This class will describe a blend of classical and  recent developments in existence and stability of shocks and periodic patterns in hyperbolic and hyperbolic-parabolic balance laws, including solutions possessing discontinuities.

Topics to be covered may include Kawashima-Shizuta dissipation conditions and nonformation of singularities in localized solutions; Chapman-Enskog expansion, subcharacteristic conditions, and existence of smooth and discontinuous profiles;  Kreiss-Majda theory of inviscid shock stability, and relations to viscous/relaxation stability; Whitham  equations and modulation of periodic patterns in 1- and multi-D; Bar-Nepomnasshchy/Barker (computer aided) analysis of stability of thin film flows; nonlinear damping as  generalization of Kreiss symmetrizers and Kawashima compensators; Eckhaus amplitude equations in the presence of conservation laws.