Stability estimates for scalar conservation laws with moving flux constraints. (English) Zbl 1432.35140
Summary: We study well-posedness of scalar conservation laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed.
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35B35 | Stability in context of PDEs |
| 35L67 | Shocks and singularities for hyperbolic equations |
References:
| [1] | Adimurthi, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux,, Comm. Pure Appl. Math., 64, 84 (2011) · Zbl 1223.35222 · doi:10.1002/cpa.20346 |
| [2] | B. Andreianov, Riemann problems with non-local point constraints and capacity drop,, Math. Biosci. Eng., 12, 259 (2015) · Zbl 1308.35134 |
| [3] | B. Andreianov, A second-order model for vehicular traffics with local point constraints on the flow,, Math. Models Methods Appl. Sci., 26, 751 (2016) · Zbl 1337.35086 · doi:10.1142/S0218202516500172 |
| [4] | B. Andreianov, Finite volume schemes for locally constrained conservation laws,, Numer. Math., 115, 609 (2010) · Zbl 1196.65151 · doi:10.1007/s00211-009-0286-7 |
| [5] | B. Andreianov, A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux,, Arch. Ration. Mech. Anal., 201, 27 (2011) · Zbl 1261.35088 · doi:10.1007/s00205-010-0389-4 |
| [6] | B. Andreianov, Well-posedness for a one-dimensional fluid-particle interaction model,, SIAM J. Math. Anal., 46, 1030 (2014) · Zbl 1302.35263 · doi:10.1137/130907963 |
| [7] | F. Bouchut, Kružkov’s estimates for scalar conservation laws revisited,, Trans. Amer. Math. Soc., 350, 2847 (1998) · Zbl 0955.65069 · doi:10.1090/S0002-9947-98-02204-1 |
| [8] | A. Bressan, <em>Hyperbolic Systems of Conservation Laws</em>, vol. 20 of Oxford Lecture Series in Mathematics and its Applications,, Oxford University Press (2000) · Zbl 0997.35002 |
| [9] | A. Bressan, Uniqueness for discontinuous ODE and conservation laws,, Nonlinear Anal., 34, 637 (1998) · Zbl 0948.34006 · doi:10.1016/S0362-546X(97)00590-7 |
| [10] | R. M. Colombo, A well posed conservation law with a variable unilateral constraint,, J. Differential Equations, 234, 654 (2007) · Zbl 1253.65122 · doi:10.1016/j.jde.2006.10.014 |
| [11] | R. M. Colombo, A Hölder continuous ODE related to traffic flow,, Proc. Roy. Soc. Edinburgh Sect. A, 133, 759 (2003) · Zbl 1052.34007 · doi:10.1017/S0308210500002663 |
| [12] | M. L. Delle Monache, Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result,, J. Differential Equations, 257, 4015 (2014) · Zbl 1302.35248 · doi:10.1016/j.jde.2014.07.014 |
| [13] | M. Garavello, The Aw-Rascle traffic model with locally constrained flow,, J. Math. Anal. Appl., 378, 634 (2011) · Zbl 1211.35018 · doi:10.1016/j.jmaa.2011.01.033 |
| [14] | M. Garavello, The Cauchy problem for the Aw- Rascle- Zhang traffic model with locally constrained flow,, Preprint (2016) · Zbl 1382.35161 |
| [15] | S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81, 228 (1970) · Zbl 0202.11203 |
| [16] | S. Villa, Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model, 2016,, URL <a href= · Zbl 1371.35173 |
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