PDE Seminar Abstracts

Dynamic isoperimetry on weighted Riemannian manifolds

Eric Kwok
University of New South Wales
Mon 8th Aug 2016, 2-3pm, Carslaw Room 829 (AGR)


Transport and mixing in dynamical systems are important mechanisms for many physical processes. We consider the detection of transport barriers using a recently developed geometric technique: the dynamic isoperimetric problem. Solutions to the dynamic isoperimetric problem are sets with persistently small boundary size relative to interior volume, as the sets are evolved by the dynamics. In the presence of small diffusion these sets have very low dispersion over finite-times because of their lasting small boundary size, and thus are natural candidates for coherent sets, bounded by transport barriers.

I will talk about the construction of a dynamic Laplacian based on the Laplace-Beltrami, and Perron-Frobenius operator on a weighted Riemannian manifold. I will present a Cheeger-type inequality for the dynamic Laplacian, and show that the level surfaces of the second eigenfunction of the dynamic Laplacian can be used to generate good solutions to the dynamic isoperimetric problem.