Hyperbolic systems of conservation laws
The Theory of Classical and Nonclassical Shock Waves
These lecture notes were prepared for a Nachdiplom-Vorlesungen course at the Forschungsinstitut für Mathematik (FIM), ETH Zurich, in Fall 2000. I am grateful to the Mathematics Department faculty, particularly Rolf Jeltsch and Michael Struwe, for the opportunity to lecture in such an inspiring setting. Some of this material was previously taught as an advanced graduate course at the École Polytechnique (Palaiseau) from 1995-1999, at the Instituto Superior Técnico (Lisbon) in Spring 1998, and at the University of Wisconsin (Madison) in Fall 1998. The project began in Summer 1995 with a series of lectures at the Tata Institute of Fundamental Research (Bangalore). A key objective of this course is to present a self-contained overview of the well-posedness theory for nonlinear hyperbolic systems of first-order partial differential equations in divergence form, also known as hyperbolic systems of conservation laws. These equations are significant in various fields of continuum physics, particularly when formulating fundamental balance laws for mass, momentum, and total energy of fluids or solids, while neglecting small-scale mechanisms such as viscosity and heat conduction. Solutions to these hyperbolic conservation laws can develop singularities, like shock waves, in finite time, even from smooth initial conditions.
