[L'évolution de l'énergie totale d'un gaz le long d'un transport optimal et applications]
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.
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@article{CRMATH_2003__337_3_173_0,
author = {Agueh, Martial and Ghoussoub, Nassif and Kang, Xiaosong},
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},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00289-9/}
}
TY - JOUR AU - Agueh, Martial AU - Ghoussoub, Nassif AU - Kang, Xiaosong 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 UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00289-9/ DO - 10.1016/S1631-073X(03)00289-9 LA - en ID - CRMATH_2003__337_3_173_0 ER -
%0 Journal Article %A Agueh, Martial %A Ghoussoub, Nassif %A Kang, Xiaosong %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 %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00289-9/ %R 10.1016/S1631-073X(03)00289-9 %G en %F CRMATH_2003__337_3_173_0
Agueh, Martial; Ghoussoub, Nassif; Kang, Xiaosong. The optimal evolution of the free energy of interacting gases and its applications. Comptes Rendus. Mathématique, Tome 337 (2003) no. 3, pp. 173-178. doi : 10.1016/S1631-073X(03)00289-9. http://www.numdam.org/articles/10.1016/S1631-073X(03)00289-9/
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☆ This work is partially supported by a grant from the Natural Science and Engineering Research Council of Canada.
