Recent results on KAM for multidimensional PDEs

Abstract

In this short overview I present some recent results about the KAM theory for multidimensional partial differential equations (PDEs) trying to avoid technicalities. In particular I will not state a precise KAM theorem but I will focus on the dynamical consequences for the PDEs: the existence and the stability (or not) of quasi periodic in time solutions. Concretely, I present the complete study of the nonlinear beam equation on the d-dimensional torus recently obtained in collaboration with H. Eliasson and S. Kuksin. When d ≥ 2 we are able to construct explicit examples where the quasi periodic solutions are linearly unstable, a new feature in Hamiltonian PDEs that could complement recent results in weak turbulence theory.