Electron. J. Differential Equations, Vol. 2019 (2019), No. 65, pp. 1-22.

Stability of Riemann solutions to pressureless Euler equations with Coulomb-type friction by flux approximation

Qingling Zhang

We study the stability of Riemann solutions to pressureless Euler equations with Coulomb-type friction under the nonlinear approximation of flux functions with one parameter. The approximated system can be seen as the generalized Chaplygin pressure Aw-Rascle model with Coulomb-type friction, which is also equivalent to the nonsymmetric system of Keyfitz-Kranzer type with generalized Chaplygin pressure and Coulomb-type friction. Compared with the original system. The approximated system is strictly hyperbolic, which has one eigenvalue genuinely nonlinear and the other linearly degenerate. Hence, the structure of its Riemann solutions is much different from the ones of the original system. However, it is proven that the Riemann solutions to the approximated system converge to the corresponding ones to the original system as the perturbation parameter tends to zero, which shows that the Riemann solutions to the nonhomogeneous pressureless Euler equations is stable under such kind of flux approximation. In a word, we not only analyze the mechanism of the occurrence of the delta shocks, but also generalize the result about the stability of Riemann solutions with respect to flux perturbation from the well-known homogeneous case to the nonhomogeneous case.

Submitted July 7, 2017. Published May 10, 2019.
Math Subject Classifications: 35L65, 35L67, 35B30.
Key Words: Stability of Riemann solutions; pressureless Euler equations; delta shock wave; Coulomb-type friction; flux approximation.

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Qingling Zhang
School of Mathematics and Computer Sciences
Jianghan University
Wuhan 430056, China
email: zhangqingling2002@163.com

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