MA8103 Non-Linear Hyperbolic Conservation Laws

General information

Background: In the course we study a class of nonlinear partial differential equation called hyperbolic conservation laws. These equations are fundamental in our understanding of continuum mechanical systems, and can be used to describe mass, momentum and enery conservation in mechanical systems. Examples of the use of conservation laws you may have seen in TMA4305 Partial differential equations and TMA4195 Mathematical modeling as well as in courses in physics and fluid mechanics. The equations share many properties that make numerical computations difficult. The equations may, for instance, develop singularities in finite time from smooth initial data. These equations have been extensively studied due to their importance in applications. Examples of applications include weather forecasting, flow of oil in a petroleum reservoir, waves breaking at a shore, and in gas dynamics.

Lecturers: Harald Hanche-Olsen and Helge Holden

Textbook: H. Holden and N. H. Risebro: Front Tracking for Hyperbolic Conservation Laws, Springer, Second edition 2015. The book exists as an eBook, and NTNU students can read and download it free of charge. There will be an inexpensive paperback edition called "MyCopy" that sells for EUR 24.95 (incl. shipping).

Lecture plan

Week Date Material covered
◐ means: Harald lectures
◑ means: Helge lectures
3 Mon Jan. 15 ◑ p. –9.
Tue Jan. 16 ◐ p. 10–15 (very briefly), p. 53–56 (approx)
4 Mon Jan. 22 ◐ p. 56–59 (note)
Tue Jan. 23 ◐ p. 60–64 (and the estimate (2.35) in the general setting, with an extra factor 2)
5 Mon Jan. 29 ◑ p. 64–71
Tue Jan. 30 ◑ p. 71-75.
6 Mon Feb. 05 ◑ p. 75–80
Thu Feb. 08 ◑ p. 80-84
7 Mon Feb. 12 ◑ p. 85–87, 95-98
Thu Feb. 15 ◐ p. 98–106 finished in the middle of a proof, at eq (3.28)
8 Mon Feb. 19 ◐ p. 106–109. For a different take on Kolmogorov–Riesz, see here.
Thu Feb. 22 ◐ p. 109–115 starting with the proof of Thm 3.8. Skipped the calculation leading up to Thm 3.10.
9 Mon Feb. 26 ◑ p. 171–175.
Thu Mar. 1 ◑ p. 175–180. Recommended exercises: Ch. 1: 8,9. Ch. 2: 1,3,5,10,17. Ch.3: 3,5.
10 Mon Mar. 5 ◑ p. 180–185.
Thu Mar. 8 ◐ p. 186–189 and 223–225. We skip the derivation of the shallow water equations from Navier–Stokes.
11 Mon Mar. 12 ◐ p. 228–235.
Thu Mar. 15 ◐ p. 235–243.
12 Mon Mar. 19 ◐ p. 243–248. I did not cover the detailed existence proof for viscous profiles in the shallow water equations. On the other hand, I have essentially covered theorem 5.16 on p. 253:
See this handwritten note on a better approach to the Hugoniot locus.
Thu Mar. 22 ◑ p. 249–252
13 Mon Mar. 26 Easter, no lectures
Thu Mar. 29
14 Mon Apr. 2
Thu Apr. 5 ◑ p. 252–260
15 Mon Apr. 9 ◑ p. 260–263
Thu Apr. 12 ◑ p. 263–265 (middle of the page)
16 Mon Apr. 16 ◐ p. 265–270 (skipping quite a bit of detail)
Thu Apr. 19 ◑ p. 283–288
17 Mon Apr. 23 ◑ p. 288-289, 293-295, convergence just barely described
Thu Apr. 26 ◐ Equivalence of Eulerian and Lagrangian formulations of conservation laws; loosely based on a paper by Dafermos (1993). It's on Springer Link – look for the small download link beneath the abstract.
This is the final lecture.
I wrote up a summary of this lecture, mostly for my own benefit – but you're welcome to it, if you find it useful.


The exam is oral. At the day of the exam you meet at 9:00 am in our office. The time for each candidate will be decided then.

Curriculum: Ch. 1, p. 1-16; Ch. 2; Ch. 3, p. 91-126; Ch. 4, p. 171-188; Ch. 5, p. 223-270; Ch. 6, p. 283-297, and notes by Harald on relation between Eulerian and Lagrangian formulation.

Regulations concerning the exam: For the first 20 min the candidate presents a lecture using blackboard only on one of the topics given below. Notes are allowed. It should be a lecture presenting the main parts of the topic, or if the topic is big, a selected part of it. Proofs are encouraged. After that, and for approximately 25 min the candidate will be asked questions about all of the curriculum. Here there will of course be no notes. The topic is decided at the beginning of the exam.

Topics: (A): Ch. 2; (B): Ch. 3; (C): Ch. 4; (D): Ch. 5 (except Sec. 5.6); (E) Sec 5.6 & notes regarding relation between Euler and Lagrangian formulation.

2022-05-15, Harald Hanche-Olsen