# Mathematical Models for Transport in Macroscopic and Mesoscopic Systems - Abstract

**Castella, François**

*Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity*

We study the limiting behavior of a nonlinear Schrödinger equation describing a three dimensional gas that is strongly confined along the vertical, $z$ direction. The confinement induces fast oscillations in time, that need to be averaged out. Since the Hamiltonian in the $z$ direction is merely assumed confining, without any further specification, the associated spectrum is discrete but arbitrary, and the fast oscillations induced by the nonlinear equation entail countably many frequencies that are arbitrarily distributed as well. For that reason, averaging can not rely on small denominator estimates or like. To overcome these difficulties, we prove that the fast oscillations are almost periodic in time, with values in a Sobolev-like space that we completely identify. We then exploit the existence of long time averages for almost periodic function to perform the necessary averaging procedure in our nonlinear problem. This is a joint work with N. Ben Abdallah (Toulouse) and F. Mehats (Rennes).