Systems of conservation laws are partial differential equations with several ap- plications coming from both physics and engineering, in particular from the fluid dynamics.
Despite recent progress, the mathematical understanding of these equations is still incomplete. In particular, no general well-posedness theory is presently available for systems of conservation laws in several space variables.
The aim of the course is to discuss existence and uniqueness results for systems of conservation laws in one space variable. Although due to time constraints I will usually give complete proofs in the scalar case only, I will use techniques that have been successfully applied to the case of systems.
The tentative schedule is as follows.
-
Classical solutions, the theory of characteristics.
-
Distributional solutions, Rankine-Hugoniot conditions.
-
Admissibility criteria for distributional solutions.
-
Existence of global in time, admissible distributional solutions. The wave-front tracking algorithm.
-
Kružkov’s theorem.
-
Lax’s solution of the Riemann problem and the Standard Riemann Semigroup.