Long-time homogenization and asymptotic ballistic transport of classical waves
Résumé
Consider an elliptic operator in divergence form with symmetric coefficients.
If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is
key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization.
When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying
correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associated
operator are unclear.
At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization.
The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an ``eigendefect''), which we express with the help of a new family of extended correctors.
We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large times
for the wave equation in terms of the spatial growth of these extended correctors.
On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions).
On the other hand, this allows us to prove asymptotic ballistic transport of classical waves
at low energies for almost periodic and random operators.
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