American Institute of Mathematical Sciences

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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

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 & 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.  doi: 10.1137/140975255.  Google Scholar

[2]

D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes' model for pedestrian flows,, J. Math. Anal. Appl., 420 (2014), 387.  doi: 10.1016/j.jmaa.2014.05.072.  Google Scholar

[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.  doi: 10.1080/03605300902892279.  Google Scholar

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics,, Birkhäuser Boston Inc., (1995).  doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[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.  doi: 10.1080/03605307908820117.  Google Scholar

[6]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512300049.  Google Scholar

[7]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling,, Numer. Math., ().  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.11.003.  Google Scholar

[9]

C. Christoforou, Systems of hyperbolic conservation laws with memory,, J. Hyperbolic Differ. Equ., 4 (2007), 435.  doi: 10.1142/S0219891607001215.  Google Scholar

[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.  doi: 10.1016/j.apm.2013.02.031.  Google Scholar

[11]

R. M. Colombo and F. Marcellini, A traffic model aware of real time data,, M3AS, ().   Google Scholar

[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).  doi: 10.1142/S0218202511500230.  Google Scholar

[13]

R. M. Colombo and G. Guerra, Hyperbolic balance laws with a non local source,, Comm. Partial Differential Equations, 32 (2007), 1917.  doi: 10.1080/03605300701318849.  Google Scholar

[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.  doi: 10.1090/S0033-569X-2012-01309-2.  Google Scholar

[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.  doi: 10.1051/cocv/2010007.  Google Scholar

[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.  doi: 10.1016/j.jde.2015.08.005.  Google Scholar

[17]

R. M. Colombo and L.-M. Mercier, Nonlocal crowd dynamics models for several populations,, Acta Mathematica Scientia, 32 (2012), 177.  doi: 10.1016/S0252-9602(12)60011-3.  Google Scholar

[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.  doi: 10.4310/CMS.2009.v7.n1.a2.  Google Scholar

[19]

R. M. Colombo and E. Rossi, Hyperbolic predators vs. parabolic prey,, Commun. Math. Sci., 13 (2015), 369.  doi: 10.4310/CMS.2015.v13.n2.a6.  Google Scholar

[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.  doi: 10.1016/S0252-9602(15)30028-X.  Google Scholar

[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.  doi: 10.1007/s00030-012-0164-3.  Google Scholar

[22]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, 3rd edition, (2010).  doi: 10.1007/978-3-642-04048-1.  Google Scholar

[23]

A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall Inc., (1964).   Google Scholar

[24]

M. S. Gross, On gas dynamics effects in the modelling of laser cutting processes,, Appl. Math. Model., 30 (2006), 307.  doi: 10.1016/j.apm.2005.03.021.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.09.003.  Google Scholar

[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).  doi: 10.1088/0022-3727/44/10/105502.  Google Scholar

[27]

S. N. Kružhkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar

[28]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas,, J. Differential Equations, 190 (2003), 439.  doi: 10.1016/S0022-0396(02)00158-4.  Google Scholar

[29]

R. J. LeVeque, Numerical Methods for Conservation Laws,, 2nd edition, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[30]

A. J. Lotka, Elements of Physical Biology,, Williams and Wilkins, (1925).   Google Scholar

[31]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Progress in Nonlinear Differential Equations and their Applications, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[32]

J. D. Murray, Mathematical Biology. II, vol. 18 of Interdisciplinary Applied Mathematics,, 3rd edition, (2003).   Google Scholar

[33]

B. Perthame, Transport Equations in Biology,, Frontiers in Mathematics, (2007).   Google Scholar

[34]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[35]

P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).   Google Scholar

[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, ().  doi: 10.1051/m2an/2015050.  Google Scholar

[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.  doi: 10.1016/S0017-9310(96)00342-0.  Google Scholar

[38]

W. Schulz, M. Nießen, U. Eppelt and K. Kowalick, Simulation of laser cutting,, in Springer Series in Materials Science, 119 (2009), 21.  doi: 10.1007/978-1-4020-9340-1_2.  Google Scholar

[39]

W. Schulz, G. Simon, H. Urbassek and I. Decker, On laser fusion cutting of metals,, J. Phys. D - Appl. Phys, 20 (1987).  doi: 10.1088/0022-3727/20/4/013.  Google Scholar

[40]

D. Serre, Systems of Conservation Laws. 1 & 2,, Cambridge University Press, (1999).  doi: 10.1017/CBO9780511612374.  Google Scholar

[41]

W. Steen, Laser Material Processing,, Springer, (2003).   Google Scholar

[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).  doi: 10.1088/0022-3727/20/9/016.  Google Scholar

[43]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,, Mem. Acad. Lincei Roma, 2 (1926), 31.   Google Scholar

[44]

G. Vossen and J. Schüttler, Mathematical modelling and stability analysis for laser cutting,, Mathematical and Computer Modelling of Dynamical Systems, 18 (2012), 439.  doi: 10.1080/13873954.2011.642387.  Google Scholar

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.  doi: 10.1137/140975255.  Google Scholar

[2]

D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes' model for pedestrian flows,, J. Math. Anal. Appl., 420 (2014), 387.  doi: 10.1016/j.jmaa.2014.05.072.  Google Scholar

[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.  doi: 10.1080/03605300902892279.  Google Scholar

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics,, Birkhäuser Boston Inc., (1995).  doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[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.  doi: 10.1080/03605307908820117.  Google Scholar

[6]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512300049.  Google Scholar

[7]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling,, Numer. Math., ().  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.11.003.  Google Scholar

[9]

C. Christoforou, Systems of hyperbolic conservation laws with memory,, J. Hyperbolic Differ. Equ., 4 (2007), 435.  doi: 10.1142/S0219891607001215.  Google Scholar

[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.  doi: 10.1016/j.apm.2013.02.031.  Google Scholar

[11]

R. M. Colombo and F. Marcellini, A traffic model aware of real time data,, M3AS, ().   Google Scholar

[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).  doi: 10.1142/S0218202511500230.  Google Scholar

[13]

R. M. Colombo and G. Guerra, Hyperbolic balance laws with a non local source,, Comm. Partial Differential Equations, 32 (2007), 1917.  doi: 10.1080/03605300701318849.  Google Scholar

[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.  doi: 10.1090/S0033-569X-2012-01309-2.  Google Scholar

[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.  doi: 10.1051/cocv/2010007.  Google Scholar

[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.  doi: 10.1016/j.jde.2015.08.005.  Google Scholar

[17]

R. M. Colombo and L.-M. Mercier, Nonlocal crowd dynamics models for several populations,, Acta Mathematica Scientia, 32 (2012), 177.  doi: 10.1016/S0252-9602(12)60011-3.  Google Scholar

[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.  doi: 10.4310/CMS.2009.v7.n1.a2.  Google Scholar

[19]

R. M. Colombo and E. Rossi, Hyperbolic predators vs. parabolic prey,, Commun. Math. Sci., 13 (2015), 369.  doi: 10.4310/CMS.2015.v13.n2.a6.  Google Scholar

[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.  doi: 10.1016/S0252-9602(15)30028-X.  Google Scholar

[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.  doi: 10.1007/s00030-012-0164-3.  Google Scholar

[22]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, 3rd edition, (2010).  doi: 10.1007/978-3-642-04048-1.  Google Scholar

[23]

A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall Inc., (1964).   Google Scholar

[24]

M. S. Gross, On gas dynamics effects in the modelling of laser cutting processes,, Appl. Math. Model., 30 (2006), 307.  doi: 10.1016/j.apm.2005.03.021.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.09.003.  Google Scholar

[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).  doi: 10.1088/0022-3727/44/10/105502.  Google Scholar

[27]

S. N. Kružhkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar

[28]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas,, J. Differential Equations, 190 (2003), 439.  doi: 10.1016/S0022-0396(02)00158-4.  Google Scholar

[29]

R. J. LeVeque, Numerical Methods for Conservation Laws,, 2nd edition, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[30]

A. J. Lotka, Elements of Physical Biology,, Williams and Wilkins, (1925).   Google Scholar

[31]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Progress in Nonlinear Differential Equations and their Applications, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[32]

J. D. Murray, Mathematical Biology. II, vol. 18 of Interdisciplinary Applied Mathematics,, 3rd edition, (2003).   Google Scholar

[33]

B. Perthame, Transport Equations in Biology,, Frontiers in Mathematics, (2007).   Google Scholar

[34]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[35]

P. Quittner and P. Souplet, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], (2007).   Google Scholar

[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, ().  doi: 10.1051/m2an/2015050.  Google Scholar

[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.  doi: 10.1016/S0017-9310(96)00342-0.  Google Scholar

[38]

W. Schulz, M. Nießen, U. Eppelt and K. Kowalick, Simulation of laser cutting,, in Springer Series in Materials Science, 119 (2009), 21.  doi: 10.1007/978-1-4020-9340-1_2.  Google Scholar

[39]

W. Schulz, G. Simon, H. Urbassek and I. Decker, On laser fusion cutting of metals,, J. Phys. D - Appl. Phys, 20 (1987).  doi: 10.1088/0022-3727/20/4/013.  Google Scholar

[40]

D. Serre, Systems of Conservation Laws. 1 & 2,, Cambridge University Press, (1999).  doi: 10.1017/CBO9780511612374.  Google Scholar

[41]

W. Steen, Laser Material Processing,, Springer, (2003).   Google Scholar

[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).  doi: 10.1088/0022-3727/20/9/016.  Google Scholar

[43]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,, Mem. Acad. Lincei Roma, 2 (1926), 31.   Google Scholar

[44]

G. Vossen and J. Schüttler, Mathematical modelling and stability analysis for laser cutting,, Mathematical and Computer Modelling of Dynamical Systems, 18 (2012), 439.  doi: 10.1080/13873954.2011.642387.  Google Scholar

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