Overview
Explore a 46-minute lecture by Michela Procesi on stability and recursive solutions in Hamiltonian Partial Differential Equations (PDEs) on compact manifolds. Delve into the existence of special recursive solutions near elliptic fixed points, examining their stability and instability properties. Investigate the complex interplay between chaotic and recursive phenomena caused by resonances and small divisors, utilizing KAM theory methods. Focus on stability properties near fixed points and the existence and stability of quasi-periodic and almost-periodic solutions. Cover topics including non-linear PDEs, dynamical systems, invariant tori, perturbation theory, small solutions, linear theory, KAM in infinite dimensions, and reversible autonomous NLS equations. Examine generic tangential sites, combinatorial theorems, and finite regularity solutions for NLS, concluding with open problems in the field.
Syllabus
Intro
Non linear PDE's
PDE examples
Dynamical systems in dimension.
Invariant tori
Infinite tori
Perturbation Theory
Small solutions
Linear theory
KAM in infinite dimension
A result on the reversible autonomous NLS Consider a reversible NLS equation
Generic tangential sites
EXAMPLE: points connected by edges
The main combinatorial Theorem
Drawbacks
Finite regularity solutions for NLS
Open problems
Taught by
International Mathematical Union