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Séminaire
Rigorous, computer assisted proofs in the dynamics of dissipative PDE's: Heteroclinic connections between fixed points for Kuramoto-Sivashinski PDE
Piotr Zgliczynski
Jagiellonian University, Cracovie, Pologne

We will discuss the method of self-consistent bounds for dissipative PDEs. This methods allows for a direct application of tools from dynamical system theory (finite dimensional) to dissipative PDE. This includes both: abstract theorems and rigorous algorithms for integration of PDEs. As an example we will discuss a computer assisted proof of the existence of some heteroclinic connections between fixed points for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions. The proof consists of the following stages: 1) the proof of the existence of two fixed points, "the source" and "the target" 2) rigorous estimates for the attracting region around the target point 3) rigorous estimates for one dimensional unstable manifold of the source point 4) rigorous integration of PDE - the propagation of the unstable manifold of the source until it enters the basin of attraction of the target point.

Date: 2012-02-15 à 09:30
Endroit: VCH-3850