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Homogenization and enhancement of the G-equation in random environments

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Date
2013
Indexation documentaire
Analyse
Subject
viscosity solutions; Hamilton-Jacobi equations; homogenization; ergodic media
Nom de la revue
Communications on Pure and Applied Mathematics
Volume
66
Numéro
10
Date de publication
2013
Pages article
1582–1628
Nom de l'éditeur
Wiley
DOI
http://dx.doi.org/10.1002/cpa.21449
URI
https://basepub.dauphine.fr/handle/123456789/7256
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Auteur
Cardaliaguet, Pierre
Souganidis, Panagiotis E.
Type
Article accepté pour publication ou publié
Résumé en anglais
We study the homogenization of a $G$-equation which is advected by a divergence free stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G-equation and we give necessary and sufficient conditions in order to have enhancement. Since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition which is necessary to apply the sub-additive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence which has both a long time averaged limit (due to the sub-additive ergodic theorem) and stays (in the same sense) asymptotically close to the minimal time.

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