2017, 10(1): 171-192. doi: 10.3934/krm.2017007

Explicit equilibrium solutions for the aggregation equation with power-law potentials

1. 

Department of Mathematics, Imperial College London London SW7 2AZ, United Kingdom

2. 

School of Mathematics, The University of Manchester Manchester M13 9PL, United Kingdom

Received  February 2016 Revised  June 2016 Published  November 2016

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernelsare constructed by inverting Fredholm integral operators or byemploying certain integral identities. These solutions are expected tobe the global energy stable equilibria and to characterize the generic behaviorsof stationary solutions for more general interactions.

Citation: José A. Carrillo, Yanghong Huang. Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinetic & Related Models, 2017, 10 (1) : 171-192. doi: 10.3934/krm.2017007
References:
[1]

M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, second edition, 2003.

[2]

G. Albi, D. Balagué, J. A. Carrillo, J. von Brecht, Stability analysis of flock and mill rings for second order models in swarming, SIAM J. Appl. Math., 74 (2014), 794-818. doi: 10.1137/13091779X.

[3]

D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088. doi: 10.1007/s00205-013-0644-6.

[4]

D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25. doi: 10.1016/j.physd.2012.10.002.

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, volume 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Revised reprint of the 1979 original.

[6]

A. J. Bernoff, C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250. doi: 10.1137/100804504.

[7]

A. L. Bertozzi, T. Kolokolnikov, H. Sun, D. Uminsky, J. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), 955-985. doi: 10.4310/CMS.2015.v13.n4.a6.

[8]

M. Bessemoulin-Chatard, F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 34 (2012), B559-B583. doi: 10.1137/110853807.

[9]

J. A. Cañizo, J. A. Carrillo, F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217. doi: 10.1007/s00205-015-0852-3.

[10]

J. A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a.

[11]

J. A. Carrillo, M. Chipot and Y. Huang, On global minimizers of repulsive-attractive power-law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 372(2014), 20130399, 13pp.

[12]

J. A. Carrillo, Y. -P. Choi and M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances, In Collective dynamics from bacteria to crowds, volume 553 of CISM Courses and Lectures, pages 1-46. Springer, Vienna, 2014.

[13]

J. A. Carrillo, M. G. Delgadino, A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys., 343 (2016), 747-781. doi: 10.1007/s00220-016-2598-7.

[14]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211.

[15]

J. A. Carrillo, M. R. D'Orsogna, V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363.

[16]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, In Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol. , pages 297-336. Birkhäuser Boston, Inc. , Boston, MA, 2010.

[17]

J. A. Carrillo, Y. Huang, S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578. doi: 10.1017/S0956792514000126.

[18]

J. A. Carrillo, Y. Huang, S. Martin, Nonlinear stability of flock solutions in second-order swarming models, Nonlinear Anal. Real World Appl., 17 (2014), 332-343. doi: 10.1016/j.nonrwa.2013.12.008.

[19]

J. A. Carrillo, A. Klar, S. Martin, S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.

[20]

J. A. Carrillo, A. Klar, A. Roth, Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136. doi: 10.4310/CMS.2016.v14.n4.a12.

[21]

J. A. Carrillo, S. Martin, V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126. doi: 10.1016/j.physd.2013.02.004.

[22]

J. A. Carrillo, R. J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376.

[23]

J. A. Carrillo and J. L. Vázquez, Some free boundary problems involving non-local diffusion and aggregation, Philos. Trans. A, 373(2015), 20140275, 16pp.

[24]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi, L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007.

[25]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi, L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Physical review letters, 96 (2006), 104302.

[26]

R. Estrada and R. P. Kanwal, Singular Integral Equations, Birkhäuser Boston, Inc. , Boston, MA, 2000.

[27]

K. Fellner, G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291. doi: 10.1142/S0218202510004921.

[28]

K. Fellner, G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), 1436-1450. doi: 10.1016/j.mcm.2010.03.021.

[29]

R. C. Fetecau, Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64. doi: 10.1016/j.physd.2012.11.004.

[30]

R. C. Fetecau, Y. Huang, T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716. doi: 10.1088/0951-7715/24/10/002.

[31]

D. D. Holm, V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles, Phys. D, 220 (2006), 183-196. doi: 10.1016/j.physd.2006.07.010.

[32]

Y. Huang, Explicit Barenblatt profiles for fractional porous medium equations, Bull. Lond. Math. Soc., 46 (2014), 857-869. doi: 10.1112/blms/bdu045.

[33]

B. D. Hughes, K. Fellner, Continuum models of cohesive stochastic swarms: The effect of motility on aggregation patterns, Phys. D, 260 (2013), 26-48. doi: 10.1016/j.physd.2013.05.001.

[34]

T. Kolokolnikov, J. A. Carrillo, A. Bertozzi, R. Fetecau, M. Lewis, Emergent behaviour in multi-particle systems with non-local interactions [Editorial], Phys. D, 260 (2013), 1-4. doi: 10.1016/j.physd.2013.06.011.

[35]

T. Kolokolnikov, H. Sun, D. Uminsky, A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011), 015203. doi: 10.1103/PhysRevE.84.015203.

[36]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 180.

[37]

A. J. Leverentz, C. M. Topaz, A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908. doi: 10.1137/090749037.

[38]

H. Levine, W.-J. Rappel, I. Cohen, Self-organization in systems of self-propelled particles, Physical Review E, 63 (2000), 017101. doi: 10.1103/PhysRevE.63.017101.

[39]

A. Mogilner, L. Edelstein-Keshet, A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570. doi: 10.1007/s002850050158.

[40]

L. Ryan, Analysis of swarm behavior in two dimensions, 2012. HMC Senior Theses.

[41]

R. Simione, D. Slepčev, I. Topaloglu, Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986. doi: 10.1007/s10955-015-1215-z.

[42]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.

show all references

References:
[1]

M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, second edition, 2003.

[2]

G. Albi, D. Balagué, J. A. Carrillo, J. von Brecht, Stability analysis of flock and mill rings for second order models in swarming, SIAM J. Appl. Math., 74 (2014), 794-818. doi: 10.1137/13091779X.

[3]

D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088. doi: 10.1007/s00205-013-0644-6.

[4]

D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25. doi: 10.1016/j.physd.2012.10.002.

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, volume 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Revised reprint of the 1979 original.

[6]

A. J. Bernoff, C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250. doi: 10.1137/100804504.

[7]

A. L. Bertozzi, T. Kolokolnikov, H. Sun, D. Uminsky, J. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), 955-985. doi: 10.4310/CMS.2015.v13.n4.a6.

[8]

M. Bessemoulin-Chatard, F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 34 (2012), B559-B583. doi: 10.1137/110853807.

[9]

J. A. Cañizo, J. A. Carrillo, F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217. doi: 10.1007/s00205-015-0852-3.

[10]

J. A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a.

[11]

J. A. Carrillo, M. Chipot and Y. Huang, On global minimizers of repulsive-attractive power-law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 372(2014), 20130399, 13pp.

[12]

J. A. Carrillo, Y. -P. Choi and M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances, In Collective dynamics from bacteria to crowds, volume 553 of CISM Courses and Lectures, pages 1-46. Springer, Vienna, 2014.

[13]

J. A. Carrillo, M. G. Delgadino, A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys., 343 (2016), 747-781. doi: 10.1007/s00220-016-2598-7.

[14]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211.

[15]

J. A. Carrillo, M. R. D'Orsogna, V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363.

[16]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, In Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol. , pages 297-336. Birkhäuser Boston, Inc. , Boston, MA, 2010.

[17]

J. A. Carrillo, Y. Huang, S. Martin, Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578. doi: 10.1017/S0956792514000126.

[18]

J. A. Carrillo, Y. Huang, S. Martin, Nonlinear stability of flock solutions in second-order swarming models, Nonlinear Anal. Real World Appl., 17 (2014), 332-343. doi: 10.1016/j.nonrwa.2013.12.008.

[19]

J. A. Carrillo, A. Klar, S. Martin, S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.

[20]

J. A. Carrillo, A. Klar, A. Roth, Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136. doi: 10.4310/CMS.2016.v14.n4.a12.

[21]

J. A. Carrillo, S. Martin, V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126. doi: 10.1016/j.physd.2013.02.004.

[22]

J. A. Carrillo, R. J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376.

[23]

J. A. Carrillo and J. L. Vázquez, Some free boundary problems involving non-local diffusion and aggregation, Philos. Trans. A, 373(2015), 20140275, 16pp.

[24]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi, L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007.

[25]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi, L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Physical review letters, 96 (2006), 104302.

[26]

R. Estrada and R. P. Kanwal, Singular Integral Equations, Birkhäuser Boston, Inc. , Boston, MA, 2000.

[27]

K. Fellner, G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291. doi: 10.1142/S0218202510004921.

[28]

K. Fellner, G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), 1436-1450. doi: 10.1016/j.mcm.2010.03.021.

[29]

R. C. Fetecau, Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64. doi: 10.1016/j.physd.2012.11.004.

[30]

R. C. Fetecau, Y. Huang, T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716. doi: 10.1088/0951-7715/24/10/002.

[31]

D. D. Holm, V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles, Phys. D, 220 (2006), 183-196. doi: 10.1016/j.physd.2006.07.010.

[32]

Y. Huang, Explicit Barenblatt profiles for fractional porous medium equations, Bull. Lond. Math. Soc., 46 (2014), 857-869. doi: 10.1112/blms/bdu045.

[33]

B. D. Hughes, K. Fellner, Continuum models of cohesive stochastic swarms: The effect of motility on aggregation patterns, Phys. D, 260 (2013), 26-48. doi: 10.1016/j.physd.2013.05.001.

[34]

T. Kolokolnikov, J. A. Carrillo, A. Bertozzi, R. Fetecau, M. Lewis, Emergent behaviour in multi-particle systems with non-local interactions [Editorial], Phys. D, 260 (2013), 1-4. doi: 10.1016/j.physd.2013.06.011.

[35]

T. Kolokolnikov, H. Sun, D. Uminsky, A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011), 015203. doi: 10.1103/PhysRevE.84.015203.

[36]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 180.

[37]

A. J. Leverentz, C. M. Topaz, A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908. doi: 10.1137/090749037.

[38]

H. Levine, W.-J. Rappel, I. Cohen, Self-organization in systems of self-propelled particles, Physical Review E, 63 (2000), 017101. doi: 10.1103/PhysRevE.63.017101.

[39]

A. Mogilner, L. Edelstein-Keshet, A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570. doi: 10.1007/s002850050158.

[40]

L. Ryan, Analysis of swarm behavior in two dimensions, 2012. HMC Senior Theses.

[41]

R. Simione, D. Slepčev, I. Topaloglu, Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986. doi: 10.1007/s10955-015-1215-z.

[42]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.

Figure 1.  The steady states exist only on the shaded region and their expressions are obtained from the reduced governing equation
Figure 2.  Comparison between the compacted supported solutions (17) and the more singular steady states (22) with unit total mass ($M_0=1$): (a) the energy; (b) the radius of support
Figure 3.  The comparison of $K*\rho$ and $K*\rho_\delta$ at $a=5/2$ and $b=2$, for two solutions (17) and (22). The black dots indicate the boundary of the support of (17) and the support of (22)
Figure 4.  The compactly supported steady states exist only in the shaded region
Figure 5.  The radial profiles of the constructed solutions for $a=4$. As $b$ increase from $2-d$ (the Newtonian potential) to $b_{max} = (2+3d-d^2)/(d+1)$, the solution becomes negative starting at $\bar{b}=(2+2d-d^2)/(d+1)$
Figure 6.  The two limits of a sectionally analytical function $\Psi$ along the line segment connecting $-R$ and $R$
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