Hyperbolic Systems of Conservation Laws 

University of Alberta Mathematical and Statistical Sciences Dr. Thomas Hillen 4923395, thillen@ualberta.ca 
Texts:
Syllabus:In
this course we will study the theory of hyperbolic systems of conservation
laws. Hyperbolic
systems arise in many areas of applied mathematics, including gas
dynamics, thermodynamics, population
dynamics, or traffic flow. In
contrast to dissipative systems (like reactiondiffusion equations),
solutions of hyperbolic systems with smooth initial data can generate
“shocks” in finite time. The solution is no longer differentiable and
weak solutions have to be studied. We
will develop the existence and uniqueness theory for solutions of
conservation laws in spaces of functions of “bounded variation”
(BVspaces). At the beginning we will recall distributions and weak
limits of measures. Then we study “broad” solutions
(solutions which do not form shocks). After that we investigate
discontinuous solutions in detail, we will derive the RankineHugoniot
conditions, the entropy conditions, the Laxcondition
and we will discuss the vanishing viscosity method. We will
classify strictly hyperbolic systems into genuinely nonlinear or linear
degenerate systems. Then we use solutions to the Riemann
problem to define a front tracking algorithm. This method is merely an
analytical tool to obtain results on local and global existence and on
uniqueness. Prerequisites: Some basic knowledge on PDE’s. 
Assignments:
Presentations:
Course Notes 
