# An introduction to hyperbolic conservation laws

**Lecturer:**Stefano Modena**Date:**Wednesday 09:15 - 10:45**Room:**MPI MiS, A3 02**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Content (Keywords):**Entropy solutions, Kružkov's theorem, Riemann problem, Wavefront

tracking, Glimm scheme**Prerequisites:**Standard basic results in mathematical analysis. No previous knowledge in PDEs or Conservation Laws is assumed.

## Abstract

Systems of conservation laws are evolutionary nonlinear PDEs with several applications coming from both physics and engineering, in particular from fluid dynamics and traffic models. Despite recent progress, the mathematical understanding of these equations is still incomplete. In particular, general well-posedness results for the Cauchy problem are presently available only for systems of conservation laws *in one space dimension*, while very little is known for systems* in several space dimensions*.

The course aims to be an introduction to the well-posedness theory for the Cauchy problem associated to a system of conservation laws in one space dimension.

The approach I would like to adopt is the following. On one side, I will try to present a comprehensive overview of the main well-posedness results available in the literature. On the other side, I will provide the details of the proofs of such results in a simplified setting (a single 1D equation): here, indeed, the same techniques which have been successfully applied in the case of systems can be understood with much less effort.

This is a tentative list of topics, to be adapted according to the wishes of the audience:

- Classical solutions: the method of characteristics.
- Weak solutions, the Rankine-Hugoniot condition, admissibility criteria.
- Existence results: the wavefront tracking algorithm and the Glimm scheme.
- Kružkov's entropy theorem.
- Some notes on the Cauchy problem for systems.

## Regular Lectures (Summer 2016)

**Information Theory in the Neurosciences**- Jürgen Jost
- Date: Friday 13:30 - 15:00, MPI MiS, A3 01

**Domain and wall pattern in ferromagnets**- Felix Otto
- Date: Tuesday 09:15 - 11:00, MPI MiS, A3 01

**Introduction to stochastic PDE**- Benjamin Gess
- Date: Thursday 16:00 - 18:00, MPI MiS, A3 01

**Categories in Algebra and Geometry**- Tobias Fritz
- Date: Monday 11:00 - 13:00, MPI MiS, A3 02

**Introduction to the Theory of Neural Networks**- Guido Montúfar
- Date: Thursday 11:15 - 12:45, MPI MiS, A3 02

**Min-max Techniques in Geometry**- Jim Portegies, Slava Matveev
- Date: Thursday 13:30 - 15:00, MPI MiS A3 02

**General Relativity**- Stefan Hollands
- Date: tba

**Advanced Statistical Mechanics**- Klaus Kroy
- Date: Tuesday 9:15 - 10:45 and Thursday 9:15 - 10:45, ThHS

**An introduction to hyperbolic conservation laws**- Stefano Modena
- Date: Wednesday 09:15 - 10:45, MPI MiS A3 02

**Mathematical Hydrodynamics**- László Székelyhidi
- Date: Tuesday and Wednesday 11:15 - 12:45, Neues Augusteum A-314

**Quantum Field Theory on Curved Spacetimes**- Rainer Verch, Thomas-Paul Hack
- Date: Wednesday 11:15 - 12:45 and Thursday 13:30 - 15:00, ITP, Brüderstr. 16, Room 114

- IMPRS-Ringvorlesung
- Peter F. Stadler, Jürgen Jost, Tobias Fritz
- Date: Tuesday 13:30 - 15:00, MPI MiS G3 10

**Hauptseminar "Rough Path and Quantum Fields III"**- Stefan Hollands, Felix Otto, Max von Renesse (Organizers)
- Date: Monday 17:00 - 18:30, MPI MiS G3 10

**Likbez**- Slava Matveev (Organizer)
- Date: Friday 14:15 - 15:45, MPI MiS A3 01