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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
2. ICM, Uniwersytet Warszawski, ul. Prosta 69, 00838 Warsaw

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.
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Keywords loss of uniqueness; Riemann problem; crowd dynamics; loss of self--similarity; capacity drop.; non--local constrained hyperbolic PDE's

Citation: Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini. Riemann problems with non--local point constraints and capacity drop. Mathematical Biosciences and Engineering, 2015, 12(2): 259-278. doi: 10.3934/mbe.2015.12.259

References

  • 1. J. Hyperbolic Differ. Equ., 9 (2012), 105-131.
  • 2. In preparation, 2014.
  • 3. Numerische Mathematik, 115 (2010), 609-645.
  • 4. Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722.
  • 5. Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.
  • 6. Fire Safety Journal, 44 (2009), 532-544.
  • 7. J. Differential Equations, 234 (2007), 654-675.
  • 8. Comm. Partial Differential Equations, 28 (2003), 1371-1389.
  • 9. Math. Methods Appl. Sci., 28 (2005), 1553-1567.
  • 10. Nonlinear Analysis: Real World Applications, 10 (2009), 2716-2728.
  • 11. J. Math. Anal. Appl., 38 (1972), 33-41.
  • 12. Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2000.
  • 13. Indiana Univ. Math. J., 31 (1982), 471-491.
  • 14. Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 223-251.
  • 15. Applied Mathematical Sciences, 18, Springer-Verlag, New York, 1996.
  • 16. SIAM J. Appl. Math., 55 (1995), 625-640.
  • 17. Mat. Sb. (N.S.), 81 (1970), 228-255.
  • 18. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002.
  • 19. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
  • 20. Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
  • 21. J. Hyperbolic Differ. Equ., 4 (2007), 729-770.
  • 22. Operations Res., 4 (1956), 42-51.
  • 23. J. Differential Equations, 246 (2009), 408-427.
  • 24. Arch. Ration. Mech. Anal., 160 (2001), 181-193.

 

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