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

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Wellposedness and scattering for the Zakharov system in four dimensions

By Kenji Nakanishi

This is joint work with Ioan Bejenaru, Zihua Guo and Sebastian Herr. We consider the Cauchy problem for the Zakharov system in four space dimensions, extending the local wellposedness by Ginibre, Tsutsumi and Velo to wider range of Sobolev exponents, together with scattering for small data. We observe distinct phenomena at two extreme points of exponents. One is at the energy space, where our nonlinear estimates suffer from divergence related to the critical Sobolev embedding. We can however prove the results for small data, without any improvement of the estimates, but from the existence in a smaller Sobolev space and the uniqueness in a larger one, together with the conservation law. At another corner point of exponents, the critical space is also intermediate, but we obtain a strong illposedness result in terms of instantaneous exit or non-existence of weak solutions.

Information about the video

Date of recording 23/05/2016
Date of publication 03/06/2016
Institution IHES
Format MP4

Domain(s)

Analysis of PDEs

MSC codes

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