Hyperbolic conservation laws: the next generation

Abstract: The study of systems of quasilinear hyperbolic partial differential equations (PDE), known as Conservation Laws, got its start in the 1940's and 50's, when engineers and physicists began to study compressible flows (high-speed flow, magnetohydrodynamics) in earnest. The theory of linear PDE was developing rapidly; and scientific computing both contributed to the visualization of solutions and provided a motivation to build a mathematical framework. In elliptic PDE, the nonlinear theory, though far from simple, follows the well-known path of linearization. But that is not true for hyperbolic PDE. It has been over forty years since the first proof (by James Glimm) of existence of solutions to a system of conservation laws in a single space variable, but the theory of multidimensional conservation laws is in its infancy.

It has been difficult to develop a theory of conservation laws, but the underlying mathematical problems are interesting, and have contributed to the discovery of new methods of analysis. Now a new generation of researchers is beginning a serious exploration multidimensional conservation laws, and is finding even more interesting analysis and challenges. This talk will try to provide a very rough road map of the mathematical results, and puzzles, of conservation laws.

Refreshments will be served at 2:45 p.m. (MATX 1115, Math Lounge).