Nous exposons ici deux remarques sur la notion de flot généralisé, introduite par DiPerna et le troisième auteur (R.J. Di Perna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (3) (1989) 511–547), pour les équations différentielles ordinaires. D'une part, nous fournissons une preuve autonome de l'unicité d'un tel flot, c'est-à-dire une preuve ne reposant pas sur l'interprétation du flot généralisé en termes de flot pour l'équation de transport associée. D'autre part, en utilisant cette fois l'équation de transport associée, nous généralisons sensiblement la preuve d'unicité fournie dans l'article cité en nous affranchissant pour le flot de l'hypothèse de structure de groupe en temps.
This Note presents two remarks on the notion of generalized flow solution to ordinary differential equations, as introduced by DiPerna and the third author (R.J. Di Perna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (3) (1989) 511–547). On the one hand, we provide a self-contained proof of the uniqueness of such a flow. By this, we mean that our new proof does not exploit the interpretation of the generalized flow in terms of flow for the associated linear transport equation. On the other hand, this time using the associated linear transport equation, we slightly extend the result of uniqueness contained in the article cited, proving it holds without the group property of the flow (in the time variable).
Accepté le :
Publié le :
Maxime Hauray 1 ; Claude Le Bris 2, 3 ; Pierre-Louis Lions 4, 5
@article{CRMATH_2007__344_12_759_0,
author = {Maxime Hauray and Claude Le Bris and Pierre-Louis Lions},
title = {Deux remarques sur les flots g\'en\'eralis\'es d'\'equations diff\'erentielles ordinaires},
journal = {Comptes Rendus. Math\'ematique},
pages = {759--764},
publisher = {Elsevier},
volume = {344},
number = {12},
year = {2007},
doi = {10.1016/j.crma.2007.05.010},
language = {fr},
}
TY - JOUR AU - Maxime Hauray AU - Claude Le Bris AU - Pierre-Louis Lions TI - Deux remarques sur les flots généralisés d'équations différentielles ordinaires JO - Comptes Rendus. Mathématique PY - 2007 SP - 759 EP - 764 VL - 344 IS - 12 PB - Elsevier DO - 10.1016/j.crma.2007.05.010 LA - fr ID - CRMATH_2007__344_12_759_0 ER -
Maxime Hauray; Claude Le Bris; Pierre-Louis Lions. Deux remarques sur les flots généralisés d'équations différentielles ordinaires. Comptes Rendus. Mathématique, Volume 344 (2007) no. 12, pp. 759-764. doi : 10.1016/j.crma.2007.05.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.05.010/
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