Date: Tuesday, September 18, 2012
Title: Ziwet Lectures: Pattern formation and partial differential equations
Abstract: 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 CahnHilliard equation, RayleighB\'enard convection and the Boussinesq approximation, rough crystal growth and the KuramotoSivashinsky equation.
These examples from different applications have in common that only a few physical mechanisms, which are modeled by simplelooking 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 selfsimilar 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 CahnHilliard 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 KuramotoSivashinsky equation, that mixes conservative and dissipative dynamics, the method relies on a new result on Burgers' equation.
Speaker: Felix Otto
Institution: Max Planck Institute for Mathematics in the Sciences, Leipzig
