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

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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

Rinaldo M. Colombo ^1, , Francesca Marcellini ^2, and Elena Rossi ^2,

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

Abstract
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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.

Keywords: predatory -- prey systems., Nonlocal conservation laws, parabolic-hyperbolic problems, laser cutting modeling.

Mathematics Subject Classification: Primary: 35L65; Secondary: 35M30, 92D2.

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.

show all references

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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