New developments of delta shock waves and its applications in systems of conservation laws

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:252

Article

Yang, Hanchun¹  Zhang, Yanyan¹ 

[1] Yunnan Univ, Dept Math, Kunming 650091, Peoples R China

关键词: Hyperbolic systems of conservation laws;    Delta shock wave;    Generalized Rankine-Hugoniot relation;    Entropy condition;    Limiting viscosity approach;    Numerical simulations;   

DOI  :  10.1016/j.jde.2012.02.015

来源: Elsevier

【 摘 要 】

A theory of delta shock waves with Dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws is established. In this paper, we solve constructively the Riemann problems for the system under consideration. In solutions, we find another kind of delta shock waves on which both state variables simultaneously contain the Dirac delta functions. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. We strictly prove it satisfies the system in the sense of distributions. The generalized Rankine-Hugoniot relation and entropy condition are proposed to solve the delta shock waves. Furthermore, we show all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations. The generalized Rankine-Hugoniot relation is also confirmed. In particular, our theory on the delta shock waves possesses the generality and practicability which can be conveniently and successfully applied to those systems investigated by Korchinski (1977), Tan, Zhang and Zheng (1991), Ercole (2000), Cheng and Yang (2011), etc. And we also give a simplified approach to solve a 2-D Riemann problem for the system studied by Tan and Zhang (1990) for the case 2J(+)2J(-) and obtain the explicit formulae of the delta shock waves. Finally, the numerical simulations completely coinciding with theoretical analysis are presented. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free