On the initial value problem for a class of discrete velocity models

Davide Bellandi; Davide Bellandi

doi:10.3934/mbe.2017003

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 31-43. doi: 10.3934/mbe.2017003

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On the initial value problem for a class of discrete velocity models

Davide Bellandi

University of Ferrara, Department of Mathematics and Computer Science, Via Machiavelli 35,44121 Ferrara, Italy

Received: 22 October 2015 Accepted: 12 January 2016 Published: 01 February 2017
MSC : Primary: 35L03, 35Q91; Secondary: 90B20

In this paper we investigate the initial value problem for a class of hyperbolic systems relating the mathematical modeling of a class of complex phenomena, with emphasis on vehicular traffic flow. Existence and uniqueness for large times of solutions, a basic requisite both for models building and for their numerical implementation, are obtained under weak hypotheses on the terms modeling the interaction among agents. The results are then compared with the existing literature on the subject.
- Complex systems,
- vehicular traffic,
- kinetic models,
- discrete velocity,
- global well posedness
Citation: Davide Bellandi. On the initial value problem for a class of discrete velocity models[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 31-43. doi: 10.3934/mbe.2017003

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Abstract

In this paper we investigate the initial value problem for a class of hyperbolic systems relating the mathematical modeling of a class of complex phenomena, with emphasis on vehicular traffic flow. Existence and uniqueness for large times of solutions, a basic requisite both for models building and for their numerical implementation, are obtained under weak hypotheses on the terms modeling the interaction among agents. The results are then compared with the existing literature on the subject.

References

[1]	[ G. Ajmone Marsan,N. Bellomo,A. Tosin, null, Complex Systems and Society: Modeling and Simulation, Springer, 2013.
[2]	[ L. Arlotti,E. De Angelis,L. Fermo,M. Lachowicz,N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl.Math. Lett., 25 (2012): 490-495.
[3]	[ N. Bellomo,A. Bellouquid, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, J. Differ. Equations, 252 (2012): 1350-1368.
[4]	[ N. Bellomo,V. Coscia,M. Delitala, On the mathematical theory of vehicular traffic fow Ⅰ -Fluid dynamic and kinetic modeling, Math. Mod. Meth. Appl. Sci., 12 (2002): 1801-1843.
[5]	[ N. Bellomo,C. Dogbé, On the modelling of traffic and crowds -a survey of models, speculations and perspectives, SIAM Rev., 53 (2011): 409-463.
[6]	[ N. Bellomo,D. Knopoff,J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Mod. Meth. Appl. Sci., 23 (2013): 1861-1913.
[7]	[ N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint Math. Mod. Meth. Appl. Sci. 22 (2012), 1230004, 29pp.
[8]	[ A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach Math. Mod. Meth. Appl. Sci. 22(2012), 1140003, 35pp.
[9]	[ A. Bellouquid,M. Delitala, null, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Birkhäuser, Boston, 2006.
[10]	[ A. Benfenati,V. Coscia, Nonlinear microscale interactions in the kinetic theory of active particles, Appl. Math. Lett., 26 (2013): 979-983.
[11]	[ V. Coscia,M. Delitala,P. Frasca, On the mathematical theory of vehicular traffic flow Ⅱ: Discrete velocity kinetic models, Int. J. Non-Linear Mech., 42 (2007): 411-421.
[12]	[ V. Coscia,L. Fermo,N. Bellomo, On the mathematical theory of living systems Ⅱ: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011): 3902-3911.
[13]	[ M. Delitala,A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Mod. Meth. Appl. Sci., 17 (2007): 901-932.
[14]	[ L. Arlotti, N. Bellomo, E. De Angelis and M. Lachowicz, Generalized Kinetic Models in Applied Sciences World Scientific, New Jersey, 2003.
[15]	[ J. Banasiak and M. Lachowicz Methods of Small Parameter in Mathematical Biology Birkhauser, 2014.
[16]	[ S. Kaniel,M. Shinbrot, The Boltzmann equation. Ⅰ. Uniqueness and local existence, Math. Phys., 58 (1978): 65-84.
[17]	[ B. S. Kerner, The Physics of Traffic, Empirical Freeway Pattern Features Engineering Applications and Theory, Springer, 2004.
[18]	[ P. Lax, Hyperbolic Partial Differential Equations Courant Lecture Notes, 2006.

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