Granular media equation with double-well external landscape: limiting steady state

Julian Tugaut

doi:10.5802/crmath.595

Research article - Partial differential equations, Probability theory

Granular media equation with double-well external landscape: limiting steady state

Julian Tugaut ¹

¹ Université Jean Monnet, CNRS UMR 5208, Institut Camille Jordan, Maison de l’Université, 10 rue Tréfilerie, CS 82301, 42023 Saint-Étienne Cedex 2, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 775-778.

Abstract
Résumé

In this paper, we give a simple condition on the initial state of the granular media equation which ensures that the limit as the time goes to infinity is the unique steady state with positive center of mass. To do so, we use functional inequalities, Laplace method and McKean–Vlasov diffusion (which corresponds to the probabilistic interpretation of the granular media equation).

Dans ce papier, nous donnons une condition simple portant sur l’état initial de l’équation des milieux granulaires qui assure que la limite en temps long est l’unique probabilité invariante avec un centre de masse strictement positif. Pour ce faire, nous utilisons des inégalités fonctionnelles, la méthode de Laplace et la diffusion de McKean–Vlasov (qui correspond à l’interprétation probabiliste de l’équation des milieux granulaires).

Received: 2023-10-21
Accepted: 2023-11-29
Published online: 2024-09-17

DOI: 10.5802/crmath.595

Classification: 35K55, 60J60, 60G10, 39B72

Author's affiliations:

Julian Tugaut ¹

¹ Université Jean Monnet, CNRS UMR 5208, Institut Camille Jordan, Maison de l’Université, 10 rue Tréfilerie, CS 82301, 42023 Saint-Étienne Cedex 2, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2024__362_G7_775_0,
     author = {Julian Tugaut},
     title = {Granular media equation with double-well external landscape: limiting steady state},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {775--778},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.595},
     language = {en},
}

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Julian Tugaut. Granular media equation with double-well external landscape: limiting steady state. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 775-778. doi : 10.5802/crmath.595. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.595/

References
Cited by

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[7] J. Tugaut Phase transitions of McKean–Vlasov processes in double-wells landscape, Stochastics, Volume 86 (2014) no. 2, pp. 257-284 | DOI | MR | Zbl

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