Hyperbolic Systems of Conservation 


Math 667 

Topics in 

Differential Equations


Fall 2007, TR 14:00-14:30,  TBA
University of Alberta
Mathematical and Statistical Sciences
Dr. Thomas Hillen
492-3395, thillen@ualberta.ca


  1. A. Bressan, Hyperbolic Systems of Conservation Laws. Oxford University Press, 2000.

  2. P. G. LeFloch, Hyperbolic Systems of Conservation Laws. Birkhaeuser, Basel, 2001.
  3. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer, Heidelberg, 2000. 
  4. Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method. Chapman & Hall, CRC Press, 2003.
  5. R. LeVeque, Numerical Methods for Conservation Laws. Birkhaeuser, Basel 1990.
  6. D. Kroener, Numerical Schemes for Conservation Laws. Wiley and Tauber, New York, 1997.


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. 

Prerequisites: Some basic knowledge on PDE’s.


 Homework 70 %, in class presentation 30 %


 Dr. Thomas Hillen, 492-3395, thillen@ualberta.ca
office hours: TR 4-5 PM, CAB 575.



  1. Bressan, p 38, problems 1, 2, 3, 4, 5
    (due: Tues, Sep. 25, 2PM)

  2. Bressan, p 71, problems 2, 3, 4, 5, 6
    (due: Thurs, Oct.11, 2PM)

  3. Bressan, p 71, problem 7,
                 p 88, problems 1, 2, 5, 7
    (due: Thurs, Oct. 25, 2PM)

  4. Bressan, p 107, problems 1, 2,  plus some more problems to be chosen later.   
    (due: Thurs, Nov. 15, 2PM)

  5. Some additinal problems      
    (due: Thurs, Nov. 29, 2PM)


  1.  Isentropic gas dynamics.  Bressan Sec. 5.5, p. 103-106.
  2. Vanishing viscosity method I. L.C. Evans, PDE, AMS Providence 1998, Sec. 11.4 p. 599-603.
  3. Vanishing viscosity method II. L.C. Evans, PDE, AMS Providence 1998, Sec. 11.4 p. 599-603.
  4. Numerical methods for linear equations. R. LeVeque,  Numerical Methods for Conservation Laws, Birkh\"auser, Basel, 1992.  p. 97-102 and 110-112. Show table 10.1 on page 101.
  5. Godumov's method. R. LeVeque, Numerical Methods for Conservation Laws   Birkh\"auser, Basel, 1992. p. 136-140 (not Courant-Isaacson-Rees). 
  6. Two out of three  Counterexamples.  Bressan-skript. Examples 9, 10, 11, p. 61 - 66.
  7. Traffic Flow. G.B. Whitham, Linear and Nonlinear Waves. Wiley, 1974 pages 68-80. 
  8. Flood Waves. G.B. Whitham, Linear and Nonlinear Waves. Wiley, 1974, pages 80-91.

Course Notes


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