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November  2015, 20(9): 3013-3027. doi: 10.3934/dcdsb.2015.20.3013

Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation

Thi-Bich-Ngoc Mac 1,

1. 

1-Université de Toulouse; UPS, INSA, UT1, UTM, 2-CNRS, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

Received  July 2014 Revised  May 2015 Published  September 2015

This paper is devoted to study the existence of solutions of hydrodynamic model for systems of self-propelled particles subject to alignment and volume exclusion interactions. On one hand, we prove the existence of solutions by using the modified Garlerkin method for quasi-linear parabolic simulations. On the other hand, we also perform simulations to compare theoretical and numerical results. The numerical results show that the numerical solutions exist for short time in some cases of coefficients.
Keywords: vortex problem., Non-conservative equations, existence of solution, numerical solution.
Mathematics Subject Classification: 35Q80, 35L60, 82C22, 82C70, 92D5.
Citation: Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013
References:
[1]

A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249-1278. doi: 10.3934/dcdsb.2014.19.1249.

[2]

A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9PP. doi: 10.1103/PhysRevE.77.011920.

[3]

E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001.

[4]

A. Cziròk and T. Vicsek, Collective behavior of interacting self-propelled particles, Physica A, 281 (2000), 17-29. doi: 10.1016/S0378-4371(00)00013-3.

[5]

P. Degond, G. Dimarco , T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Communications in Mathematical Sciences, 13 (2015), 1615-1638. doi: 10.4310/CMS.2015.v13.n6.a12.

[6]

P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456. doi: 10.1007/s00332-012-9157-y.

[7]

P. Degond, A. Frouvelle, J.-G. Liu, S. Motsch and L. Navoret, Macroscopic models of collective motion and self-organization,, Séminaire Laurent Schwartz — EDP et applications, (): 2012.  doi: 10.5802/slsedp.32.

[8]

P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Archive for Rational Mechanics and Analysis, 216 (2015), 63-115. doi: 10.1007/s00205-014-0800-7.

[9]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[10]

P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114. doi: 10.4310/MAA.2013.v20.n2.a1.

[11]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Mod. Meth. Appl. Sci., 22 (2012), 1250011, 40pp. doi: 10.1142/S021820251250011X.

[12]

A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal., 44 (2012), 791-826. doi: 10.1137/110823912.

[13]

S. Henkes, Y. Fily and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density, Phys. Rev. E, 84 (2011), 040301. doi: 10.1103/PhysRevE.84.040301.

[14]

S. Motsch and L. Navoret, Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simul., 9 (2011), 1253-1275. doi: 10.1137/100794067.

[15]

F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods, Phys. Rev. E, 74 (2006), 030904(R). doi: 10.1103/PhysRevE.74.030904.

[16]

V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self propelling particles with kinematic constraints: the relations between the discrete and the continuous description, Phys. A, 381 (2007), 39-46. doi: 10.1016/j.physa.2007.03.045.

[17]

M. E. Taylor, Partial Differential Equations III, Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[18]

Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Phys. Rev. Lett., 80 (1998), 4819-4822. doi: 10.1103/PhysRevLett.80.4819.

[19]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249-1278. doi: 10.3934/dcdsb.2014.19.1249.

[2]

A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9PP. doi: 10.1103/PhysRevE.77.011920.

[3]

E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001.

[4]

A. Cziròk and T. Vicsek, Collective behavior of interacting self-propelled particles, Physica A, 281 (2000), 17-29. doi: 10.1016/S0378-4371(00)00013-3.

[5]

P. Degond, G. Dimarco , T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Communications in Mathematical Sciences, 13 (2015), 1615-1638. doi: 10.4310/CMS.2015.v13.n6.a12.

[6]

P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456. doi: 10.1007/s00332-012-9157-y.

[7]

P. Degond, A. Frouvelle, J.-G. Liu, S. Motsch and L. Navoret, Macroscopic models of collective motion and self-organization,, Séminaire Laurent Schwartz — EDP et applications, (): 2012.  doi: 10.5802/slsedp.32.

[8]

P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Archive for Rational Mechanics and Analysis, 216 (2015), 63-115. doi: 10.1007/s00205-014-0800-7.

[9]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[10]

P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114. doi: 10.4310/MAA.2013.v20.n2.a1.

[11]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Mod. Meth. Appl. Sci., 22 (2012), 1250011, 40pp. doi: 10.1142/S021820251250011X.

[12]

A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal., 44 (2012), 791-826. doi: 10.1137/110823912.

[13]

S. Henkes, Y. Fily and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density, Phys. Rev. E, 84 (2011), 040301. doi: 10.1103/PhysRevE.84.040301.

[14]

S. Motsch and L. Navoret, Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simul., 9 (2011), 1253-1275. doi: 10.1137/100794067.

[15]

F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods, Phys. Rev. E, 74 (2006), 030904(R). doi: 10.1103/PhysRevE.74.030904.

[16]

V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self propelling particles with kinematic constraints: the relations between the discrete and the continuous description, Phys. A, 381 (2007), 39-46. doi: 10.1016/j.physa.2007.03.045.

[17]

M. E. Taylor, Partial Differential Equations III, Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[18]

Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Phys. Rev. Lett., 80 (1998), 4819-4822. doi: 10.1103/PhysRevLett.80.4819.

[19]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

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