Publication Cover
Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 8
84
Views
0
CrossRef citations to date
0
Altmetric
Research Article

The perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction

ORCID Icon &
Pages 1418-1446 | Received 11 May 2023, Accepted 15 Aug 2023, Published online: 31 Aug 2023

References

  • Chaplygin S. On gas jets. Sci Mem Moscow Univ Math Phys. 1904;21:1–121.
  • Tsien H. Two dimensional subsonic flow of compressible fluids. J Aeron Sci. 1939;6:399–407. doi: 10.2514/8.916
  • Bilic N, Tupper G, Viollier R. Unification of dark matter and dark energy: the inhomogeneous Chaplygin gas. Phys Lett B. 2002;535:17–21. doi: 10.1016/S0370-2693(02)01716-1
  • Gorini V, Kamenshchik A, Moschella U, et al. The Chaplygin gas as a model for dark energy. arXiv: gr-qc/0403062.
  • Brenier Y. Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations. J Math Fluid Mech. 2005;7:S326–S331. doi: 10.1007/s00021-005-0162-x
  • Guo L, Sheng W, Zhang T. The two-dimensional Riemann problelm for isentropic Chaplygin gas dynamic system. Comm Pure Appl Anal. 2010;9(2):431–458. doi: 10.3934/cpaa.2010.9.431
  • Serre D. Multidimensional shock interaction for a Chaplygin gas. Arch Rational Mech Anal. 2009;191:539–577. doi: 10.1007/s00205-008-0110-z
  • Wang Z, Zhang Q. The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Acta Math Sci. 2012;32B(3):825–841.
  • Benaoum H. Accelerated universe from modified Chaplygin gas and tachyonic fluid. arXiv: hep-th/0205140.
  • Setare M. Interacting holographic generalized Chaplygin gas model. Phys Lett B. 2007;654:1–6. doi: 10.1016/j.physletb.2007.08.038
  • Sheng W, Wang G, Yin G. Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes. Nonlinear Anal Real World Appl. 2015;22:115–128. doi: 10.1016/j.nonrwa.2014.08.007
  • Wang G. The Riemann problem for one dimensional generalized Chaplygin gas dynamics. J Math Anal Appl. 2013;403:434–450. doi: 10.1016/j.jmaa.2013.02.026
  • Pan L, Han X. The Aw-Rascle traffic model with Chaplygin pressure. J Math Anal Appl. 2013;401:379–387. doi: 10.1016/j.jmaa.2012.12.022
  • Sheng W, Zeng Y. Generalized δ-entropy condition to Riemann solutions for Chaplygin gas in traffic model. Appl Math Mech Engl Ed. 2015;36(3):353–364. doi: 10.1007/s10483-015-1915-6
  • Aw A, Rascle M. Resurrection of second order models of traffic flow. SIAM J Appl Math. 2000;60:916–938. doi: 10.1137/S0036139997332099
  • Daganzo C. Requiem for second order fluid approximations of traffic flow. Transp Res Part B. 1995;29:277–286. doi: 10.1016/0191-2615(95)00007-Z
  • Zhang H. A non-equilibrium traffic model devoid of gas-like behavior. Transp Res Part B. 2002;36:275–290. doi: 10.1016/S0191-2615(00)00050-3
  • Greenberg J. Extensions and amplifications of a traffic model of Aw-Rascle. SIAM J Appl Math. 2001;62(3):729–745. doi: 10.1137/S0036139900378657
  • Shen C, Sun M. Formation of delta-shocks and vacuum states in the vanishing pressue limit of solutions to the Aw-Rascle model. J Differ Equ. 2010;249:3024–3051. doi: 10.1016/j.jde.2010.09.004
  • Sun M. Interactions of elementary waves for the Aw-Rascle model. SIAM J Appl Math. 2009;69(6):1542–1558. doi: 10.1137/080731402
  • Yin G, Chen JJ. Existence and stability of Riemann solution of the Aw-Rascle model with friction. Indian J Pure Appl Math. 2018;49(4):671–688. doi: 10.1007/s13226-018-0294-3
  • Chen TT, Jiang WF, Li T. On the stability of the improved Aw-Rascle-Zhang model with Chaplygin pressure. Nonlinear Anal: Real World Appl. 2021;62:103351. doi: 10.1016/j.nonrwa.2021.103351
  • Jiang WF, Wang Z. Developing an Aw-Rascle model of traffic flow. J Eng Math. 2016;97:135–146. doi: 10.1007/s10665-015-9801-2
  • Cheng H. Delta shock waves for a linearly dengenerate hyperbolic system of conservation laws of Keyfitz-Kranzer type. Adv Math Phys. 2013;10. Article ID 958120.
  • Cheng H. On a nonsymmetric Keyfitz-Kranzer system of conservation laws with generalized and modified Chaplygin gas pressure law. Adv Math Phys. 2013;14.
  • Lu Y-G. Existence of global entropy solution to general system of Keyfitz-Kranzer type. J Funct Anal. 2013;264:2457–2468. doi: 10.1016/j.jfa.2013.02.021
  • Abreu E, De la cruz R, Lambert W. Riemann problems and delta-shock solutions for a Keyfitz-Kranzer system with a forcing term. J Math Anal Appl. 2021;502:125267. doi: 10.1016/j.jmaa.2021.125267
  • De la cruz R, Santos M. Delta shock wave for a system of Keyfitz-Kranzer type. Z Angew Math Mech. 2019;99(3):e201700251. doi: 10.1002/zamm.v99.3
  • De la cruz R, Abreu E, Santos M. Interaction of delta shock waves for a nonsymmetric Keyfitz-Kranzer system of conservation laws. Monatsh Math. 2021;194(4):737–766. doi: 10.1007/s00605-021-01524-w
  • Bouchut F. On zero-pressure gas dynamics//Advances in Kinetic Theory and Computing. Ser Adv Math Appl Sci. 1994;22:171–190. doi: 10.1142/SAMAS
  • Weinan E, Rykov YG, Sinai YG. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun Math Phys. 1996;177:349–380. doi: 10.1007/BF02101897
  • Huang F, Wang Z. Well posedness for pressureless flow. Commun Math Phys. 2001;222:117–146. doi: 10.1007/s002200100506
  • Sheng W, Zhang T. The Riemann problem for transportation equations in gas dynamics. Mem Amer Math Soc. 1999;137(654).
  • Shen C. The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IAM J Appl Math. 2016;81:76–99.
  • Korchinski D. Solutions of a Riemann problem for a system of conservation laws possessing no classical weak solution [thesis]. Adelphi University, 1977.
  • Tan D, Zhang T. Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I. Four-J cases, II. Initial data involving some rarefaction waves. J Differ Equ. 1994;111:203–254. doi: 10.1006/jdeq.1994.1081
  • Tan D, Zhang T, Zheng Y. Delta-shock wave as limits of vanishing viscosity for hyperbolic system of conservation laws. J Differ Equ. 1994;112:1–32. doi: 10.1006/jdeq.1994.1093
  • Savage S, Hutter K. The motion of finite mass of granular material down a rough incline. J Fluid Mech. 1989;199:177–215. doi: 10.1017/S0022112089000340
  • Shen C. The Riemann problem for the Chaplygin gas equations with a source term. Z Angew Math Mech. 2016;96:681–695. doi: 10.1002/zamm.v96.6
  • Sun M. The exact Riemann solutions to the generalized Chaplygin gas equations with friction. Commun Nonlinear Sci Numer Simulat. 2016;36:342–353. doi: 10.1016/j.cnsns.2015.12.013
  • Guo L, Li T, Pan L, et al. The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term. Nonlinear Anal Real World Appl. 2018;41:588–606. doi: 10.1016/j.nonrwa.2017.11.013
  • Faccanoni G, Mangeney A. Exact solution for granular flows. Int J Numer Anal Mech Geomech. 2012;37:1408–1433. doi: 10.1002/nag.v37.10
  • Smoller J. Shock waves and reaction-diffusion equation. New York: Springer-Verlag; 1994.
  • Chen GQ, Liu H. Formation of δ−shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J Math Anal. 2003;34:925–938. doi: 10.1137/S0036141001399350
  • Danilvo VG, Shelkovich VM. Dynamics of propagation and interaction of δ-shock waves in conservation law system. J Differ Equ. 2005;211:333–381. doi: 10.1016/j.jde.2004.12.011
  • Danilvo VG, Shelkovich VM. Delta-shock waves type solution of hyperbolic systems of conservation laws. Q Appl Math. 2005;63:401–427. doi: 10.1090/qam/2005-63-03
  • Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proc Edinb Math Soc. 2012;55:711–729. doi: 10.1017/S0013091512000065
  • Kalisch H, Mitrovic D. Singular solutions for shallow water equations. IMA J Appl Math. 2012;77:340–350. doi: 10.1093/imamat/hxs014
  • Shen C, Sun M. Interactions of delta shocks for transport equations with split delta functions. J Math Anal Appl. 2009;351:747–755. doi: 10.1016/j.jmaa.2008.11.005

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.