Homogenization and enhancement of the $G-$equation in random environmentsReport as inadecuate




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1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision 2 Department of Mathematics Chicago

Abstract : 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.

Keywords : Hamilton-Jacobi equations viscosity solutions homogenization ergodic media





Author: Pierre Cardaliaguet - Panagiotis Souganidis -

Source: https://hal.archives-ouvertes.fr/



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