WS 20/21: Hyperbolic Conservation Laws (tutorials)

This course has come to the end. See student evaluation reports (in Polish).

Lecutre: Agnieszka Świerczewska-Gwiazda (Wednesday, 14:15 - 15:45)
Tutorial: Kuba Skrzeczkowski (Wednesday, 16:00 - 17:30)
USOS: link

This is a "topics" course for 4-5th year and PhD students discussing current state of the art in hyperbolic conservation laws. Although they are first order PDEs, their mathematical analysis is much more complicated comparing to elliptic or parabolic problems that usually enjoy some nice regularity properties (especially smoothness of solutions). On the contrary, solutions to hyperbolic equations form discontinuities even for smooth initial conditions. The course will concentrate on very few general results that are currenty available (Kruzkhov theory, compensated compactness, semigroup approach, kinetic formulation). Time permitting, we discuss recent developments concerning recently solved Onsager's conjecture and applications of the presented theory to some singular limits in reaction-diffusion equations.

Preliminaries: first courses in Functional Analysis and PDEs form sufficient background to attend this course. In particular, one should have experience with Banach spaces, Sobolev spaces, weak formulations and weak convergence.

Grading: There will be one (small) problem to be submitted each weak. The course will be concluded with an oral exam.

Homeworks are here.

Topic notes:

• Topic 1. Introduction. Examples of CL. Essentials from mathematical analysis.
• Topic 2. Some properties of entropy solutions.
• Topic 3. Strong continuity in time of entropy solutions.
• Topic 4. Continuity estimates for vanishing viscosity method. Existence of entropy solutions.
• Topic 5. Comments on Kruzkhov doubling variables method.
• Topic 6. Toolbox for compensated compactness.
• Topic 7. Introduction to kinetic functions.
• Topic 8. Kinetic approximation for SCL.

Videos and blackboard prints from each class are uploaded in the faculty moodle.