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

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

Boris Andreianov 1, , Carlotta Donadello 1, , Ulrich Razafison 1, and  Massimiliano D. Rosini 2,

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.
Keywords: loss of uniqueness, Riemann problem, crowd dynamics, loss of self--similarity, capacity drop., non--local constrained hyperbolic PDE's.
Mathematics Subject Classification: Primary: 35L65, 90B2.
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.  Google Scholar

[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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[17]

S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[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.  Google Scholar

[19]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

[24]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.  Google Scholar

show all references

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[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.  Google Scholar

[19]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

[24]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.  Google Scholar

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