[Un principe variationnel pour les équations de flot gradient dans les espaces métriques]
Nous présentons une nouvelle approche variationnelle pour lʼétude dʼévolution de flot gradient dans des espaces métriques. En particulier, nous proposons une fonctionnelle définie sur des trajectoires entières. Nous démontrons que les minimums de cette fonctionnelle convergent vers des courbes de descente maximale dans le cas dʼune énergie géodésiquement convexe. Le point crucial de lʼargument est la reformulation de lʼapproche variationnelle en terms du principe de la programmation dynamique. Ce resultat peut sʼappliquer à une large classe dʼévolution nonlineaires qui peuvent être reformulées comme des flots gradient dans des espaces métriques de Wasserstein.
We present a novel variational approach to gradient-flow evolution in metric spaces. In particular, we advance a functional defined on entire trajectories, whose minimizers converge to curves of maximal slope for geodesically convex energies. The crucial step of the argument is the reformulation of the variational approach in terms of a dynamic programming principle, and the use of the corresponding Hamilton–Jacobi equation. The result is applicable to a large class of nonlinear evolution PDEs including nonlinear drift-diffusion, Fokker–Planck, and heat flows on metric-measure spaces.
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@article{CRMATH_2011__349_23-24_1225_0,
author = {Rossi, Riccarda and Savar\'e, Giuseppe and Segatti, Antonio and Stefanelli, Ulisse},
title = {A variational principle for gradient flows in metric spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {1225--1228},
publisher = {Elsevier},
volume = {349},
number = {23-24},
year = {2011},
doi = {10.1016/j.crma.2011.11.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.11.002/}
}
TY - JOUR AU - Rossi, Riccarda AU - Savaré, Giuseppe AU - Segatti, Antonio AU - Stefanelli, Ulisse TI - A variational principle for gradient flows in metric spaces JO - Comptes Rendus. Mathématique PY - 2011 SP - 1225 EP - 1228 VL - 349 IS - 23-24 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.11.002/ DO - 10.1016/j.crma.2011.11.002 LA - en ID - CRMATH_2011__349_23-24_1225_0 ER -
%0 Journal Article %A Rossi, Riccarda %A Savaré, Giuseppe %A Segatti, Antonio %A Stefanelli, Ulisse %T A variational principle for gradient flows in metric spaces %J Comptes Rendus. Mathématique %D 2011 %P 1225-1228 %V 349 %N 23-24 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.11.002/ %R 10.1016/j.crma.2011.11.002 %G en %F CRMATH_2011__349_23-24_1225_0
Rossi, Riccarda; Savaré, Giuseppe; Segatti, Antonio; Stefanelli, Ulisse. A variational principle for gradient flows in metric spaces. Comptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1225-1228. doi : 10.1016/j.crma.2011.11.002. http://www.numdam.org/articles/10.1016/j.crma.2011.11.002/
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