Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term

Yu Zhang; Yanyan Zhang

doi:10.3934/cpaa.2019073

Article Contents

2019, Volume 18, Issue 3: 1523-1545. Doi: 10.3934/cpaa.2019073

This issue Previous Article Applications of generalized trigonometric functions with two parameters Next Article Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential

Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term

Yu Zhang^1, , and
Yanyan Zhang^2,

1.
Department of Mathematics, Yunnan Normal University, Kunming 650500, China
2.
College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

* Corresponding author
* Corresponding author

Received: February 2017

Revised: February 2018

Published: May 2019

This work is supported by NSF of China (11501488), Yunnan Applied Basic Research Projects (2018FD015), the Scientific Research Foundation Project of Yunnan Education Department (2018JS150), Nan Hu Young Scholar Supporting Program of XYNU

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Abstract

The Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term are studied. The Riemann solutions exactly include two kinds: delta-shock solutions and vacuum solutions. In order to see more clearly the influence of the source term on Riemann solutions, the generalized Rankine-Hugoniot relations of delta shock waves are derived in detail, and the position, propagation speed and strength of delta shock wave are given. It is also shown that, as the source term vanishes, the Riemann solutions converge to the corresponding ones of the homogeneous system, which is just the generalized zero-pressure flow model and contains the one-dimensional zero-pressure flow as a prototypical example. Furthermore, the generalized balance relations associated with the generalized mass and momentum transportation are established for the delta-shock solution. Finally, two typical examples are presented to illustrate the application of our results.

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Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 76N10.

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Figure 1. The Riemann solution of (1) and (2) when $u_-<0<u_+$ and $\beta>0$ for a given time $t$ before the time $(f^{-1}(0)-u_-)/\beta$. The left is the $(u, v)$-phase plane, and the right is the corresponding $(x, t)$-characteristic plane

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Figure 2. The delta-shock solution of (1) and (2) for $\beta>0$, where the propagation speed of delta shock wave is positive on the left and negative on the right when $t = 0$

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