Dia: Dimecres, 14 de novembre de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Filippo Giuliani, Universitat Politècnica de Catalunya
Títol: KAM for quasi-linear PDEs
Resum: The KAM for PDEs is the mathematical theory developed for the search of quasi-periodic (in time) solutions of partial differential equations on compact (spatial) domains.
Many notable PDEs (NLS, KdV, Klein-Gordon...) possess a Hamiltonian structure and behave, in a neighborhood of the origin, like an infinite chain of harmonic oscillators weakly coupled by the nonlinear terms. Then it is natural to look at these equations as infinite dimensional dynamical systems and use perturbative arguments to find finite dimensional invariant tori close to the origin.
The main issues arising in this kind of problems are related to the geometry/dimension of the spatial domain, the dispersive effects of the PDE and the number of the derivatives appearing in the nonlinearities. In this talk we will focus on PDEs on the circle and we will give an overview of the strategy for proving KAM results by using generalized implicit function theorems.
Although the KAM theory for PDEs on spatial 1-d domains is now well understood, the progress for quasi-linear cases, namely when the derivatives contained in the linear and nonlinear terms have the same order, is quite recent.
In this framework, we present a new result (in collaboration with R.Feola and M. Procesi) of existence and stability of quasi-periodic solutions for perturbations of the Degasperis-Procesi equation on the circle, which is a model for nonlinear shallow water phenomena. In this work we developed new techniques to deal with quasi-linear equations that have very weak dispersive effects and a complicated resonant structure.
Last updated: Fri Nov 9 08:24:26 2018