Hyperbolic Conservation Laws: More Questions than Answers
by
Barbara Lee Keyfitz
Fields Institute and University of Houston

The analysis of quasilinear hyperbolic partial differential equations presents a number of challenges. Although equations of this type are important in a number of applications, ranging from high-speed aerodynamics, through magnetohydrodynamics, to multiphase flows important in industrial technology, there is little theory against which even to check the reliability of numerical simulations.

Development of a theory for conservation laws in a single space variable has led to remarkable advances in analysis, including the theory of compensated compactness and the study of novel function spaces. Recently, a number of groups have begun to approach multidimensional systems via self-similar solutions.

In this talk, I will give some history of the development of conservation law theory, including an indication of why the applications are important. I will describe some of the recent results on self-similar solutions, and the interesting results in analysis that they involve. Finally, I will outline some of the paradoxical questions that remain. My research in this area has been joint with Suncica Canic, Katarina Jegdic, Eun Heui Kim, Gary Lieberman, and Allen Tesdall.

Date received: September 11, 2007