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Abstract
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The study of quasilinear hyperbolic partial differential equations (also known as conservation laws) presents formidable technical challenges. For example, the solutions to most initial-value 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 self-similar 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.
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