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A spectral ansatz for the long-time homogenization of the wave equation

Mitia Duerinckx, Antoine Gloria, Matthias Ruf

Abstract

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).

A spectral ansatz for the long-time homogenization of the wave equation

Abstract

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).
Paper Structure (30 sections, 19 theorems, 283 equations)

This paper contains 30 sections, 19 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Main results: long-time homogenization
  4. Periodic setting
  5. Random setting
  6. Well-posedness of homogenized equations
  7. Spectral approach and two-scale expansion
  8. Geometric approach and hyperbolic two-scale expansion
  9. Ill-prepared data
  10. Spectral approach and two-scale expansion
  11. Definition of spectral correctors
  12. Spectral two-scale expansion
  13. Well-posedness of homogenized equation
  14. Proof of Proposition \ref{['prop:form-spectral']}
  15. Proof of Lemma \ref{['lem:apriori-baru-sp']}
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\boldsymbol a$ be $Q$-periodic. There exist sequences of spectral correctors $\{\psi^{n}\}_{n}$ and $\{\zeta^{n,m}\}_{n,m}$ obtained as solutions of elliptic problems on the periodic cell $Q$, a sequence of homogenized tensors $\{\bar{{\boldsymbol b}}^{n}\}_{n}$, and a sequence of Fourier multi where $\bar{u}_\varepsilon^\ell$ is an ancient solution of the following formal homogenized equatio

Theorems & Definitions (32)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1
  • Theorem 2
  • Definition 1.3
  • Theorem 3
  • Proposition 1.5
  • proof : Proof of Proposition \ref{['prop:Bloch']}
  • Definition 2.1: Spectral correctors
  • ...and 22 more