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 $d\ge 2$ 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.

DOI : https://doi.org/10.5802/jedp.107
Keywords: Multidimensional PDEs, Quasi periodic solutions, KAM theory, stable and unstable tori
@article{JEDP_2014____A4_0,
author = {Gr\'ebert, Beno\^\i t},
title = {Recent results on KAM for multidimensional PDEs},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2014},
doi = {10.5802/jedp.107},
language = {en},
url = {http://www.numdam.org/item/JEDP_2014____A4_0}
}

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/item/JEDP_2014____A4_0/

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