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KAM theory for PDEs

Course Type: 
PhD Course
Academic Year: 
2015-2016
Period: 
Nov. - Mar.
Duration: 
60 h
Description: 

1 - Introduction:  Quasi-periodic solutions,  small divisor problem, 

  • Examples of Hamiltonian and Reversible PDEs: nonlinear wave and Schrodinger equations, KdV, Hamiltonian formulation of the water waves equations.
  • A Nash-Moser implicit function theorem,  
  • General tools: scales of Banach spaces (Sobolev, C^k, Holder, analytic), interpolation, smoothing operators, 
  • KAM theorem for finite dimensional systems

2 - Quasi-periodic solutions of the water waves equations

  • General tools: pseudo-differential operators, composition, adjoint, commutator, Egorov theorem, 

3 - Quasi-periodic solutions of nonlinear wave equations in any dimension. Bourgain's multiscale approach and extensions.  

Location: 
A-133

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