Mathematical Physics Seminar
Monday 19.IV.2010
Time: 16:00-17:00
Room: 0089
Instytut Matematyki, ul. Łojasiewicza 6, 30-348 Kraków
Piotr Zgliczyński
(Uniwersytet Jagielloński, Cracow, Poland)
Heteroclinic orbits for the Kuramoto-Sivashinsky equation
Abstract:
We will discuss the method of self-consistent bounds for dissipative PDEs.
This method allows for a direct application of tools from dynamical systems
theory (in finite dimensions) to dissipative PDE; this includes both abstract
theorems and rigorous algorithms for integration of PDEs. As an example, we
shall discuss a computer-assisted proof of the existence of some heteroclinic
orbits between fixed points for the Kuramoto-Sivashinsky PDE on the line with
odd and periodic boundary conditions. The proof consists of the following stages:
(1) 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: propagation of the unstable manifold of the
source until it enters the basin of attraction of the target point.
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