Series: Mathematics Colloquium

Date: Thursday, September 18, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Hosts: Augustin Banyaga, Yuxi Zheng

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Barbara Lee Keyfitz, University of Houston

Title: 

Multidimensional Conservation Laws: How to Solve Hyperbolic Problems
with Elliptic Methods

Abstract:  

We begin with the basic classification of partial differential
equations (PDE), in which many time-dependent problems are of
hyperbolic type; their solutions are characterized by wave
propagation, finite domain of dependence, and focussing of
singularities.  The mathematical tools (which will NOT be the focus of
this talk) developed to analyse linear and semilinear hyperbolic
equations do not help much with nonlinear problems.  By contrast, in
elliptic PDE (which typically govern time-independent problems), the
quasilinear theory is a relatively simple variant of the linear
theory.  Despite impressive recent advances in the theory of
hyperbolic conservation laws in a single space variable, there is
little theory for multidimensional conservation laws.  One approach to
this problem is to study self-similar solutions.  It turns out that
one ends up with quasilinear elliptic equations --- and can take
advantage of the advanced state of elliptic theory.  The talk will
conclude with some recent results of Suncica Canic, Eun Heui Kim and
myself related to weak shock reflection and the von Neumann paradox.