Communications in Mathematical Sciences

Volume 11 (2013)

Number 4

Existence and large time stability of traveling wave solutions to nonlinear balance laws in traffic flows

Pages: 1011 – 1037

DOI: https://dx.doi.org/10.4310/CMS.2013.v11.n4.a6

Authors

Shih-Wei Chou (Department of Mathematics, National Central University, Jhongli City, Taiwan)

John M. Hong (Department of Mathematics, National Central University, Jhongli City, Taiwan)

Ying-Chieh Lin (Department of Mathematics, National Central University, Jhongli City, Taiwan)

Abstract

In this paper, we consider a nonlinear hyperbolic system of balance laws in Eulerian coordinates arising from a continuum traffic flow model whose source term consists of a relaxation and an extra term related to the non-uniform road widths. We establish the existence and large-time stability of traveling wave solutions for the initial value problem of such system. Contrast to previous results, there are four types of traveling waves according to the stability of the equilibria at $x=±1$. Under the entropy condition, the original and modified subcharacteristic conditions, together with a subsonic condition, we show by the weighted energy method that each type of traveling waves is asymptotically stable under small perturbations.

Keywords

conservation laws, nonlinear balance laws, traveling waves, stability, relaxation, weighted energy estimates, traffic flow

2010 Mathematics Subject Classification

35B20, 35B40, 35B45, 35L65, 76L05, 90B20

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Published 15 June 2013