A spectral ansatz for the long-time homogenization of the wave equation
[Un ansatz spectral pour l’homogénéisation de l’équation des ondes en temps long]
Duerinckx, Mitia1 ; Gloria, Antoine2 ; Ruf, Matthias3
1Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
2Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, 75005 Paris, France & Institut Universitaire de France & Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
3École Polytechnique Fédérale de Lausanne, Section de mathématiques, 1015 Lausanne, Switzerland
Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 523-587

Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present work is a new ansatz for the long-time two-scale expansion inspired by spectral analysis. Based on this spectral ansatz, we extend and refine all previous results in the field, proving homogenization up to optimal timescales with optimal error estimates, and covering all the standard assumptions on heterogeneities (both periodic and stationary random settings).

On considère l’équation des ondes en milieux hétérogènes dans le régime d’homogénéisation. En temps long, l’onde interagit de façon non triviale avec les hétérogénéités, donnant lieu à des effets dispersifs. Le résultat principal de ce travail est un nouvel ansatz pour le développement à deux échelles en temps long, inspiré par une analyse spectrale. Sur la base de cet ansatz spectral, nous étendons et raffinons tous les résultats précédents du domaine : nous obtenons un résultat d’homogénéisation valable jusqu’à l’échelle de temps optimale avec des estimations d’erreur optimales, et nous couvrons à la fois le cas d’hétérogénéités périodiques et aléatoires stationnaires.

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Accepté le :
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DOI : 10.5802/jep.259
Classification : 35B27, 35B40, 35C20, 35L05, 74Q10, 74Q15, 74H40, 35B30, 35P05
Keywords: Wave equation, long-time homogenization, heterogeneous medium, effective equations, two-scale expansions, spectral correctors
Mots-clés : Équation des ondes, homogénéisation en temps long, milieux hétérogènes, équations effectives, développements à deux échelles, correcteurs spectraux

Duerinckx, Mitia  1   ; Gloria, Antoine  2   ; Ruf, Matthias  3

1 Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
2 Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, 75005 Paris, France & Institut Universitaire de France & Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
3 École Polytechnique Fédérale de Lausanne, Section de mathématiques, 1015 Lausanne, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {A spectral ansatz for the long-time homogenization of the wave equation},
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Duerinckx, Mitia; Gloria, Antoine; Ruf, Matthias. A spectral ansatz for the long-time homogenization of the wave equation. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 523-587. doi: 10.5802/jep.259

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