Applied Maths Seminar: Wechselberger -- Rankine-Hugoniot, Lax and folds: the geometry of advection-reaction-diffusion systems

Martin Wechselberger, School of Mathematics and Statistics, The University of Sydney 

Wednesday 19th May 14:05-14:55pm, New Law School Seminar 030 (Building F10).  

Hyperbolic balance laws (conservation laws with source terms) have attracted much
attention in the biosciences because they play an important role in modeling
tactically-driven cell migration.  In particular, sharp interfaces, or shocks, in the
wave form of cell migration within tissues are observed which motivates the study of
advection-reaction-diffusion models where the diffusion is considered a viscous small
perturbation.  (Sharp interfaces are, of course, well-known in classical physical
applications such as in gas dynamics, MHD theory and traffic flows.)  

In this talk, we will give a twist to the existing hyperbolic PDE theory on viscous
balance laws and apply recently developed geometric singular perturbation methods to
this class of problems.  In particular, I will identify the underlying geometric
structures, folds and canards, that lead to the existence of travelling waves with sharp
interfaces which provides us with a geometric interpretation of the famous
Rankine-Hugoniot jump condition and the Lax entropy condition.