This semester we will cover the basic shock wave theory of hyperbolic conservation laws, including mathematical and numerical aspects of the theory. A new feature this semester is that we will emphasize some deep and profound mathematical connections between kinetic theory and conservation laws, as first developed by P.-L. Lions, B. Perthame, and E. Tadmor, and more recently studied and extended by many others. Kinetic equations go back to the nineteenth century and the work of Boltzmann, which unified various perspectives on fluid mechanics. The kinetic equations are characterized by a density function that satisfies a nonlinear conservation law in the phase space. The kinetic approach allows nonlinear conservation laws to be written as linear kinetic (or semi-kinetic) equations acting on nonlinear quantities. Moreover, it will allow us to use the (linear) Fourier transform, regularization and moments methods to provide new approaches for proving uniqueness, regularizing effects, a priori bounds, and convergence of numerical methods.
Book: Kinetic Formulation of Conservation Laws by by Benoit Perthame
Hardcover: 216 pages, Publisher: Oxford University Press, USA, ISBN-10: 0198509138, ISBN-13: 978-0198509134
Supporting literature: L.C. Evans: Partial Differential Equations, 1998. Amer. Math. Soc.. ISBN: 0-8218-0772-2. Graduate Studies in Mathematics, Volume 19.