An introduction to hyperbolic conservation laws

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.

Date and time info
Wednesday 09:15 - 10:45

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.

Audience
MSc students, PhD students, Postdocs

Language
English