Search Advanced

Mathematical Biosciences and Engineering

2015, Volume 12, Issue 2: 259-278. doi: 10.3934/mbe.2015.12.259
Previous Article Next Article

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
  • Received: 01 April 2014 Accepted: 29 June 2018 Published: 01 December 2014

  • MSC : Primary: 35L65, 90B20.

  • 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[J]. Mathematical Biosciences and Engineering, 2015, 12(2): 259-278. doi: 10.3934/mbe.2015.12.259

    Related Papers:

    [1] Shanshan Pan, Jinbao Jian, Linfeng Yang . Solution to dynamic economic dispatch with prohibited operating zones via MILP. Mathematical Biosciences and Engineering, 2022, 19(7): 6455-6468. doi: 10.3934/mbe.2022303
    [2] Dawei Li, Suzhen Lin, Xiaofei Lu, Xingwang Zhang, Chenhui Cui, Boran Yang . IMD-Net: Interpretable multi-scale detection network for infrared dim and small objects. Mathematical Biosciences and Engineering, 2024, 21(1): 1712-1737. doi: 10.3934/mbe.2024074
    [3] Jianjun Paul Tian . The replicability of oncolytic virus: Defining conditions in tumor virotherapy. Mathematical Biosciences and Engineering, 2011, 8(3): 841-860. doi: 10.3934/mbe.2011.8.841
    [4] Sunwoo Hwang, Seongwon Lee, Hyung Ju Hwang . Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model. Mathematical Biosciences and Engineering, 2021, 18(6): 8524-8534. doi: 10.3934/mbe.2021421
    [5] Francesca Marcellini . The Riemann problem for a Two-Phase model for road traffic with fixed or moving constraints. Mathematical Biosciences and Engineering, 2020, 17(2): 1218-1232. doi: 10.3934/mbe.2020062
    [6] Souâd Yacheur, Ali Moussaoui, Abdessamad Tridane . Modeling the imported malaria to north Africa and the absorption effect of the immigrants. Mathematical Biosciences and Engineering, 2019, 16(2): 967-989. doi: 10.3934/mbe.2019045
    [7] Veronika Schleper . A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences and Engineering, 2015, 12(2): 393-413. doi: 10.3934/mbe.2015.12.393
    [8] Nikolaos Kazantzis, Vasiliki Kazantzi . Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors. Mathematical Biosciences and Engineering, 2010, 7(2): 401-419. doi: 10.3934/mbe.2010.7.401
    [9] Ying Xu, Jinyong Cheng . Secondary structure prediction of protein based on multi scale convolutional attention neural networks. Mathematical Biosciences and Engineering, 2021, 18(4): 3404-3422. doi: 10.3934/mbe.2021170
    [10] Yi Hu, Petia M. Vlahovska, Michael J. Miksis . Electrohydrodynamic assembly of colloidal particles on a drop interface. Mathematical Biosciences and Engineering, 2021, 18(3): 2357-2371. doi: 10.3934/mbe.2021119
  • 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.


    [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.

  • This article has been cited by:

    1. Marco Di Francesco, Simone Fagioli, Massimiliano Daniele Rosini, Giovanni Russo, Deterministic particle approximation of the Hughes model in one space dimension, 2017, 10, 1937-5077, 215, 10.3934/krm.2017009
    2. Edda Dal Santo, Carlotta Donadello, Sabrina F. Pellegrino, Massimiliano D. Rosini, Representation of capacity drop at a road merge via point constraints in a first order traffic model, 2019, 53, 0764-583X, 1, 10.1051/m2an/2019002
    3. Massimiliano D. Rosini, Julien Y. Rolland, Ulrich Razafison, Carlotta Donadello, Boris P. Andreianov, Solutions of the Aw-Rascle-Zhang system with point constraints, 2016, 11, 1556-1801, 29, 10.3934/nhm.2016.11.29
    4. Boris Andreianov, Carlotta Donadello, Massimiliano Daniele Rosini, A second-order model for vehicular traffics with local point constraints on the flow, 2016, 26, 0218-2025, 751, 10.1142/S0218202516500172
    5. Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini, Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux, 2018, 116, 00217824, 309, 10.1016/j.matpur.2018.01.005
    6. E. Dal Santo, M. D. Rosini, N. Dymski, M. Benyahia, General phase transition models for vehicular traffic with point constraints on the flow, 2017, 40, 01704214, 6623, 10.1002/mma.4478
    7. Maria Laura Delle Monache, Paola Goatin, Stability estimates for scalar conservation laws with moving flux constraints, 2017, 12, 1556-181X, 245, 10.3934/nhm.2017010
    8. Mohamed Benyahia, Massimiliano D. Rosini, A macroscopic traffic model with phase transitions and local point constraints on the flow, 2017, 12, 1556-181X, 297, 10.3934/nhm.2017013
    9. Mohamed Benyahia, Massimiliano D. Rosini, Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic, 2020, 43, 0170-4214, 10381, 10.1002/mma.6304
    10. Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano Daniele Rosini, 2018, Chapter 5, 978-3-030-05128-0, 103, 10.1007/978-3-030-05129-7_5
    11. Jennifer Weissen, Oliver Kolb, Simone Göttlich, A combined first and second order model for a junction with ramp buffer, 2022, 8, 2426-8399, 349, 10.5802/smai-jcm.90
  • Reader Comments
  • © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(3010) PDF downloads(483) Cited by(11)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog