A well posed conservation law with a variable unilateral constraint.

*(English)*Zbl 1116.35087Summary: This paper considers the Cauchy problem for a conservation law with a variable unilateral constraint, its motivation being, for instance, the modeling of a toll gate along a highway. This problem is solved by means of nonclassical shocks and its well posedness is proved. Then, the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem.

##### MSC:

35L65 | Hyperbolic conservation laws |

90B20 | Traffic problems in operations research |

35L45 | Initial value problems for first-order hyperbolic systems |

35L67 | Shocks and singularities for hyperbolic equations |

##### Keywords:

continuum traffic models; Riemann problem; nonclassical shocks; discontinuous flux function
PDF
BibTeX
XML
Cite

\textit{R. M. Colombo} and \textit{P. Goatin}, J. Differ. Equations 234, No. 2, 654--675 (2007; Zbl 1116.35087)

Full Text:
DOI

##### References:

[1] | Berthelin, F.; Bouchut, F., Weak solutions for a hyperbolic system with unilateral constraint and mass loss, Ann. inst. H. Poincaré anal. non linéaire, 20, 6, 975-997, (2003) · Zbl 1079.76063 |

[2] | Bressan, A., Hyperbolic systems of conservation laws, Rev. mat. complut., 12, 1, 135-200, (1999) · Zbl 0943.35051 |

[3] | Bressan, A., The one-dimensional Cauchy problem, () |

[4] | Coclite, G.M.; Risebro, N.H., Conservation laws with time dependent discontinuous coefficients, SIAM J. math. anal., 36, 4, 1293-1309, (2005) · Zbl 1078.35071 |

[5] | Colombo, R.M., Hyperbolic phase transitions in traffic flow, SIAM J. appl. math., 63, 2, 708-721, (2002) · Zbl 1037.35043 |

[6] | Colombo, R.M.; Priuli, F.S., Characterization of Riemann solvers for the two phase p-system, Comm. partial differential equations, 28, 7-8, 1371-1390, (2003) · Zbl 1030.35117 |

[7] | Dubois, F.; Lefloch, P., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. differential equations, 71, 1, 93-122, (1988) · Zbl 0649.35057 |

[8] | Karlsen, K.H.; Risebro, N.H.; Towers, J.D., \(L^1\) stability for entropy solutions of nonlinear degenerate parabolic convection – diffusion equations with discontinuous coefficients, Skr. K. nor. vidensk. selsk. (3), 1-49, (2003) · Zbl 1036.35104 |

[9] | Karlsen, K.H.; Towers, J.D., Convergence of the lax – friedrichs scheme and stability for conservation laws with a discontinous space – time dependent flux, Chinese ann. math. ser. B, 25, 3, 287-318, (2004) · Zbl 1112.65085 |

[10] | Klingenberg, C.; Risebro, N.H., Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior, Comm. partial differential equations, 20, 1959-1990, (1995) · Zbl 0836.35090 |

[11] | Kružkov, S.N., First order quasilinear equations with several independent variables, Mat. sb. (N.S.), 81, 123, 228-255, (1970) · Zbl 0202.11203 |

[12] | Lax, P.D., Hyperbolic systems of conservation laws. II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803 |

[13] | 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, 229, 317-345, (1955) · Zbl 0064.20906 |

[14] | Liu, T.P., The Riemann problem for general systems of conservation laws, J. differential equations, 18, 218-234, (1975) · Zbl 0297.76057 |

[15] | Richards, P.I., Shock waves on the highway, Oper. res., 4, 42-51, (1956) · Zbl 1414.90094 |

[16] | Temple, B., Global solution of the Cauchy problem for a class of \(2 \times 2\) nonstrictly hyperbolic conservation laws, Adv. appl. math., 3, 3, 335-375, (1982) · Zbl 0508.76107 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.