Lecture Notes on Hyperbolic Conservation Laws

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of the conservation equations and definition of weak solutions. 2. Shocks, Rankine-Hugoniot equations and admissibility conditions. 3. Genuinely nonlinear and linearly degenerate characteristic fields. Solution to the Riemann problem. Wave interaction estimates. 4. Weak solutions to the Cauchy problem, with initial data having small total variation. Proof of global existence via front-tracking approximations. 5. Continuous dependence of solutions w.r.t. the initial data, in the L distance. 6. Uniqueness of entropy-admissible weak solutions. 7. Approximate solutions constructed by the Glimm scheme. 8. Vanishing viscosity approximations. 9. Counter-examples to global existence, uniqueness, and continuous dependence of solutions, when some key hypotheses are removed. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous chapters.