# American Institute of Mathematical Sciences

2015, 12(2): 259-278. doi: 10.3934/mbe.2015.12.259

## Riemann problems with non--local point constraints and capacity drop

 1 Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France, France 2 ICM, Uniwersytet Warszawski, ul. Prosta 69, 00838 Warsaw, Poland

Received  April 2014 Revised  October 2014 Published  December 2014

In the present note we discuss in details the Riemann problem for a one-dimensional hyperbolic conservation law subject to a point constraint. We investigate how the regularity of the constraint operator impacts the well--posedness of the problem, namely in the case, relevant for numerical applications, of a discretized exit capacity. We devote particular attention to the case in which the constraint is given by a non--local operator depending on the solution itself. We provide several explicit examples.
We also give the detailed proof of some results announced in the paper [Andreianov, Donadello, Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non--local point constraints.
Citation: Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini. Riemann problems with non--local point constraints and capacity drop. Mathematical Biosciences & Engineering, 2015, 12 (2) : 259-278. doi: 10.3934/mbe.2015.12.259
##### References:
 [1] D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., 9 (2012), 105-131. doi: 10.1142/S0219891612500038. [2] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Numerical simulations for conservation laws with non-local point constraints in crowd dynamics, In preparation, 2014. [3] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numerische Mathematik, 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7. [4] B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722. doi: 10.1142/S0218202514500341. [5] A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. [6] E. M. Cepolina, Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows, Fire Safety Journal, 44 (2009), 532-544. doi: 10.1016/j.firesaf.2008.11.002. [7] 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. [8] R. M. Colombo and F. S. Priuli, Characterization of Riemann solvers for the two phase p-system, Comm. Partial Differential Equations, 28 (2003), 1371-1389. doi: 10.1081/PDE-120024372. [9] R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624. [10] R. M. Colombo and M. D. Rosini, Existence of nonclassical solutions in a Pedestrian flow model, Nonlinear Analysis: Real World Applications, 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002. [11] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41. doi: 10.1016/0022-247X(72)90114-X. [12] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14. [13] C. M. Dafermos and L. Hsiao, Hyperbolic systems and balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J., 31 (1982), 471-491. doi: 10.1512/iumj.1982.31.31039. [14] N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes' model for pedestrian motion, Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 223-251. doi: 10.1007/s00033-012-0232-x. [15] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, 18, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [16] E. Isaacson and B. Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640. doi: 10.1137/S0036139992240711. [17] S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. [18] P. G. Lefloch, Hyperbolic Systems of Conservation Laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8150-0. [19] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [20] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [21] E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343. [22] P. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [23] M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, J. Differential Equations, 246 (2009), 408-427. doi: 10.1016/j.jde.2008.03.018. [24] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.

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##### References:
 [1] D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., 9 (2012), 105-131. doi: 10.1142/S0219891612500038. [2] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Numerical simulations for conservation laws with non-local point constraints in crowd dynamics, In preparation, 2014. [3] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numerische Mathematik, 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7. [4] B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722. doi: 10.1142/S0218202514500341. [5] A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000. [6] E. M. Cepolina, Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows, Fire Safety Journal, 44 (2009), 532-544. doi: 10.1016/j.firesaf.2008.11.002. [7] 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. [8] R. M. Colombo and F. S. Priuli, Characterization of Riemann solvers for the two phase p-system, Comm. Partial Differential Equations, 28 (2003), 1371-1389. doi: 10.1081/PDE-120024372. [9] R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624. [10] R. M. Colombo and M. D. Rosini, Existence of nonclassical solutions in a Pedestrian flow model, Nonlinear Analysis: Real World Applications, 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002. [11] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41. doi: 10.1016/0022-247X(72)90114-X. [12] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14. [13] C. M. Dafermos and L. Hsiao, Hyperbolic systems and balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J., 31 (1982), 471-491. doi: 10.1512/iumj.1982.31.31039. [14] N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes' model for pedestrian motion, Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 223-251. doi: 10.1007/s00033-012-0232-x. [15] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, 18, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [16] E. Isaacson and B. Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640. doi: 10.1137/S0036139992240711. [17] S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. [18] P. G. Lefloch, Hyperbolic Systems of Conservation Laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8150-0. [19] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [20] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [21] E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343. [22] P. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [23] M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, J. Differential Equations, 246 (2009), 408-427. doi: 10.1016/j.jde.2008.03.018. [24] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.
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