January  2017, 14(1): 31-43. doi: 10.3934/mbe.2017003

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

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

Received  October 22, 2015 Accepted  January 12, 2016 Published  October 2016

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.

Citation: Davide Bellandi. On the initial value problem for a class of discrete velocity models. Mathematical Biosciences & Engineering, 2017, 14 (1) : 31-43. doi: 10.3934/mbe.2017003
References:
[1] G. Ajmone MarsanN. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation, Springer, 2013.  doi: 10.1007/978-1-4614-7242-1.
[2]

L. ArlottiE. De AngelisL. FermoM. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl.Math. Lett., 25 (2012), 490-495.  doi: 10.1016/j.aml.2011.09.043.

[3]

N. Bellomo and A. Bellouquid, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, J. Differ. Equations, 252 (2012), 1350-1368.  doi: 10.1016/j.jde.2011.09.005.

[4]

N. BellomoV. Coscia and 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.  doi: 10.1142/S0218202502002343.

[5]

N. Bellomo and C. Dogbé, On the modelling of traffic and crowds -a survey of models, speculations and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677.

[6]

N. BellomoD. Knopoff and J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Mod. Meth. Appl. Sci., 23 (2013), 1861-1913.  doi: 10.1142/S021820251350053X.

[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. doi: 10.1142/S0218202512300049.

[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. doi: 10.1142/S0218202511400033.

[9] A. Bellouquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Birkhäuser, Boston, 2006. 
[10]

A. Benfenati and V. Coscia, Nonlinear microscale interactions in the kinetic theory of active particles, Appl. Math. Lett., 26 (2013), 979-983.  doi: 10.1016/j.aml.2013.04.007.

[11]

V. CosciaM. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow Ⅱ: Discrete velocity kinetic models, Int. J. Non-Linear Mech., 42 (2007), 411-421.  doi: 10.1016/j.ijnonlinmec.2006.02.008.

[12]

V. CosciaL. Fermo and N. Bellomo, On the mathematical theory of living systems Ⅱ: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011), 3902-3911.  doi: 10.1016/j.camwa.2011.09.043.

[13]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Mod. Meth. Appl. Sci., 17 (2007), 901-932.  doi: 10.1142/S0218202507002157.

[14]

L. Arlotti, N. Bellomo, E. De Angelis and M. Lachowicz, Generalized Kinetic Models in Applied Sciences World Scientific, New Jersey, 2003. doi: 10.1142/5359.

[15]

J. Banasiak and M. Lachowicz Methods of Small Parameter in Mathematical Biology Birkhauser, 2014. doi: 10.1007/978-3-319-05140-6.

[16]

S. Kaniel and 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. doi: 10.1090/cln/014.

show all references

References:
[1] G. Ajmone MarsanN. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation, Springer, 2013.  doi: 10.1007/978-1-4614-7242-1.
[2]

L. ArlottiE. De AngelisL. FermoM. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl.Math. Lett., 25 (2012), 490-495.  doi: 10.1016/j.aml.2011.09.043.

[3]

N. Bellomo and A. Bellouquid, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, J. Differ. Equations, 252 (2012), 1350-1368.  doi: 10.1016/j.jde.2011.09.005.

[4]

N. BellomoV. Coscia and 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.  doi: 10.1142/S0218202502002343.

[5]

N. Bellomo and C. Dogbé, On the modelling of traffic and crowds -a survey of models, speculations and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677.

[6]

N. BellomoD. Knopoff and J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Mod. Meth. Appl. Sci., 23 (2013), 1861-1913.  doi: 10.1142/S021820251350053X.

[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. doi: 10.1142/S0218202512300049.

[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. doi: 10.1142/S0218202511400033.

[9] A. Bellouquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Birkhäuser, Boston, 2006. 
[10]

A. Benfenati and V. Coscia, Nonlinear microscale interactions in the kinetic theory of active particles, Appl. Math. Lett., 26 (2013), 979-983.  doi: 10.1016/j.aml.2013.04.007.

[11]

V. CosciaM. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow Ⅱ: Discrete velocity kinetic models, Int. J. Non-Linear Mech., 42 (2007), 411-421.  doi: 10.1016/j.ijnonlinmec.2006.02.008.

[12]

V. CosciaL. Fermo and N. Bellomo, On the mathematical theory of living systems Ⅱ: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011), 3902-3911.  doi: 10.1016/j.camwa.2011.09.043.

[13]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Mod. Meth. Appl. Sci., 17 (2007), 901-932.  doi: 10.1142/S0218202507002157.

[14]

L. Arlotti, N. Bellomo, E. De Angelis and M. Lachowicz, Generalized Kinetic Models in Applied Sciences World Scientific, New Jersey, 2003. doi: 10.1142/5359.

[15]

J. Banasiak and M. Lachowicz Methods of Small Parameter in Mathematical Biology Birkhauser, 2014. doi: 10.1007/978-3-319-05140-6.

[16]

S. Kaniel and 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. doi: 10.1090/cln/014.

Figure 1.  Initial data corresponding to $\rho_c=5$. The class $\overline{f}_1$ of velocity $v_1=0$ is on the right in red color, the class $\overline{f}_2$ of velocity $v_2=1$ is represented by the bimodal distribution on the left in blue color
Figure 2.  Final distribution functions corresponding to $\Delta=0.5, 0.4,0.35,0.25$, and initial data as in Figure 1
Table 1.  Maximum $\rho_{t^*}$ of the density reached at time $t^*$ as function of the size $\Delta$ of the interaction domain $D_x$. Observe that in any case $\rho_{t^*}$ is greater that $\rho_c$
$\Delta$ $t^*$ $\rho_{t^*}$
1.05 4.5425 5.1890
1.00 4.5885 5.2214
0.95 4.6365 5.2660
0.90 4.6855 5.3269
0.85 4.7365 5.4092
0.80 4.7925 5.5192
0.75 4.8555 5.6652
0.70 4.9255 5.8575
0.65 5.0025 6.1090
0.60 5.4595 6.4658
0.55 5.4985 6.9726
0.50 5.5406 7.5731
0.45 5.5786 8.3056
0.40 5.6106 9.2196
0.35 5.6346 10.3606
0.30 5.6536 11.7122
0.25 5.6706 13.0508
0.20 5.6846 13.7962
0.15 5.6926 13.3582
$\Delta$ $t^*$ $\rho_{t^*}$
1.05 4.5425 5.1890
1.00 4.5885 5.2214
0.95 4.6365 5.2660
0.90 4.6855 5.3269
0.85 4.7365 5.4092
0.80 4.7925 5.5192
0.75 4.8555 5.6652
0.70 4.9255 5.8575
0.65 5.0025 6.1090
0.60 5.4595 6.4658
0.55 5.4985 6.9726
0.50 5.5406 7.5731
0.45 5.5786 8.3056
0.40 5.6106 9.2196
0.35 5.6346 10.3606
0.30 5.6536 11.7122
0.25 5.6706 13.0508
0.20 5.6846 13.7962
0.15 5.6926 13.3582
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