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
|Date:||10 / 23 / 2013|
|Time:||03:35pm - 04:35pm|