Search Advanced

Networks and Heterogeneous Media

2016, Volume 11, Issue 1: 49-67. doi: 10.3934/nhm.2016.11.49
Previous Article Next Article

Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results

  • 1.
    Unità INdAM, c/o DII, Università degli Studi di Brescia, Via Branze, 38; 25123 Brescia
  • 2.
    Dip. di Matematica e Applicazioni, Università di Milano - Bicocca, Via Cozzi, 55; 20125 Milano
  • Received: 01 April 2015 Revised: 01 September 2015

  • Primary: 35L65; Secondary: 35M30, 92D25.

  • This paper is devoted to the overview of recent results concerning nonlocal systems of conservation laws. First, we present a predator -- prey model and, second, a model for the laser cutting of metals. In both cases, these equations lead to interesting pattern formation.

    Citation: Rinaldo M. Colombo, Francesca Marcellini, Elena Rossi. Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results[J]. Networks and Heterogeneous Media, 2016, 11(1): 49-67. doi: 10.3934/nhm.2016.11.49

    Related Papers:

    [1] Rinaldo M. Colombo, Francesca Marcellini, Elena Rossi . Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results. Networks and Heterogeneous Media, 2016, 11(1): 49-67. doi: 10.3934/nhm.2016.11.49
    [2] Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang . Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks and Heterogeneous Media, 2015, 10(4): 749-785. doi: 10.3934/nhm.2015.10.749
    [3] Jan Friedrich, Sanjibanee Sudha, Samala Rathan . Numerical schemes for a class of nonlocal conservation laws: a general approach. Networks and Heterogeneous Media, 2023, 18(3): 1335-1354. doi: 10.3934/nhm.2023058
    [4] Alexandre M. Bayen, Alexander Keimer, Nils Müller . A proof of Kirchhoff's first law for hyperbolic conservation laws on networks. Networks and Heterogeneous Media, 2023, 18(4): 1799-1819. doi: 10.3934/nhm.2023078
    [5] Raimund Bürger, Harold Deivi Contreras, Luis Miguel Villada . A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux. Networks and Heterogeneous Media, 2023, 18(2): 664-693. doi: 10.3934/nhm.2023029
    [6] Dong Li, Tong Li . Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2011, 6(4): 681-694. doi: 10.3934/nhm.2011.6.681
    [7] Frederike Kissling, Christian Rohde . The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks and Heterogeneous Media, 2010, 5(3): 661-674. doi: 10.3934/nhm.2010.5.661
    [8] Boris Andreianov, Mohamed Karimou Gazibo . Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks and Heterogeneous Media, 2016, 11(2): 203-222. doi: 10.3934/nhm.2016.11.203
    [9] Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina . Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2(1): 159-179. doi: 10.3934/nhm.2007.2.159
    [10] Steinar Evje, Kenneth H. Karlsen . Hyperbolic-elliptic models for well-reservoir flow. Networks and Heterogeneous Media, 2006, 1(4): 639-673. doi: 10.3934/nhm.2006.1.639
  • This paper is devoted to the overview of recent results concerning nonlocal systems of conservation laws. First, we present a predator -- prey model and, second, a model for the laser cutting of metals. In both cases, these equations lead to interesting pattern formation.


    [1] A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983. doi: 10.1137/140975255
    [2] D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes' model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406. doi: 10.1016/j.jmaa.2014.05.072
    [3] D. Amadori and W. Shen, Global existence of large BV solutions in a model of granular flow, Comm. Partial Differential Equations, 34 (2009), 1003-1040. doi: 10.1080/03605300902892279
    [4] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1995, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6
    [5] C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117
    [6] N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), 1230004, 29pp. doi: 10.1142/S0218202512300049
    [7] Numer. Math., To appear. doi: 10.1007/s00211-015-0717-6
    [8] J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003
    [9] C. Christoforou, Systems of hyperbolic conservation laws with memory, J. Hyperbolic Differ. Equ., 4 (2007), 435-478. doi: 10.1142/S0219891607001215
    [10] R. M. Colombo, G. Guerra, M. Herty and F. Marcellini, A hyperbolic model for the laser cutting process, Appl. Math. Model., 37 (2013), 7810-7821. doi: 10.1016/j.apm.2013.02.031
    [11] M3AS, To appear.
    [12] R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34p. doi: 10.1142/S0218202511500230
    [13] R. M. Colombo and G. Guerra, Hyperbolic balance laws with a non local source, Comm. Partial Differential Equations, 32 (2007), 1917-1939. doi: 10.1080/03605300701318849
    [14] R. M. Colombo, G. Guerra and W. Shen, Lipschitz semigroup for an integro-differential equation for slow erosion, Quart. Appl. Math., 70 (2012), 539-578. doi: 10.1090/S0033-569X-2012-01309-2
    [15] R. M. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379. doi: 10.1051/cocv/2010007
    [16] R. M. Colombo and F. Marcellini, Nonlocal systems of balance laws in several space dimensions with applications to laser technology, Journal of Differential Equations, 259 (2015), 6749-6773. doi: 10.1016/j.jde.2015.08.005
    [17] R. M. Colombo and L.-M. Mercier, Nonlocal crowd dynamics models for several populations, Acta Mathematica Scientia, 32 (2012), 177-196. doi: 10.1016/S0252-9602(12)60011-3
    [18] R. M. Colombo, M. Mercier and M. D. Rosini, Stability and total variation estimates on general scalar balance laws, Commun. Math. Sci., 7 (2009), 37-65. doi: 10.4310/CMS.2009.v7.n1.a2
    [19] R. M. Colombo and E. Rossi, Hyperbolic predators vs. parabolic prey, Commun. Math. Sci., 13 (2015), 369-400. doi: 10.4310/CMS.2015.v13.n2.a6
    [20] R. M. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains, Acta Math. Sci. Ser. B Engl. Ed., 35 (2015), 906-944. doi: 10.1016/S0252-9602(15)30028-X
    [21] G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523-537. doi: 10.1007/s00030-012-0164-3
    [22] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1
    [23] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.
    [24] M. S. Gross, On gas dynamics effects in the modelling of laser cutting processes, Appl. Math. Model., 30 (2006), 307-318. doi: 10.1016/j.apm.2005.03.021
    [25] G. Guerra and W. Shen, Existence and stability of traveling waves for an integro-differential equation for slow erosion, J. Differential Equations, 256 (2014), 253-282. doi: 10.1016/j.jde.2013.09.003
    [26] K. Hirano and R. Fabbro, Experimental investigation of hydrodynamics of melt layer during laser cutting of steel, Journal of Physics D: Applied Physics, 44 (2011), 105502. doi: 10.1088/0022-3727/44/10/105502
    [27] S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
    [28] C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439-465. doi: 10.1016/S0022-0396(02)00158-4
    [29] R. J. LeVeque, Numerical Methods for Conservation Laws, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1
    [30] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, 1925.
    [31] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6
    [32] J. D. Murray, Mathematical Biology. II, vol. 18 of Interdisciplinary Applied Mathematics, 3rd edition, Springer-Verlag, New York, 2003, Spatial models and biomedical applications.
    [33] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2007.
    [34] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y
    [35] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007, Blow-up, global existence and steady states.
    [36] ESAIM: Mathematical Modelling and Numerical Analysis, To appear. doi: 10.1051/m2an/2015050
    [37] W. Schulz, V. Kostrykin, H. Zefferer, D. Petring and R. Poprawe, A free boundary problem related to laser beam fusion cutting: Ode approximation, Int. J. Heat Mass Transfer, 40 (1997), 2913-2928. doi: 10.1016/S0017-9310(96)00342-0
    [38] W. Schulz, M. Nießen, U. Eppelt and K. Kowalick, Simulation of laser cutting, in Springer Series in Materials Science, The theory of laser materials processing: Heat and mass transfer in modern technology, Springer Publishers, 119 (2009), 21-69. doi: 10.1007/978-1-4020-9340-1_2
    [39] W. Schulz, G. Simon, H. Urbassek and I. Decker, On laser fusion cutting of metals, J. Phys. D - Appl. Phys, 20 (1987), p481. doi: 10.1088/0022-3727/20/4/013
    [40] D. Serre, Systems of Conservation Laws. 1 & 2, Cambridge University Press, Cambridge, 1999, Translated from the 1996 French original by I. N. Sneddon. doi: 10.1017/CBO9780511612374
    [41] W. Steen, Laser Material Processing, Springer, 2003, URL http://books.google.it/books?id=8E_Ruj0hvzwC.
    [42] M. Vicanek and G. Simon, Momentum and heat transfer of an inert gas jet to the melt in laser cutting, Journal of Physics D: Applied Physics, 20 (1987), p1191. doi: 10.1088/0022-3727/20/9/016
    [43] V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei Roma, 2 (1926), 31-113.
    [44] G. Vossen and J. Schüttler, Mathematical modelling and stability analysis for laser cutting, Mathematical and Computer Modelling of Dynamical Systems, 18 (2012), 439-463. doi: 10.1080/13873954.2011.642387

  • This article has been cited by:

    1. Alexandre Bayen, Jean-Michel Coron, Nicola De Nitti, Alexander Keimer, Lukas Pflug, Boundary Controllability and Asymptotic Stabilization of a Nonlocal Traffic Flow Model, 2021, 49, 2305-221X, 957, 10.1007/s10013-021-00506-7
    2. Kuang Huang, Qiang Du, Stability of a Nonlocal Traffic Flow Model for Connected Vehicles, 2022, 82, 0036-1399, 221, 10.1137/20M1355732
    3. Rinaldo M. Colombo, Elena Rossi, Well‐posedness and control in a hyperbolic–parabolic parasitoid–parasite system, 2021, 147, 0022-2526, 839, 10.1111/sapm.12402
    4. Xiaoxuan Yu, Yan Xu, Qiang Du, Asymptotically compatible approximations of linear nonlocal conservation laws with variable horizon, 2022, 38, 0749-159X, 1948, 10.1002/num.22849
    5. Alexandre Bayen, Jan Friedrich, Alexander Keimer, Lukas Pflug, Tanya Veeravalli, Modeling Multilane Traffic with Moving Obstacles by Nonlocal Balance Laws, 2022, 21, 1536-0040, 1495, 10.1137/20M1366654
    6. Alberto Bressan, Wen Shen, On Traffic Flow with Nonlocal Flux: A Relaxation Representation, 2020, 237, 0003-9527, 1213, 10.1007/s00205-020-01529-z
    7. Jereme Chien, Wen Shen, Stationary wave profiles for nonlocal particle models of traffic flow on rough roads, 2019, 26, 1021-9722, 10.1007/s00030-019-0601-7
    8. Alexander Keimer, Lukas Pflug, 2023, 15708659, 10.1016/bs.hna.2022.11.001
    9. Francesca Marcellini, On the stability of a model for the cutting of metal plates by means of laser beams, 2017, 68, 08939659, 143, 10.1016/j.aml.2017.01.002
    10. Xiaoxuan Yu, Yan Xu, Qiang Du, Numerical Simulation of Singularity Propagation Modeled by Linear Convection Equations with Spatially Heterogeneous Nonlocal Interactions, 2022, 92, 0885-7474, 10.1007/s10915-022-01915-7
    11. Aekta Aggarwal, Helge Holden, Ganesh Vaidya, Well-posedness and error estimates for coupled systems of nonlocal conservation laws, 2024, 0272-4979, 10.1093/imanum/drad101
    12. Marco Inversi, Giorgio Stefani, 2025, 809, 978-1-4704-7860-5, 123, 10.1090/conm/809/16205
    13. Qunyu Yuan, Xiaoyu Cheng, Qing Huang, General Solutions of a Class of nth-Order Nonlocal Differential Equations, 2025, 24, 1575-5460, 10.1007/s12346-025-01268-0
  • Reader Comments
  • © 2016 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
Networks and Heterogeneous Media
1.2 1.8

Metrics

Article views(4728) PDF downloads(72) Cited by(13)

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