Recent results on KAM for multidimensional PDEs

Grébert, Benoît

Journées équations aux dérivées partielles (2014), article no. 4, 12 p.

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 $d\ge 2$ 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.

Mots clés : Multidimensional PDEs, Quasi periodic solutions, KAM theory, stable and unstable tori

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