A variational principle for gradient flows in metric spaces

Riccarda Rossi; Giuseppe Savaré; Antonio Segatti; Ulisse Stefanelli

doi:10.1016/j.crma.2011.11.002

Mathematical Analysis/Calculus of Variations

A variational principle for gradient flows in metric spaces
[Un principe variationnel pour les équations de flot gradient dans les espaces métriques]

Présenté par : the Editorial Board

Riccarda Rossi ¹ ; Giuseppe Savaré ² ; Antonio Segatti ² ; Ulisse Stefanelli ^3,⁴

¹ Dipartimento di Matematica, Università di Brescia, v. Valotti 9, 25133 Brescia, Italy
² Dipartimento di Matematica, Università di Pavia, v. Ferrata 1, 27100 Pavia, Italy
³ IMATI – CNR, v. Ferrata 1, 27100 Pavia, Italy
⁴ WIAS, Mohrenstr. 39, 10117 Berlin, Germany

Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1225-1228.

Résumé
Abstract

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.

Reçu le : 2011-07-02
Accepté le : 2011-11-02
Publié le : 2011-11-16

DOI : 10.1016/j.crma.2011.11.002

Affiliations des auteurs :

Riccarda Rossi ¹ ; Giuseppe Savaré ² ; Antonio Segatti ² ; Ulisse Stefanelli ^{3, 4}

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     author = {Riccarda Rossi and Giuseppe Savar\'e and Antonio Segatti and Ulisse Stefanelli},
     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},
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Riccarda Rossi; Giuseppe Savaré; Antonio Segatti; Ulisse Stefanelli. A variational principle for gradient flows in metric spaces. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1225-1228. doi : 10.1016/j.crma.2011.11.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.002/

Bibliographie
Cité par

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[3] L. Ambrosio, N. Gigli, G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, preprint, , 2011. | arXiv

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