Hyperbolic Systems of Conservation Laws

Thomas Hillen (University of Alberta)

Jan 1, 2024 — Apr 30, 2024

About the course

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 reaction-diffusion 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" (BV-spaces). 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 Rankine-Hugoniot conditions, the entropy conditions, the Lax-condition 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.


This course is available for registration under the Western Dean's Agreement. To register, you must obtain the approval of the course instructor and you must complete the Western Dean's agreement form , using the details below. The completed form should be signed by your home institution department and school of graduate studies, then returned to the host institution of the course.

Enrollment Details

Course Name
Hyperbolic Systems of Conservation Laws
Jan 1, 2024 — Apr 30, 2024
Course Number
* UofA students: MATH 667 Topics in Differential Equations I (local students) * All other students: MATH 667 Topics in Differential Equations I (external students)
Section Number
Section Code


For help with completing the Western Dean’s agreement form, please contact the graduate student program coordinator at your institution. For more information about the agreement, please see the Western Dean's Agreement website

Other Course Details

Lecture Times

Lectures will take place Monday, Wednesday and Friday from 13:00-13:50 (Mountain Time).

Remote Access

Lectures are online on zoom.