Bloch-wave homogenization on large time scales and dispersive effective wave equations

Abstract

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in ${\mathbb{R}}^n$, $n\in \{1,2,3\}$. Standard homogenization theory provides, for the limit of a small periodicity length $\eps>0$, an effective second order wave equation that describes solutions on time intervals $[0,T]$. In order to approximate solutions on large time intervals $[0,T\eps^{-2}]$, one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation, and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.

29 pages, 4 figures; changes: assumption on $H^1$ convergence of the Bloch expansion of initial data removed (follows automatically); coefficient $a^{\epsilon }$ allowed to be matrix-valued; proof of Theorem 2.4 corrected and Lemma 2.3 added