Abstract
We study wave interactions and stability of Riemann solutions to the inhomogeneous Aw-Rascle (AR) model with Coulomb-like friction term for modified Chaplygin gas. First, the Riemann problem with initial data of two piecewise constants is technically solved by introducing some variable transformation. It is found that the Riemann solutions for the inhomogeneous system are no longer self-similar, and all the elementary waves are bent into parabolic shapes. Second, by investigating the interactions of elementary waves, the global structures of Riemann solutions to the inhomogeneous AR model with perturbed three-piecewise-constant initial data are established constructively. Moreover, we show that the solutions are stable under the small perturbation of initial data.
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References
Aw, A., Rascle, M.: Resurrection of “second order’’ models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000)
Aw, A., Klar, A., Rascle, M., Materne, T.: Derivation of continuum traffic flow models from microscopic follow-the-leader model. SIAM J. Appl. Math. 63(1), 259–278 (2002)
Benaoum, H.B.: Accelerated universe from modified Chaplygin gas and tachyonic fluid, arXiv: hep-th/0205140 (2002)
Bressan, A.: Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170(2), 414–432 (1992)
Chang, T., Hsiao, L.: The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 41. Longman Scientific and Technical, Harlow (1989)
Chen, G., Liu, H.: Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34, 925–938 (2003)
Cheng, H., Yang, H.: Approaching Chaplygin pressure limit of solutions to the Aw-Rascle model. J. Math. Anal. Appl. 416(2), 839–854 (2014)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience Publishers Inc., New York (1948)
Daganzo, C.: Requiem for second order fluid approximations of traffic flow. Transport. Res. B 29, 277–286 (1995)
Fabris, J.C., Velten, H.E.S., Ogouyandjou, C., Tossa, J.: Ruling out the modified Chaplygin gas cosmologies. Phys. Lett. B 694, 289–293 (2011)
Faccanoni, G., Mangeney, A.: Exact solution for granular flows. Int. J. Numer. Anal. Methods Geomech. 37, 1408–1433 (2012)
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)
Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 89, 271–306 (1959)
Guo, L., Li, T., Pan, L., Han, X.: The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term. Nonlinear Anal. Real World Appl. 41, 588–606 (2018)
Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws. Springer, Berlin (2002)
Li, S., Wang, Q., Yang, H.: Riemann problem for the Aw-Rascle model of traffic flow with general pressure. Bull. Malays. Math. Sci. Soc. 43, 3757–3775 (2020)
Liu, Y., Sun, W.: Wave interactions and stability of Riemann solutions of the Aw-Rascle model for generalized Chaplygin gas. Acta Appl. Math. 154, 1–15 (2017)
Liu, Y., Sun, W.: The perturbed Riemann problem for the Aw-Rascle model with modified Chaplygin gas pressure. Adv. Math. Phys. 7104527, 7 (2018)
Pan, L., Han, X.: The Aw-Rascle traffic model with Chaplygin pressure. J. Math. Anal. Appl. 401, 379–387 (2013)
Pang, Y., Hu, M.: The non-self-similar Riemann solutions to a compressible fluid described by the generalized Chaplygin gas. Int. J. Nonlinear Mech. 107, 56–63 (2018)
Santos, F.C., Bedran, M.L., Soares, V.: On the thermodynamic stability of the modified Chaplygin gas. Phys. Lett. B 646, 215–221 (2007)
Shao, Z., Huang, M.: Interactions of delta shock waves for the Aw-Rascle traffic model with split delta functions. J. Appl. Anal. Comput. 7(1), 119–133 (2017)
Shen, C.: The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IMA J. Appl. Math. 81, 76–99 (2016a)
Shen, C.: The Riemann problem for the Chaplygin gas equations with a source term. Z. Angew. Math. Mech. 96, 681–695 (2016b)
Shen, C., Sun, M.: Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. J. Differ. Equ. 249(12), 3024–3051 (2010)
Sun, M.: Interactions of elementary waves for the Aw-Rascle model. SIAM J. Appl. Math. 69(6), 1542–1558 (2009)
Sun, M.: A note on the interactions of elementary waves for the AR traffic flow model without vacuum. Acta Math. Sci. 31(B), 1503–1512 (2011)
Sun, M.: The exact Riemann solutions to the generalized Chaplygin gas equations with friction. Commun. Nonlinear Sci. Numer. Simul. 36, 342–353 (2016)
Wang, G.: The Riemann problem for Aw-Rascle traffic flow with negative pressure. Chin. Ann. Math., Ser. A 35(1), 73–82 (2014)
Wang, J., Liu, J., Yang, H.: Vanishing pressure limit of solutions to the Aw-Rascle model for modified Chaplygin gas, arXiv: 1410.1110 (2014)
Yin, G., Chen, J.: Existence and stability of Riemann solution to the Aw-Rascle model with friction, Indian J. Pure. Appl. Math. 49, 671–688 (2018)
Zhang, M.: A non-equilibrium traffic model devoid of gas-like behavior. Transport. Res. B 36, 275–290 (2002)
Zhang, Q.: Stability of Riemann solutions to pressureless Euler equations with Coulomb-like friction by flux approximation. Electron. J. Differ. Equ. 2019(65), 1–22 (2019)
Zhang, Y., Zhang, Y.: The Riemann problem for the Suliciu relaxation system with the double-coefficient Coulomb-like friction terms. Int. J. Nonlinear Mech. 116, 200–210 (2019a)
Zhang, Y., Zhang, Y.: Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Commun. Pure Appl. Anal. 18(3), 1523–1545 (2019b)
Zhang, Y., Zhang, Y.: The Riemann problem for the Eulerian droplet model with buoyancy and gravity forces. Eur. Phys. J. Plus 135, 171 (2020a)
Zhang, Y., Zhang, Y.: Delta-Shock solution to the Eulerian droplet model by variable substitution method. Z. Naturforsch. A 75, 201–210 (2020b)
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Supported by Yunnan Fundamental Research Projects (202101AT070128, 2018FD015) and Scientific Research Foundation Project of Yunnan Education Department (2018JS150).
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Fan, S., Zhang, Y. Wave Interactions and Stability of Riemann Solutions to the Aw-Rascle Model with Friction for Modified Chaplygin Gas. Bull Braz Math Soc, New Series 53, 765–785 (2022). https://doi.org/10.1007/s00574-021-00282-5
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DOI: https://doi.org/10.1007/s00574-021-00282-5