Long time homogenization of classical waves

During this talk we will consider the waves equation in highly inhomogeneous media. More precisely we will be interested in the homogeneization theory that is derive a space independent equation which is a good approximation of the initial waves equation when the scale of the inhomogeneity, namely $ϵ$, goes to infinity.

It is a classical and well-known result that homogeneization process holds on time sales of order T. Moreover, up to the appearance of some dispersive phenomenon this result can be extended to time scales ${ϵ}^{-2}$T. This analysis is based on Blocs waves that diagonalize the spatial part in the d'Alembertien.

Unfortunatly, Blocs waves theory is restricted to the periodic framework. However, independently of the periodicity assomption, a truncaded version of these waves can be defined, the so-called Taylor-Bloch waves. These waves "almost" diagonalize the spatial operator. This construction is based upon the introduction of a new set of extended correctors.

Thanks to Taylor-Bloch waves we will extend the homogeneization result to polynomial time scales.