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Open PDE & Analysis Seminar

Dr Joackim Bernier (Institut de Mathématique de Toulouse)

Long time dynamics of some nonlinear resonant dispersive PDEs

Friday 12th June 2020, 3pm (Paris time)

Abstract : In confined domains, dispersive PDEs can have quite complex and turbulent long time dynamics. For example, there can be a migration of the energy to arbitrarily small spacial scales or, at least, some energy exchanges. Nevertheless, for non-resonant systems (i.e. the eigenvalues of the linearized system have to satisfy some kinds of diophantine conditions), some extensions of the Birkhoff normal form theory prove that the smooth and small solutions are stable for very long times. I will present a recent extension of this theory, based on rational transformations, to get a similar result for generic solutions of some resonant systems including the generalized KdV equations, the generalized Benjamin-Ono equations and some nonlinear Schrödinger equations on the one dimension torus. This talk is based on some joint works with Erwan Faou and Benoît Grébert.

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Thomas Alazard (ENS Paris-Saclay)         Nicolas Burq (Orsay)         Iván Moyano (Nice)