Partial Differential Equations
General entropy equations for structured population models and scattering
Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 697-702.

We consider several structured population models (age structured, size structured, maturity structured) and the general scattering equation. These models are not conservation laws, nevertheless, we show that they admit a common relative entropy structure which uses the first eigenelements of the problem. In case of scattering, it is more general than the usual ‘detailed balance principle’. Three types of consequences are deduced from this entropy structure: a priori bounds, large time convergence to the steady state and in some cases, exponential rates of convergence.

Nous considérons divers modèles de populations structurées (en âge, en taille ou en maturité) et aussi l'équation de scattering. Ces modèles ne sont pas conservatifs, néammoins nous montrons qu'ils vérifient tous une structure d'entropie relative commune qui utilise les premiers éléments propres du problème et qui, dans le cas du scattering, généralise le « principe d'équilibre en détail » habituel. Trois types de conséquences découlent de cette structure entropique : des estimations a priori, la convergence en temps grand vers un état stationnaire et parfois des taux exponentiels de convergence.

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DOI: 10.1016/j.crma.2004.03.006
Michel, Philippe 1, 2; Mischler, Stéphane 1; Perthame, Benoı̂t 2

1 CEREMADE, Université Paris 9 Dauphine, pl. de Lattre de Tassigny, 75775 Paris cedex 16, France
2 Département de mathématiques et applications, UMR 8553, École normale supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France
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Michel, Philippe; Mischler, Stéphane; Perthame, Benoı̂t. General entropy equations for structured population models and scattering. Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 697-702. doi : 10.1016/j.crma.2004.03.006. http://www.numdam.org/articles/10.1016/j.crma.2004.03.006/

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