# American Institute of Mathematical Sciences

June  2017, 12(2): 245-258. doi: 10.3934/nhm.2017010

## Stability estimates for scalar conservation laws with moving flux constraints

 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

* Corresponding author: Paola Goatin

Received  October 2016 Revised  January 2017 Published  May 2017

Fund Project: This research was supported by the Inria Associated Team "Optimal REroute Strategies for Traffic managEment" (ORESTE)

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.

Citation: Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010
##### References:
 [1] Adimurthi, R. Dutta, S. S. Ghoshal and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math., 64 (2011), 84-115. doi: 10.1002/cpa.20346. Google Scholar [2] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non-local point constraints and capacity drop, Math. Biosci. Eng., 12 (2015), 259-278. Google Scholar [3] B. Andreianov, C. Donadello and M. D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci., 26 (2016), 751-802. doi: 10.1142/S0218202516500172. Google Scholar [4] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645, With supplementary material available online. doi: 10.1007/s00211-009-0286-7. Google Scholar [5] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4. Google Scholar [6] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model, SIAM J. Math. Anal., 46 (2014), 1030-1052. doi: 10.1137/130907963. Google Scholar [7] F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1. Google Scholar [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. Google Scholar [9] A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal., 34 (1998), 637-652. doi: 10.1016/S0362-546X(97)00590-7. Google Scholar [10] R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014. Google Scholar [11] R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663. Google Scholar [12] M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J. Differential Equations, 257 (2014), 4015-4029. doi: 10.1016/j.jde.2014.07.014. Google Scholar [13] M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648. doi: 10.1016/j.jmaa.2011.01.033. Google Scholar [14] M. Garavello and S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, Preprint, 2016.Google Scholar [15] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar [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.Google Scholar

show all references

##### References:
 [1] Adimurthi, R. Dutta, S. S. Ghoshal and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math., 64 (2011), 84-115. doi: 10.1002/cpa.20346. Google Scholar [2] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non-local point constraints and capacity drop, Math. Biosci. Eng., 12 (2015), 259-278. Google Scholar [3] B. Andreianov, C. Donadello and M. D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci., 26 (2016), 751-802. doi: 10.1142/S0218202516500172. Google Scholar [4] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645, With supplementary material available online. doi: 10.1007/s00211-009-0286-7. Google Scholar [5] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4. Google Scholar [6] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model, SIAM J. Math. Anal., 46 (2014), 1030-1052. doi: 10.1137/130907963. Google Scholar [7] F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1. Google Scholar [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. Google Scholar [9] A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal., 34 (1998), 637-652. doi: 10.1016/S0362-546X(97)00590-7. Google Scholar [10] R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014. Google Scholar [11] R. M. Colombo and A. Marson, A Hölder continuous ODE related to traffic flow, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 759-772. doi: 10.1017/S0308210500002663. Google Scholar [12] M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J. Differential Equations, 257 (2014), 4015-4029. doi: 10.1016/j.jde.2014.07.014. Google Scholar [13] M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648. doi: 10.1016/j.jmaa.2011.01.033. Google Scholar [14] M. Garavello and S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, Preprint, 2016.Google Scholar [15] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar [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.Google Scholar
Graphical representation of the constraint action in the fixed (left) and moving (right) reference frames
The set $\mathcal{G}_\alpha (\dot{y})$ (thick lines) in the case of a flux function of the form $f(\rho)=V\rho(1-\rho/R)$, as in Section 3
Bus and cars speed
Different solutions of the Riemann problem (17). Each subfigure illustrates a point of the Definition 3.2: fundamental diagram representation (left) and space-time diagram (right)
Case 1
Case 2
Case 3
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