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Operator aspects of wave propagation through periodic media

Kirill Cherednichenko, Yi-Sheng Lim

TL;DR

The paper addresses the problem of quantitatively homogenising the wave equation in periodic media on long time scales. It adopts an operator-theoretic framework, combining the spectral germ, two-scale, criminal and spectral ansatzes, Gelfand/Bloch theory, and Ryzhov boundary triples with Krein’s formula to derive norm-resolvent and hyperbolic evolution estimates. It delivers both first- and second-order homogenised dynamics, proving operator-norm convergence and extending valid times to longer scales via innovative expansions and dilations. The results provide rigorous, quantitative control of wave propagation in periodic media and illuminate the connection between Bloch data and classical two-scale information, with implications for accurate long-time wave modeling. Overall, the work advances the theoretical toolkit for hyperbolic homogenisation and sets the stage for further extensions beyond the classical $O(\varepsilon^{-2})$ regime.

Abstract

Recent results in quantitative homogenisation of the wave equation with rapidly oscillating coefficients are discussed from the operator-theoretic perspective, which views the solution as the result of applying the operator of hyperbolic dynamics, i.e. the unitary group of a self-adjoint operator on a suitable Hilbert space. A prototype one-dimensional example of utilising the framework of Ryzhov boundary triples is analysed, where operator-norm resolvent estimates for the problem of classical moderate-contrast homogenisation are obtained. By an appropriate "dilation" procedure, these are shown to upgrade to second-order (and more generally, higher-order) estimates for the resolvent and the unitary group describing the evolution for the related wave equation.

Operator aspects of wave propagation through periodic media

TL;DR

The paper addresses the problem of quantitatively homogenising the wave equation in periodic media on long time scales. It adopts an operator-theoretic framework, combining the spectral germ, two-scale, criminal and spectral ansatzes, Gelfand/Bloch theory, and Ryzhov boundary triples with Krein’s formula to derive norm-resolvent and hyperbolic evolution estimates. It delivers both first- and second-order homogenised dynamics, proving operator-norm convergence and extending valid times to longer scales via innovative expansions and dilations. The results provide rigorous, quantitative control of wave propagation in periodic media and illuminate the connection between Bloch data and classical two-scale information, with implications for accurate long-time wave modeling. Overall, the work advances the theoretical toolkit for hyperbolic homogenisation and sets the stage for further extensions beyond the classical regime.

Abstract

Recent results in quantitative homogenisation of the wave equation with rapidly oscillating coefficients are discussed from the operator-theoretic perspective, which views the solution as the result of applying the operator of hyperbolic dynamics, i.e. the unitary group of a self-adjoint operator on a suitable Hilbert space. A prototype one-dimensional example of utilising the framework of Ryzhov boundary triples is analysed, where operator-norm resolvent estimates for the problem of classical moderate-contrast homogenisation are obtained. By an appropriate "dilation" procedure, these are shown to upgrade to second-order (and more generally, higher-order) estimates for the resolvent and the unitary group describing the evolution for the related wave equation.

Paper Structure

This paper contains 23 sections, 12 theorems, 138 equations, 1 table.

Table of Contents

  1. Introduction
  2. Homogenisation as a tool to study long waves
  3. Operator perspective
  4. Improving the basic homogenisation result
  5. The spectral germ approach
  6. Two-scale expansion and the secular growth problem
  7. Beyond the classical two-scale expansion
  8. Criminal ansatz
  9. Interlude
  10. Spectral ansatz
  11. A summary table
  12. Prototype one-dimensional problem and operator-norm resolvent estimates
  13. Gelfand transform
  14. Ryzhov triples and Krein's formula
  15. Operator-norm estimates in homogenisation via Krein's formula
  16. ...and 8 more sections

Key Result

Theorem 1.1

cioranescu_donato. Let $\Omega \subset \mathbb{R}^d$ be a bounded Lipschitz domain. Suppose that $\mathbb{A}$ is positive-definite uniformly in $x \in \mathbb{R}^d$, and has $L^\infty$ entries. Suppose that $f \in L^2((0,T) \times \Omega)$, $u_{\text{init}} \in H_0^1(\Omega)$, $v_{\text{init}} \in L where $u_{\hom}$ is the solution to the homogenised problem

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 15 more