FORMATION OF SINGULARITIES FOR AW-RASCLE TRAFFIC MODEL WITH RELAXATION

doi:10.1007/s10473-026-0205-3

Acta mathematica scientia,Series B ›› 2026, Vol. 46 ›› Issue (2): 595-604.doi: 10.1007/s10473-026-0205-3

Previous Articles Next Articles

FORMATION OF SINGULARITIES FOR AW-RASCLE TRAFFIC MODEL WITH RELAXATION

Min DING^*, Xiaohui LI, Jianlin XIANG

Department of Mathematics, School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China

Received:2025-03-18 Revised:2025-05-29 Published:2026-05-22
Contact: ^*Min DING, E-mail: minding@whut.edu.cn
About author:Xiaohui LI, E-mail: lixiaohuizlh@163.com; Jianlin XIANG, E-mail: jianlin.xiang@whut.edu.cn
Supported by:
Min Ding's research was supported by the NSFC (12371226), the Natural Science Foundation of Hubei province (2021CFB452) and the Fundamental Research Funds for the Central Universities (104972025KFYjc0092).

Abstract

Abstract: We study the singularity formation of smooth solutions for Cauchy problem of the Aw-Rascle traffic model with relaxation. Under the subcharacteristic assumption and general law of the velocity deviation, we construct a set of large initial data, and prove that the corresponding smooth solutions blow up in a finite time, and form a cusp singularity in the direction of genuinely nonlinear characteristic. Moreover, under the generic nondegenerate condition on initial data, we give precise description on the blowup time and location.

Key words: Aw-Rascle traffic model with relaxation, subcharacteristic condition, Riemann invariant, blow up

CLC Number:

35L03

Min DING, Xiaohui LI, Jianlin XIANG. FORMATION OF SINGULARITIES FOR AW-RASCLE TRAFFIC MODEL WITH RELAXATION[J].Acta mathematica scientia,Series B, 2026, 46(2): 595-604.

Trendmd

References

[1] Aw A, Klar A, Materne T, Rascle M. Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J Appl Math, 2002, 63: 259-278
[2] Aw A, Rascle M. Resurrection of "second order" models of traffic flow. SIAM J Appl Math, 2000, 60: 916-938
[3] Curr$\acute{\rm o}$ C, Mangananaro N. Riemann problems and exact solutions to a traffic flow model. J Math Phys, 2013, 54: 071503, 16.
[4] Curr$\acute{\rm o}$ C, Fusco D, Mangananaro N. Differential constraints and exact solution to Riemann problem for a traffic flow model. Acta Appl Math, 2012, 122: 167-178
[5] Deng Z J, Peng W J, Wang T Y, Zhang H R. Well-posedness of contact discontinuity solutions and vanishing pressure limit for the Aw-Rascle traffic flow model. Math Eng, 2025, 7: 316-349
[6] Douglis A. Existence theorems for hyperbolic systems. Comm Pure Appl Math, 1952, 5: 119-154
[7] Goatin P, Laurent-Brouty N. The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model. Z Angew Math Math Phys, 2019, 70: Paper 31
[8] Jannelli A, Mangananaro N, Rizzo A. Riemann problems for the nonhomogeneous Aw-Rascle model. Commun Nonlinear Sci Numer Simul, 2023, 118: Paper 107010
[9] Lax P D. Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J Math Phys, 1964, 5: 611-613
[10] Li T, Liu H L. Critical thresholds in a relaxation model for traffic flows. Indiana Univ Math J, 2008, 57: 1409-1430
[11] Li T, Liu, H L. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete Contin Dyn Syst, 2009, 24: 511-521
[12] Lighthill M J, Whitham G B. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc Roy Soc London Ser A, 1955, 229: 317-345
[13] Liu T P. Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations. J Differential Equations, 1979, 33: 92-111
[14] Liu T P. Hyperbolic conservation laws with relaxation. Comm Math Phys, 1987, 108: 153-175
[15] John F. Formation of singularities in one-dimensional nonlinear wave propagation. Comm Pure Appl Math, 1974, 27: 377-405
[16] Ou Z H, Dai S Q, Zhang P, Dong L Y. Nonlinear analysis in the Aw-Rascle anticipation model of traffic flow. SIAM J Appl Math, 2007, 67: 605-618
[17] Rascle M. An improved macroscopic model of traffic flow: Derivation and links with the Lighthill-Whitham model. Math Comput Modelling, 2002, 35: 581-590
[18] Shao Z Q. Blow-up of solutions to the initial-boundary value problem for quasilinear hyperbolic systems of conservation laws. Nonlinear Anal, 2008, 68: 716-740
[19] Yang T, Zhu C J. Existence and non-existence of global smooth solutions for p-system with relaxation. J Differential Equations, 2000, 161: 321-336

FORMATION OF SINGULARITIES FOR AW-RASCLE TRAFFIC MODEL WITH RELAXATION

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Yifei WU, Zhibo YANG, Qi ZHOU. WELL-POSEDNESS OF THE DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS AND THE KLEIN-GORDON EQUATIONS [J]. Acta mathematica scientia,Series B, 2025, 45(6): 2447-2477.
[2]	Zhong TAN, Yiying WANG. THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH [J]. Acta mathematica scientia,Series B, 2025, 45(6): 2510-2535.
[3]	Siyan GUO, Jiangbo HAN, Runzhang XU. WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION [J]. Acta mathematica scientia,Series B, 2025, 45(4): 1438-1470.
[4]	Ala A. TALAHMEH, Salim A. MESSAOUDI, Mohamed ALAHYANE. THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY [J]. Acta mathematica scientia,Series B, 2022, 42(1): 141-154.
[5]	Suxia XIA. ON BLOW-UP PHENOMENON OF THE SOLUTION TO SOME WAVE-HARTREE EQUATION IN d ≥ 5 [J]. Acta mathematica scientia,Series B, 2020, 40(3): 782-794.
[6]	Huafei DI, Yadong SHANG. BLOW-UP PHENOMENA FOR A CLASS OF GENERALIZED DOUBLE DISPERSION EQUATIONS [J]. Acta mathematica scientia,Series B, 2019, 39(2): 567-579.
[7]	Shun-Tang WU. BLOW-UP OF SOLUTION FOR A VISCOELASTIC WAVE EQUATION WITH DELAY [J]. Acta mathematica scientia,Series B, 2019, 39(1): 329-338.
[8]	Zhensheng GAO, Zhong TAN. BLOW-UP CRITERION OF CLASSICAL SOLUTIONS FOR THE INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS [J]. Acta mathematica scientia,Series B, 2017, 37(6): 1632-1638.
[9]	Mohamed BEN AYED, Habib FOURTI. SCALAR CURVATURE TYPE PROBLEM ON THE THREE DIMENSIONAL BOUNDED DOMAIN [J]. Acta mathematica scientia,Series B, 2017, 37(1): 139-173.
[10]	Nikolay KUTEV, Natalia KOLKOVSKA, Milena DIMOVA. FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY [J]. Acta mathematica scientia,Series B, 2016, 36(3): 881-890.
[11]	Yi JIANG, Chuan LUO. OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION [J]. Acta mathematica scientia,Series B, 2016, 36(3): 872-880.
[12]	Amir PEYRAVI. LOWER BOUNDS OF BLOW UP TIME FOR A SYSTEM OF SEMI-LINEAR HYPERBOLIC PETROVSKY EQUATIONS [J]. Acta mathematica scientia,Series B, 2016, 36(3): 683-688.
[13]	Siwar AMMAR, Mokhles HAMMAMI. BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY [J]. Acta mathematica scientia,Series B, 2015, 35(6): 1511-1546.
[14]	XU Run-Zhang, DING Yun-Hua. GLOBAL SOLUTIONS AND FINITE TIME BLOW UP FOR DAMPED KLEIN-GORDON EQUATION [J]. Acta mathematica scientia,Series B, 2013, 33(3): 643-652.
[15]	GUO Hong-Xia, CHEN Guo-Wang. A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS” [J]. Acta mathematica scientia,Series B, 2013, 33(2): 375-392.