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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