Stability estimates for scalar conservation laws with moving flux constraints

Maria Laura Delle Monache; Paola Goatin; Maria Laura Delle Monache; Paola Goatin

doi:10.3934/nhm.2017010

Networks and Heterogeneous Media

2017, Volume 12, Issue 2: 245-258. doi: 10.3934/nhm.2017010

Previous Article Next Article

Stability estimates for scalar conservation laws with moving flux constraints

Maria Laura Delle Monache ^{1,2
,},
Paola Goatin ^{3
,
,}

1.
Department of mathematical sciences, Rutgers University -Camden, 311 N. 5th Street, Camden, NJ 08102, USA
2.
Inria, Univ. Grenoble Alpes, CNRS, GIPSA-lab, F-38000 Grenoble, France
3.
Inria Sophia Antipolis -Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004, route des Lucioles -BP 93, 06902 Sophia Antipolis Cedex, France

Received: 01 October 2016 Revised: 01 January 2017
Primary: 35L65; Secondary: 35B35

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.
- Scalar conservation laws,
- moving flux constraints,
- PDE-ODE system
Citation: Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints[J]. Networks and Heterogeneous Media, 2017, 12(2): 245-258. doi: 10.3934/nhm.2017010

Related Papers:

Abstract

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.

References

[1]	Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. Comm. Pure Appl. Math. (2011) 64: 84-115.
[2]	Riemann problems with non-local point constraints and capacity drop. Math. Biosci. Eng. (2015) 12: 259-278.
[3]	A second-order model for vehicular traffics with local point constraints on the flow. Math. Models Methods Appl. Sci. (2016) 26: 751-802.
[4]	Finite volume schemes for locally constrained conservation laws. Numer. Math. (2010) 115: 609-645, With supplementary material available online.
[5]	A theory of \begin{document}$ L^1$\end{document}-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. (2011) 201: 27-86.
[6]	Well-posedness for a one-dimensional fluid-particle interaction model. SIAM J. Math. Anal. (2014) 46: 1030-1052.
[7]	Kružkov's estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc. (1998) 350: 2847-2870.
[8]	A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem.
[9]	Uniqueness for discontinuous ODE and conservation laws. Nonlinear Anal. (1998) 34: 637-652.
[10]	A well posed conservation law with a variable unilateral constraint. J. Differential Equations (2007) 234: 654-675.
[11]	A Hölder continuous ODE related to traffic flow. Proc. Roy. Soc. Edinburgh Sect. A (2003) 133: 759-772.
[12]	Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result. J. Differential Equations (2014) 257: 4015-4029.
[13]	The Aw-Rascle traffic model with locally constrained flow. J. Math. Anal. Appl. (2011) 378: 634-648.
[14]	M. Garavello and S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, Preprint, 2016.
[15]	First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) (1970) 81: 228-255.
[16]	S. Villa, P. Goatin and C. Chalons, Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model, 2016, URL https://hal.archives-ouvertes.fr/hal-01347925, preprint.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)