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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.

[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.

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.

[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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