Hyperbolic Conservation Laws
A conservation law is a first-order partial differential equation which asserts that the rate of change of the total amount of a conserved quantity within a domain is governed by a flux function, which controls the loss or increase over the boundary of said domain. The theory of (hyperbolic) conservation laws has widespread applications: Water flow in canals, traffic flows, gas dynamics in pipes, supply chains, etc.
Some prototypical equations that fall under this category include:
- Burgers' equation
- (nonlinear) Wave equation
- Isentropic Euler equations
Rough plan of the lecture:
- Scalar Conservation Laws: Characteristics, shocks, weak solutions and non-uniqueness, entropy conditions, uniqueness (Kruzkov), existence via compensated compactness
- Systems: Weak solutions, entropy, existence for systems of two equations via compensated compactness
- Isentropic Euler in 1D: Existence via compensated compactness
Lecture: The course will be held online in its entirety. Lecture notes and videos will be provided via Moodle.
Exercises: There will be a weekly exercise sheet which will be discussed during a weekly BBB-session. The BBB-session is scheduled for Thursday: 4pm - 6pm. Attendance is voluntary though and if desired by the participants, the sessions can be moved to a different date. Written solutions will be uploaded as well.
Exam: At the end of the semester, there will be an oral exam.
Exam admission: Some sort of participation in the exercises will be required in order to be admitted to the exam. The precise requirements will be announced at the beginning of the course.
The course is aimed at students with an interest and some experience in Analysis. Some knowledge in PDE-theory and/or Functional Analysis could be useful. However, important theorems and tools from both areas will be reviewed in the lecture and further discussed in the exercises.
- Evans, L. C. (2010). Partial differential equations. Providence, R.I.: American Mathematical Society.
- Evans, L. C. (2009): Weak Convergence Methods For Nonlinear Partial Differential Equations (Cbms Regional Conference Series in Mathematics)
- R. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Diff. Equat. 20 (1976), 187–212.
- R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27–70.