Nonlinear Waves Trimester - May Conference

Collection Nonlinear Waves Trimester - May Conference

Organizer(s) Frank MERLE (Université de Cergy-Pontoise & IHÉS), Pierre RAPHAËL (Université Nice Sophia-Antipolis), Nikolay TZVETKOV (Université de Cergy-Pontoise)

Date(s) 23/05/2016 - 27/05/2016

linked URL https://indico.math.cnrs.fr/event/889/

00:00:00 / 00:00:00

20 21

Birkhoff normal form for nonlinear wave equations

By Walter Craig

Many theorems on global existence of small amplitude solutions of nonlinear wave equations in ${\mathbb R}^n$ depend upon a competition between the time decay of solutions and the degree of the nonlinearity. Decay estimates are more effective when inessential nonlinear terms are able to be removed through a well-chosen transformation. Most wave equations that arise in a physical context can be considered as Hamiltonian PDEs, that is, partial differential equations that can be formulated as a Hamiltonian system. In this talk, we construct Birkhoff normal forms transformations for the class of wave equations which are Hamiltonian PDEs and null forms, giving a new proof via canonical transformations of the global existence theorems for null form wave equations of S. Klainerman, J. Shatah and other, in space dimensions $n \geq 3$.The critical case $n = 2$ is also under consideration. These results are work-in-progress with A. French and C. - R. Yang

Information about the video

Date of recording 27/05/2016
Date of publication 03/06/2016
Institution IHES
Format MP4

Domain(s)

Analysis of PDEs

MSC codes

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