May 31 (final lecture): I showed that the p-system and the Euler equations are strictly hyperbolic systems (under natural assumptions). The solution of the Riemann problem for convex scalar conservation laws was derived. Finally, I discussed briefly the Kruzkov entropy condition and the correpsonding existence and uniqueness result.
Updated Syllabus/achievement requirements: Material covered in Evans' book: Chapter 8 (Calculus of variations) except 8.3, 8.4.4 (b), Chapter 9 (Nonvariational techniques) except 9.4, 9.5, Chapters 11 + 3 (Conservation laws): 11.1, 11.4.2, 11.4.3 (including Theorem 3 but not the proof), 3.4.2, 3.4.3, 3.4.5.
Written lecture notes on weak convergence methods available here
May 24: I continued with the theory of systems of conservation laws; The focus was on the notion of strict hyperbolicity and the derivation of some useful results about matrices that satisfy the strict hyperbolicity condition.
Next week: The plan is to solve the (scalar) Riemann problem.
May 10/11: This week I started with conservation laws. I introduced the equations, defined weak solutions and explained why we need them, and derived the Rankine-Hugoniot jump condition. Finally, I introduced the (traveling wave/vanishing viscosity) entropy condition for scalar conservation laws, which is a condition that encodes the missing physics in conservation laws and consequently restores the uniqueness of weak solutions.