Stability of large-amplitude traveling fronts for a traffic flow model with moving barriers

Jingyu Li; Lina Wang; Yaping Wu

doi:10.3934/dcdss.2023087

Article Contents

2024, Volume 17, Issue 2: 720-741. Doi: 10.3934/dcdss.2023087

This issue Previous Article The linear stability and basic reproduction numbers for autonomous FDEs Next Article A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅲ : General case

Stability of large-amplitude traveling fronts for a traffic flow model with moving barriers

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.
School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
3.
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

^*Corresponding author: Lina Wang
^*Corresponding author: Lina Wang

Dedicated to Professor Yihong Du on the occasion of his 60th birthday

Received: February 2023

Early access: April 24, 2023

Published online: April 24, 2023

This work is supported by NSF of China (No.11871048), NSF of Jilin Province (20210101144JC) and Beijing NSF (No.1232004, No.1212002).

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Abstract

In this paper, we study the existence and stability of traveling front solutions to a nonlinear hyperbolic system of balance law, which models the dynamics of a heterogeneous traffic flow with moving barriers (e.g. slow vehicles). We first prove via shooting method that, the balance law admits a family of monotone traveling fronts, if the barriers have small strength regardless of their shapes. Then we show that, under some assumptions on the nonlinearities and by applying detailed spectral analysis and $ C_0 $ semigroup theories, the traveling fronts are spectrally stable, linearly and nonlinearly exponentially stable in some exponentially weighted spaces, where the wave strengths can be large.

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Mathematics Subject Classification: Primary: 35C07, 35B35, 35P05, 90B20; Secondary: 35L40.

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References

[1]	B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux, J. Math. Pures Appl., 116 (2018), 309-346. doi: 10.1016/j.matpur.2018.01.005.
[2]	A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.
[3]	N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.
[4]	S.-W. Chou, J. M. Hong and Y.-C. Lin, Existence and large time stability of traveling wave solutions to nonlinear balance laws in traffic flows, Comm. Math. Sci., 11 (2013), 1011-1037. doi: 10.4310/CMS.2013.v11.n4.a6.
[5]	R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014.
[6]	C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, second edition, Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-29089-3.
[7]	M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result, J. Differential Equations, 257 (2014), 4015-4029. doi: 10.1016/j.jde.2014.07.014.
[8]	J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math., 62 (2002), 729-745. doi: 10.1137/S0036139900378657.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, New York/Berlin, 1981.
[10]	F. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
[11]	S. Jin and J. Liu, Relaxation and diffusion enhanced dispersive waves, Proc. Royal Soc. London Ser. A, 446 (1994), 555-563.
[12]	S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276. doi: 10.1002/cpa.3160480303.
[13]	B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E, 50 (1994), 54-83.
[14]	D. A. Kurtze and D. Hong, Traffic jams, granular flow, and soliton selection, Phys. Rev. E, 52 (1995), 218-221.
[15]	C. Lattanzio, A. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM J. Math. Anal., 43 (2011), 50-67. doi: 10.1137/090767224.
[16]	H. Y. Lee, H. W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models, Phys. Rev. E, 69 (2004), 016118.
[17]	T. Li, Well-posedness theory of an inhomogeneous traffic flow model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 401-414. doi: 10.3934/dcdsb.2002.2.401.
[18]	T. Li, Mathematical modelling of traffic flows, Hyperbolic Problems: Theory, Numerics, Applications, 695-704, Springer, Berlin, 2003.
[19]	T. Li, Nonlinear dynamics of traffic jams, Phys. D, 207 (2005), 41-51. doi: 10.1016/j.physd.2005.05.011.
[20]	T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. doi: 10.1137/070690638.
[21]	T. Li, Qualitative analysis of some PDE models of traffic flow, Netw. Heterog. Media, 8 (2013), 773-781. doi: 10.3934/nhm.2013.8.773.
[22]	T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Comm. Math. Sci., 3 (2005), 101-118. doi: 10.4310/CMS.2005.v3.n2.a1.
[23]	T. Li and Y. Wu, Linear and nonlinear exponential stability of traveling waves for hyperbolic systems with relaxation, Comm. Math. Sci., 7 (2009), 571-593. doi: 10.4310/CMS.2009.v7.n3.a3.
[24]	M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. R. Soc. London Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.
[25]	H. Liu, C. W. Woo and T. Yang, Decay rate for traveling waves of a relaxation model, J. Differential Equations, 134 (1997), 343-367. doi: 10.1006/jdeq.1996.3220.
[26]	T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707.
[27]	C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J., 51 (2002), 773-904. doi: 10.1512/iumj.2002.51.2212.
[28]	C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal., 172 (2004), 93-131. doi: 10.1007/s00205-003-0293-2.
[29]	M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edin. Sec A, 128 (1998), 1053-1068. doi: 10.1017/S0308210500030067.
[30]	T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Mathématiques d'Orsay, Département de Mathématique, Université de Paris-Sud, Orsay, 1978.
[31]	H. Payne, Models of freeway traffic and control, Mathematical Models of Public Systems Simulation Council, 1 (1971), 51-60.
[32]	P. I. Richards, Shock waves on highway, Operat. Res., 4 (1956), 42-51.
[33]	G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
[34]	H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B, 36 (2002), 275-290.
[35]	P. Zhang and S. Wong, Essence of conservation forms in the traveling wave solutions of higher-order traffic flow models, Phys. Rev. E, 74 (2006), 026109. doi: 10.1103/PhysRevE.74.026109.