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 -dimensional torus recently obtained in collaboration with H. Eliasson and S. Kuksin. When 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.
@incollection{JEDP_2014____A4_0,
author = {Beno{\^\i}t Gr\'ebert},
title = {Recent results on {KAM} for multidimensional {PDEs}},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {4},
pages = {1--12},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2014},
doi = {10.5802/jedp.107},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.107/}
}
TY - JOUR AU - Benoît Grébert TI - Recent results on KAM for multidimensional PDEs JO - Journées équations aux dérivées partielles PY - 2014 SP - 1 EP - 12 PB - Groupement de recherche 2434 du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.107/ DO - 10.5802/jedp.107 LA - en ID - JEDP_2014____A4_0 ER -
%0 Journal Article %A Benoît Grébert %T Recent results on KAM for multidimensional PDEs %J Journées équations aux dérivées partielles %D 2014 %P 1-12 %I Groupement de recherche 2434 du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.107/ %R 10.5802/jedp.107 %G en %F JEDP_2014____A4_0
Benoît Grébert. Recent results on KAM for multidimensional PDEs. Journées équations aux dérivées partielles (2014), article no. 4, 12 p. doi : 10.5802/jedp.107. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.107/
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