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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 |
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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