# 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. You can purchase a paperback edition called MyCopy for € 24.95 (incl. shipping). See the upper right on the linked page.

### Time and place

For regular weeks the lectures are

Weekday Time Room
Monday 12:15-14 822, Sentralbygg 2
Friday 10:15-12 734, Sentralbygg 2

The lectures will be in English, and there will be an oral exam at the end of the semester.

### Lecture plan

Week Date Who Material covered
means: Harald lectures
means: Helge lectures
3 Fri 17 Jan pp. –8.
4 Mon 20 Jan pp. 9–15, pp. 53–55
Fri 24 Jan pp. 55–59
5 Mon 27 Jan pp. 60–66
Fri 31 Jan pp. 67–74
6 Mon 3 Feb pp. 74–79
Fri 7 Feb pp. 80–83 (note)
7 Mon 10 Feb pp. 84–87. Recommended exercises: Ch.2: 1,3,10,16
Fri 14 Feb pp. 427–430 (note on BV · updated 2020-02-21), 95–97
8 Mon 17 Feb pp. 98–105
Fri 21 Feb pp. 105–109
9 Mon 24 Feb exercises; Kolmogorov–Riesz; pp. 109–110
Fri 28 Feb pp. 110–115; 171–173
10 Mon 2 Mar pp. 174–176
Fri 6 Mar pp. 177–185
11 Mon 9 Mar pp. 186–188, 223–229. We skip the derivation of the shallow water equations from Navier–Stokes.
Fri 13 Mar Due to the new coronavirus situation, we are now experimenting with digital solutions.
pp. 229–231: part 1part 2
12 Mon 16 Mar pp. 231–238: “slides”; pencast: part 1 · part 2 · part 3
Tue 17 Mar pp. 238–240: “slides”; pencast: part 4 (improved resolution)
Fri 20 Mar pp. 241–248: part a; part b; part c; part d
13 Mon 23 Mar pp. 249–256 (except Lemma 5.18): “slides”; pencast: part 1 · part 2
Fri 27 Mar pp. 256–277: part a; part b; part c
14 Mon 30 Mar pp. 283–312 (no proofs): part a; part b
Fri 3 Apr Correspondence between Eulerian and Lagrangian formulations of conservation laws
Note · “slides” · pencast: part 1 · part 2 · part 3
15 Mon 6 Apr Easter, no lectures
Fri 10 Apr
16 Mon 13 Apr
Fri 17 Apr Exercises: 3.1, 3.3, 3.10; 4.1; 5.8, 5.9, 5.10
Exercise 3.1 is wrong. To fix, assume $u^0_j=[j\le0]$ instead of $u^0_j=[j\ge0]$ (using the Iverson bracket)
For Exercise 3.10, read this before starting on it.
And for Exercise 5.9, read this first.
Slides” · pencasts: 3.1 · 3.3 · 3.10 · 4.1 (newest) · 5.8 · 5.9 (new) · 5.10 (newer)
17 Mon 20 Apr Sec. 4.5. pp. 212–217. lecture
And that's it!