Structural stability of solutions to the Riemann problem for a scalar conservation law

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:389

Article

Shen, Chun

关键词: Delta standing wave;    Vacuum state;    Riemann problem;    Wave interaction;    Discontinuous flux function;    Nonstrictly hyperbolicity;   

DOI  :  10.1016/j.jmaa.2011.12.044

来源: Elsevier

【 摘 要 】

The aim of this paper is to study the structural stability of solutions to the Riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients. It is proved that the Riemann solution is possibly instable when one of the Riemann initial data is at the vacuum. Furthermore, we point out that the Riemann solution is also possibly instable even when the Riemann initial data stay far away from vacuum. In order to deal with it, we perturb the Riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively. It is also proved that the Riemann solutions are unstable in some certain situations under the local small perturbations of the Riemann initial data by letting the perturbed parameter epsilon tend to zero. In addition, the interaction of the delta standing wave and the contact vacuum state is considered which appear in the Riemann solutions. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free