Recent results on KAM for multidimensional PDEs
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 d2 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.

DOI: 10.5802/jedp.107
Keywords: Multidimensional PDEs, Quasi periodic solutions, KAM theory, stable and unstable tori
Grébert, Benoît 1

1 Laboratoire de Mathématiques Jean Leray Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France
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Grébert, Benoît. Recent results on KAM for multidimensional PDEs. Journées équations aux dérivées partielles (2014), article  no. 4, 12 p. doi : 10.5802/jedp.107. http://www.numdam.org/articles/10.5802/jedp.107/

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