Well-posedness of contact discontinuity solutions and vanishing pressure limit for the Aw–Rascle traffic flow model

Zijie Deng; Wenjian Peng; Tian-Yi Wang; Haoran Zhang; Zijie Deng; Wenjian Peng; Tian-Yi Wang; Haoran Zhang

doi:10.3934/mine.2025014

Mathematics in Engineering

2025, Volume 7, Issue 3: 316-349. doi: 10.3934/mine.2025014

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Well-posedness of contact discontinuity solutions and vanishing pressure limit for the Aw–Rascle traffic flow model

1.
School of Mathematics and Statistics, Wuhan University of Technology, Wuhan, China
2.
Department of Mathematics, City University of Hong Kong, Hong Kong, China
3.
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China

Received: 23 April 2025 Revised: 17 May 2025 Accepted: 19 May 2025 Published: 09 June 2025

This paper investigates the well-posedness of contact discontinuity solutions and the vanishing pressure limit for the Aw–Rascle traffic flow model with general pressure functions. The well-posedness problem is formulated as a free boundary problem, where initial discontinuities propagate along linearly degenerate characteristics. To address vacuum degeneracy, a condition at density jump points is introduced, ensuring a uniform lower bound for density. The Lagrangian coordinate transformation is applied to fix the contact discontinuity.The well-posedness of contact discontinuity solutions is established, showing that compressive initial data leads to finite-time blow-up of the velocity gradient, while rarefactive initial data ensures global existence. For the vanishing pressure limit, uniform estimates of velocity gradients and density are derived via level set argument. The contact discontinuity solutions of the Aw–Rascle system are shown to converge to those of the pressureless Euler equations, with matched convergence rates for characteristic triangles and discontinuity lines. Furthermore, under the conditions of pressure, enhanced regularity in non-discontinuous regions yields convergence of blow-up times.
- Aw–Rascle traffic flow model,
- well-posdness,
- blow-up,
- lower bound of density,
- vanishing pressure limit,
- convergence rate
Citation: Zijie Deng, Wenjian Peng, Tian-Yi Wang, Haoran Zhang. Well-posedness of contact discontinuity solutions and vanishing pressure limit for the Aw–Rascle traffic flow model[J]. Mathematics in Engineering, 2025, 7(3): 316-349. doi: 10.3934/mine.2025014

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Abstract

This paper investigates the well-posedness of contact discontinuity solutions and the vanishing pressure limit for the Aw–Rascle traffic flow model with general pressure functions. The well-posedness problem is formulated as a free boundary problem, where initial discontinuities propagate along linearly degenerate characteristics. To address vacuum degeneracy, a condition at density jump points is introduced, ensuring a uniform lower bound for density. The Lagrangian coordinate transformation is applied to fix the contact discontinuity.The well-posedness of contact discontinuity solutions is established, showing that compressive initial data leads to finite-time blow-up of the velocity gradient, while rarefactive initial data ensures global existence. For the vanishing pressure limit, uniform estimates of velocity gradients and density are derived via level set argument. The contact discontinuity solutions of the Aw–Rascle system are shown to converge to those of the pressureless Euler equations, with matched convergence rates for characteristic triangles and discontinuity lines. Furthermore, under the conditions of pressure, enhanced regularity in non-discontinuous regions yields convergence of blow-up times.

References

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