[Propriétés d'homogénéisation en temps long et transport balistique asymptotique des ondes classiques]
Considérons un opérateur elliptique sous forme divergence à coefficients symétriques non constants. Si ces coefficients sont périodiques, la théorie de Floquet-Bloch permet de diagonaliser l'opérateur elliptique, ce qui est crucial pour l'étude des propriétés spectrales de l'opérateur et le point de départ usuel pour l'étude des propriétés d'homogénéisation en temps long de l'opérateur des ondes associé. Quand les coefficients ne sont pas périodiques (disons quasi-périodiques, presque périodiques, ou aléatoires stationnaires ergodiques), la théorie de Floquet-Bloch ne s'applique plus et les propriétés spectrales ainsi que le comportement en temps long de l'opérateur des ondes associé ne sont pas claires a priori. Aux basses fréquences, nous pouvons cependant considérer un développement de Taylor formel des ondes de Bloch (que celles-ci existent ou non) en se basant sur des correcteurs introduits en homogénéisation elliptique. Ces ondes de Taylor-Bloch diagonalisent l'opérateur elliptique à un terme d'erreur près (un “défaut propre”), que nous exprimons à l'aide d'une nouvelle famille de correcteurs étendus. Nous utilisons cette formulation des défauts propres pour quantifier les propriétés de transport et d'homogénéisation en temps long pour l'équation des ondes associée en termes de croissance spatiale des correcteurs étendus. D'une part, cela quantifie la validité de l'homogénéisation en temps long (à la fois pour l'opérateur homogénéisé standard et pour des versions d'ordre supérieur). D'autre part, cela nous permet d'établir le transport balistique asymptotique des ondes classiques aux basses énergies pour des opérateurs presque périodiques et aléatoires.
Consider an elliptic operator in divergence form with symmetric coefficients. If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization. When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associated operator are unclear. At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization. The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an “eigendefect”), which we express with the help of a new family of extended correctors. We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large times for the wave equation in terms of the spatial growth of these extended correctors. On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions). On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.
Keywords: Homogenization, periodic, quasiperiodic, random, waves, long-time, ballistic transport.
Mot clés : Homogénéisation, périodique, presque périodique, aléatoire, ondes, temps long, transport balistique.
@article{ASENS_2019__52_3_703_0,
author = {Benoit, Antoine and Gloria, Antoine},
title = {Long-time homogenization and asymptotic ballistic transport of classical waves},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {703--759},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 52},
number = {3},
year = {2019},
doi = {10.24033/asens.2395},
mrnumber = {3982871},
zbl = {1437.35030},
language = {en},
url = {http://www.numdam.org/articles/10.24033/asens.2395/}
}
TY - JOUR AU - Benoit, Antoine AU - Gloria, Antoine TI - Long-time homogenization and asymptotic ballistic transport of classical waves JO - Annales scientifiques de l'École Normale Supérieure PY - 2019 SP - 703 EP - 759 VL - 52 IS - 3 PB - Société Mathématique de France. Tous droits réservés UR - http://www.numdam.org/articles/10.24033/asens.2395/ DO - 10.24033/asens.2395 LA - en ID - ASENS_2019__52_3_703_0 ER -
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Benoit, Antoine; Gloria, Antoine. Long-time homogenization and asymptotic ballistic transport of classical waves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 703-759. doi : 10.24033/asens.2395. http://www.numdam.org/articles/10.24033/asens.2395/
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