Recent results on KAM for multidimensional PDEs

Benoît Grébert[1]

  • [1] Laboratoire de Mathématiques Jean Leray Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France

Journées Équations aux dérivées partielles (2014)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Grébert, Benoît. "Recent results on KAM for multidimensional PDEs." Journées Équations aux dérivées partielles (2014): 1-12. <http://eudml.org/doc/275420>.

@article{Grébert2014,
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 $d\ge 2$ 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.},
affiliation = {Laboratoire de Mathématiques Jean Leray Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France},
author = {Grébert, Benoît},
journal = {Journées Équations aux dérivées partielles},
keywords = {Multidimensional PDEs; Quasi periodic solutions; KAM theory; stable and unstable tori},
language = {eng},
pages = {1-12},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Recent results on KAM for multidimensional PDEs},
url = {http://eudml.org/doc/275420},
year = {2014},
}

TY - JOUR
AU - Grébert, Benoît
TI - Recent results on KAM for multidimensional PDEs
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 12
AB - 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.
LA - eng
KW - Multidimensional PDEs; Quasi periodic solutions; KAM theory; stable and unstable tori
UR - http://eudml.org/doc/275420
ER -

References

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