classiques.

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00289-9

Martial Agueh 1 ; Nassif Ghoussoub 1 ; Xiaosong Kang 1

1 Pacific Institute for the Mathematical Sciences and Department of Mathematics, The University of British Columbia, Vancouver, BC V6T 1Z2, Canada
@article{CRMATH_2003__337_3_173_0,
     author = {Martial Agueh and Nassif Ghoussoub and Xiaosong Kang},
     title = {The optimal evolution of the free energy of interacting gases and its applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {173--178},
     publisher = {Elsevier},
     volume = {337},
     number = {3},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00289-9},
     language = {en},
}
TY  - JOUR
AU  - Martial Agueh
AU  - Nassif Ghoussoub
AU  - Xiaosong Kang
TI  - The optimal evolution of the free energy of interacting gases and its applications
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 173
EP  - 178
VL  - 337
IS  - 3
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00289-9
LA  - en
ID  - CRMATH_2003__337_3_173_0
ER  - 
%0 Journal Article
%A Martial Agueh
%A Nassif Ghoussoub
%A Xiaosong Kang
%T The optimal evolution of the free energy of interacting gases and its applications
%J Comptes Rendus. Mathématique
%D 2003
%P 173-178
%V 337
%N 3
%I Elsevier
%R 10.1016/S1631-073X(03)00289-9
%G en
%F CRMATH_2003__337_3_173_0
Martial Agueh; Nassif Ghoussoub; Xiaosong Kang. The optimal evolution of the free energy of interacting gases and its applications. Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 173-178. doi : 10.1016/S1631-073X(03)00289-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00289-9/

[1] M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge–Kantorovich theory, Preprint, 2002

[2] M. Agueh, N. Ghoussoub, X. Kang, Geometric inequalities via a general comparison principle for interacting gases, Preprint, 2002

[3] W. Beckner Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math., Volume 11 (1999) no. 1, pp. 105-137

[4] J. Carrillo, R. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, in press

[5] D. Cordero-Erausquin, W. Gangbo, C. Houdré, Inequalities for generalized entropy and optimal transportation, in: Proceedings of the Workshop: Mass transportation Methods in Kinetic Theory and Hydrodynamics, in press

[6] D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities, Preprint, 2002

[7] M. Del Pino, J. Dolbeault, The optimal Euclidean Lp-Sobolev logarithmic inequality, J. Funct. Anal. (2002), in press

[8] I. Gentil, The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton–Jacobi equations, Preprint 2002

[9] R. McCann A convexity principle for interacting gases, Adv. Math., Volume 128 (1997) no. 1, pp. 153-179

[10] F. Otto, Doubly degenerate diffusion equations as steepest descent, Preprint, Univ. Bonn, 1996

[11] F. Otto; C. Villani Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal., Volume 173 (2000) no. 2, pp. 361-400

Cité par Sources :

This work is partially supported by a grant from the Natural Science and Engineering Research Council of Canada.

Commentaires - Politique