The Shock Reflection Phenomenon for Scalar Conservation Law with Dirac Measure Source Term

Abstract

This paper is mainly concerned with the shock reflection phenomenon for convex scalar conservation law with Dirac measure source term. The Riemann solutions are constructed completely and then the impact of the strength of source term on the Riemann solutions is considered in detail. In order to illustrate the shock reflection phenomenon, the initial data with three pieces of constant states are considered and the interactions between a backward shock wave plus a stationary wave discontinuity and a rarefaction wave are displayed in all kinds of situations. Furthermore, the global solutions are constructed completely and the large time asymptotic states are obtained. In some certain situations, the shock reflection phenomenon is captured when a rarefaction wave interacts with a stationary wave discontinuity and then is divided into a transmitted rarefaction wave and a reflected shock wave at the critical point. In addition, it is shown that the Riemann solutions are stable with respect to the particular small perturbations of the initial data.