PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Leonid Berlyand.

Title:Pattern formation and partial differential equations
Seminar:Marker Lecture Series
Speaker:Dr. Felix Otto, Max-Planck-Institute for Mathematics
In three specific examples, we shall demonstrate how the theory of partial differential equations (PDEs) relates to pattern formation in nature: Spinodal decomposition and the Cahn-Hilliard equation, Rayleigh-Benard convection and the Boussinesq approximation, rough crystal growth and the Kuramoto-Sivashinsky equation. These examples from different applications have in common that only a few physical mechanisms, which are modeled by simple-looking evolutionary PDEs, lead to complex patterns. These mechanisms will be explained, numerical simulation shall serve as a visual experiment. Numerical simulations also reveal that generic solutions of these deterministic equations have stationary or self-similar statistics that are independent of the system size and of the details of the initial data. We show how PDE methods, i. e. a priori estimates, can be used to understand some aspects of this universal behavior. In case of the Cahn-Hilliard equation, the method makes use of its gradient flow structure and a property of the energy landscape. In case of the Boussinesq equation, a "driven gradient flow", the background field method is used. In case of the Kuramoto-Sivashinsky equation, that mixes conservative and dissipative dynamics, the method relies on a new result on Burgers' equation.

Room Reservation Information

Room Number:MB114
Date:10 / 23 / 2013
Time:03:35pm - 04:35pm