The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system

Chun Shen; Wancheng Sheng; Meina Sun

doi:10.3934/cpaa.2018022

Article Contents

2018, Volume 17, Issue 2: 391-411. Doi: 10.3934/cpaa.2018022

This issue Previous Article Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction Next Article The regularity of some vector-valued variational inequalities with gradient constraints

The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system

1.
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China
2.
Department of Mathematics, Shanghai University, Shanghai 200444, China
3.
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

* Corresponding author: Chun Shen.
* Corresponding author: Chun Shen.

Received: May 31, 2017

Revised: July 31, 2017

Published: June 2018

Chun Shen is supported by NSFC (11441002) and Shandong Provincial Natural Science Foundation (ZR2014AM024), Wancheng Sheng is supported by NSFC (11371240) and Meina Sun is supported by NSFC (11271176).

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Abstract

The Riemann problem for the scaled Leroux system is considered. It is proven rigorously that the Riemann solutions for the scaled Leroux system converge to the corresponding ones for a non-strictly hyperbolic system of conservation laws when the perturbation parameter tends to zero. In addition, some interesting phenomena are displayed in the limiting process, such as the formation of delta shock wave and a rarefaction (or shock) wave degenerates to be a contact discontinuity.

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Mathematics Subject Classification: Primary:35L65, 35L67;Secondary:35B25.

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Figure 1. The phase plane for the scaled Leroux system (1.2) when $u_{-}<0$, left for $\varepsilon>0$ and right for the limit $\varepsilon\rightarrow0$ situation.

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Figure 2. The phase plane for the scaled Leroux system (1.2) when $u_{-}>0$, left for $\varepsilon>0$ and right for the limit $\varepsilon\rightarrow0$ situation.

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