Electron. J. Differential Equations, Vol. 2017 (2017), No. 156, pp. 1-18.

Axisymmetric solutions of a two-dimensional nonlinear wave system with a two-constant equation of state

Guodong Wang, Yanbo Hu, Huayong Liu

Abstract:
We study a special class of Riemann problem with axisymmetry for two-dimensional nonlinear wave equations with the equation of state $p=A_1\rho^{\gamma_1}+A_2\rho^{\gamma_2}$, $A_i<0$, $-3<\gamma_i<-1$ (i=1,2). The main difficulty lies in that the equations can not be directly reduced to an autonomous system of ordinary differential equations. To solve it, we use the axisymmetry and self-similarity assumptions to reduce the equations to a decoupled system which includes three components of solution. By solving the decoupled system, we obtain the structures of the corresponding solutions and their existence.

Submitted February 17, 2017. Published June 28, 2017.
Math Subject Classifications: 35L65, 35J70, 35R35.
Key Words: Nonlinear wave system; generalized Chaplygin gas; axisymmetry; decoupled system.

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Guodong Wang
School of Mathematics and Physics
Anhui Jianzhu University of China
Hefei 230601, China
email: gdwang@163.com
  Yanbo Hu
Department of Mathematics
Hangzhou Normal University of China
Hangzhou 310036, China
email: yanbo.hu@hotmail.com
  Huayong Liu
School of Mathematics and Physics
Anhui Jianzhu University of China
Hefei 230601, China
email: liuhy@ahjzu.edu.cn

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