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Homogenization of waves in periodic media : Long time behavior and dispersive effective equations

We study second order linear wave equations in periodic media with a small periodicity length epsilon>0,

\partial_t^2 u^\epsilon(x,t) = \nabla.(a(x/ \epsilon) \nabla u^\epsilon(x,t)), (1)

aiming at the derivation of effective equations in R^n. Standard homogenization theory provides, for the limit epsilon\rightarrow 0, an effective second order wave equation that describes solutions of (1) on time intervals [0,T]. In this talk a refinement of this classical result is presented : We investigate the behavior of solutions u^\epsilon on large time intervals [0,T\epsilon^-2] and show that in order to approximate the solutions for all t\in[0,T\epsilon^-2] one has to use a dispersive, higher order wave equation. We provide a well-posed fourth order effective constant coefficient equation of the form

\partial_t^2 w^\epsilon = A D^2 w^\epsilon + \epsilon^2 E D^2 \partial_t^2 w^\epsilon - \epsilon^2 F D^4 w^\epsilon (2)

and estimate the errors between the solution u^\epsilon of the original heterogeneous problem and the solution w^\epsilon of the dispersive wave equation. We demonstrate that the main difficulty lies in the derivation of (2), which is performed in two steps. In the first step we identify a family of relevant, not necessarily well-posed limit models via classical Bloch wave analysis and present also an alternative approach : adaption operators. In the second step a method to select the well-posed effective model (2) is provided.


CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46