Distribution solutions of nonlinear systems of conservation laws
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 2007: Volume 190, Number 889
ISBNs: 978-0-8218-3990-4 (print); 978-1-4704-0495-6 (online)
MathSciNet review: 2355635
MSC: Primary 35L65
The local structure of solutions of initial value problems for nonlinear systems of conservation laws is considered. Given large initial data, there exist systems with reasonable structural properties for which standard entropy weak solutions cannot be continued after finite time, but for which weaker solutions, valued as measures at a given time, exist. At any given time, the singularities thus arising admit representation as weak limits of suitable approximate solutions in the space of measures with respect to the space variable.
Two distinct classes of singularities have emerged in this context, known as delta-shocks and singular shocks. Notwithstanding the similar form of the singularities, the analysis of delta-shocks is very different from that of singular shocks, as are the systems for which they occur. Roughly speaking, the difference is that for delta-shocks, the density approximations majorize the flux approximations, whereas for singular shocks, the flux approximations blow up faster. As against that admissible singular shocks have viscous structure.
Table of Contents
- 1. General distribution solutions
- 2. Delta-shocks
- 3. Singular shocks