Abstract

The study of quasilinear hyperbolic partial differential equations (also known as conservation laws) presents formidable technical challenges. For example, the solutions to most initialvalue problems have rather low regularity, and are found in function spaces which are themselves not easy to analyze. In a single space dimension, there is now a satisfactory theory, although it is limited to small data. In more than one space dimension, there is almost no theory.
In this presentation, I will give an overview of how technical difficulties in one space dimension have been overcome, emphasizing the underlying concepts that distinguish nonlinear from linear problems. This sets the stage for a description of the small amount of analysis that has been completed for conservation laws in two space dimensions, where the study of selfsimilar problems has yielded some rigorous results. I will illustrate with an exposition of a model problem involving Mach stems for a nonlinear wave equation. This example is joint work with Suncica Canic and Eun Heui Kin.
