Analysis Seminar

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Almost global existence for Hamiltonian PDEs on boundaryless compact manifolds

Speaker: Dario Bambusi, Università degli Studi di Milano

Location: Warren Weaver Hall 1314

Date: Tuesday, April 29, 2025, 2:30 p.m.

Synopsis:

Taking the nonlinear Klein Gordon equation as a model problem, I will present a result on the qualitative behaviour of solutions of Hamiltonian PDEs on compact boundaryless Riemannian manifolds. Precisely, one has that solutions corresponding to smooth and small initial data remain small and smooth for times of order \(\epsilon^{-r}\), for all \(r\). Here \(\epsilon\) is the size of the initial datum. The proofs is based on variants of Birkhoff normal form.

I will start by reviewing the classical method of Birkhoff normal form for finite dimensional Hamiltonian systems, then I will recall the theory for equations in one space dimensions and finally I will present the ideas leading to the theory in higher space dimension.

Joint work with J. Bernier, B. Grebert, R. Imekratz.