MA8103 Non-Linear Hyperbolic Conservation Laws
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.
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).
|◐ 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||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.