Multi-dimensional Systems of Conservation Laws: An Introductory Lecture

Abstract

These notes are written after the crash course given at the ICMS conference on Hyperbolic conservation laws. We intend to review several aspects of the theory of the Cauchy problem and the Initial-boundary value problem (IBVP). On the one hand, we give a thorough account of the theory for linear, constant coefficient operators, following Gårding, Hersch, Kreiss and others. Hyperbolicity raises interesting questions in real algebraic geometry, a topic to which Petrowski’s school (in particular Oleĭnik) contributed. Next, we turn towards quasilinear systems and recall the interplay between entropies and symmetrizability. This leads us to the local existence of a classical solution. The global-in-time Cauchy problem necessitates weak solutions; these must be selected by admissibility criteria. We give a review of the various criteria that have been elaborated so far. Some of them lead us to the ‘viscous’ approximation of hyperbolic systems. We review the structural properties of these models, whose paradigm is the Navier-Stokes-Fourier (NSF) system of gas dynamics. This is more or less Kawashima’s theory, in the simplified description that we have given in recent papers. We end with results about singular limits, such as the convergence of NSF towards Euler-Fourier when Newtonian viscosity tends to zero, and the analysis of the principal sub-systems introduced by Boillat and Ruggeri. Despite the length of these notes, they contain only very few proofs. We focus instead on the concepts and the theorems of the theory.KeywordsWeak SolutionCauchy ProblemRiemann ProblemFree Boundary ProblemEntropy InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.