Long time behavior of the solutions of NLW on the d-dimensional torus

Joackim Bernier, Erwan Faou, Benoit Grebert

Published 2019-06-12Version 1

We consider the non linear wave equation (NLW) on the d-dimensional torus with an smooth nonlinearity of order at least two at the origin. We prove that, for almost all mass, small smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolves according to a time dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semi-linear Hamiltonian PDEs whose linear frequencies satisfy a very general non resonance condition. In particular it also applies straightforwardly to a full dispersion Whitham-Boussinesq system in water waves theory.