The time horizon for stochastic homogenization of the one-dimensional wave equation

Article type: Research Article

Authors: Schäffner, M.^a | Schweizer, B.^{b; *}

Affiliations: [a] Institut für Mathematik, MLU Halle-Wittenberg, Theodor-Lieser-Straße 5, 06120 Halle (Saale), Germany | [b] Fakultät für Mathematik, TU Dortmund, Vogelspothsweg 87, 44227 Dortmund, Germany

Correspondence: [*] Corresponding author. E-mail: ben.schweizer@tu-dortmund.de.

Abstract: The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε−2. We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε−β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε−1.

Keywords: Stochastic homogenization, wave equation

DOI: 10.3233/ASY-241923

Journal: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-39, 2024