# American Institute of Mathematical Sciences

March  2016, 11(1): 49-67. doi: 10.3934/nhm.2016.11.49

## 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, Italy 2 Dip. di Matematica e Applicazioni, Università di Milano - Bicocca, Via Cozzi, 55; 20125 Milano, Italy, Italy

Received  April 2015 Revised  September 2015 Published  January 2016

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. Networks and Heterogeneous Media, 2016, 11 (1) : 49-67. doi: 10.3934/nhm.2016.11.49
##### References:
 [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] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, 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] R. M. Colombo and F. Marcellini, A traffic model aware of real time data, 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] E. Rossi and V. Schleper, Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions, 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.

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##### References:
 [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] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, 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] R. M. Colombo and F. Marcellini, A traffic model aware of real time data, 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] E. Rossi and V. Schleper, Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions, 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.
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