Grégoire Allaire (CMAP): Homogénéisation de l'équation des ondes en temps longs et effets dispersifs

We report on a joint work with A. Lamacz-Keymling and J. Rauch.
We study the homogenization of the wave equation in a periodic medium for long times of the order of any inverse power of the period. The unknown can be either a scalar or a vector field, while the coefficients can be purely periodic or locally periodic tensors. We obtain high order homogenized equations which include dispersive corrections that are crucial for long time accuracy. Our main tools are (i) a so-called "criminal ansatz", which generalizes to the hyperbolic setting an idea of Bakhavalov and Panasenko in the elliptic setting, (ii) an elimination process for the higher order time derivatives in the high order homogenization equation, (iii) a stability estimate for the corresponding homogenized solutions, based on frequency filtering, (iv) an error estimate valid for any long times. The importance of considering high order homogenized equations to catch dispersive effects in the context of the wave equation was first recognized by Santosa and Symes and rigorously analyzed by Lamacz. Our work gives a systematic and complete analysis for all time scales and all high order corrective terms.