July  2014, 19(5): 1227-1248. doi: 10.3934/dcdsb.2014.19.1227

Confinement for repulsive-attractive kernels

1. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States

2. 

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

3. 

Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States

Received  August 2012 Revised  November 2012 Published  April 2014

We investigate the confinement properties of solutions of the aggregation equation with repulsive-attractive potentials. We show that solutions remain compactly supported in a large fixed ball depending on the initial data and the potential. The arguments apply to the functional setting of probability measures with mildly singular repulsive-attractive potentials and to the functional setting of smooth solutions with a potential being the sum of the Newtonian repulsion at the origin and a smooth suitably growing at infinity attractive potential.
Citation: Daniel Balagué, José A. Carrillo, Yao Yao. Confinement for repulsive-attractive kernels. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1227-1248. doi: 10.3934/dcdsb.2014.19.1227
References:
[1]

L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217.  doi: 10.1016/j.anihpc.2010.11.006.  Google Scholar

[2]

L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495.  doi: 10.1002/cpa.20223.  Google Scholar

[3]

L. A. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, (2005).   Google Scholar

[4]

D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials,, In Hyperbolic problems-theory, (2012), 136.   Google Scholar

[5]

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

[6]

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

[7]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615.   Google Scholar

[8]

A. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683.  doi: 10.1088/0951-7715/22/3/009.  Google Scholar

[9]

A. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pure Appl. Math., 64 (2011), 45.  doi: 10.1002/cpa.20334.  Google Scholar

[10]

A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202511400057.  Google Scholar

[11]

J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Confinement in nonlocal interaction equations,, Nonlinear Anal., 75 (2012), 550.  doi: 10.1016/j.na.2011.08.057.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Ration. Mech. Anal., 179 (2006), 217.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[16]

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

[17]

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

[18]

M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[19]

Q. Du and P. Zhang, Existence of weak solutions to some vortex density models,, SIAM J. Math. Anal., 34 (2003), 1279.  doi: 10.1137/S0036141002408009.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

E. Geigant, K. Ladizhansky and A. Mogilner, An integrodifferential model for orientational distributions of f-actin in cells,, SIAM Journal on Applied Mathematics, 59 (1998), 787.  doi: 10.1088/0951-7715/24/10/002.  Google Scholar

[25]

K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An integro-differential equation model for alignment and orientational aggregation,, J. Differential Equations, 264 (2009), 1387.  doi: 10.1016/j.jde.2008.11.006.  Google Scholar

[26]

T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential,, Phys. D, 260 (2013), 65.  doi: 10.1016/j.physd.2012.10.009.  Google Scholar

[27]

T. Kolokonikov, H. Sun, D. Uminsky, and A. Bertozzi., Stability of ring patterns arising from 2d particle interactions,, Phys. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.015203.  Google Scholar

[28]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows,, Arch. Ration. Mech. Anal., 172 (2004), 407.  doi: 10.1007/s00205-004-0307-8.  Google Scholar

[29]

F. Lin and P. Zhang, On the hydrodynamic limit of ginzburg-landau vortices,, Discrete Contin. Dynam. Systems, 6 (2000), 121.  doi: 10.3934/dcds.2000.6.121.  Google Scholar

[30]

E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity,, Boll. Unione Mat. Ital. (9), 2 (2009), 509.   Google Scholar

[31]

N. Masmoudi and P. Zhang, Global solutions to vortex density equations arising from sup-conductivity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 441.  doi: 10.1016/j.anihpc.2004.07.002.  Google Scholar

[32]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534.  doi: 10.1007/s002850050158.  Google Scholar

[33]

I. Primi, A. Stevens and J. J. L. Velázquez, Mass-selection in alignment models with non-deterministic effects,, Comm. Partial Differential Equations, 34 (2009), 419.  doi: 10.1080/03605300902797171.  Google Scholar

[34]

G. Raoul, Nonlocal interaction equations: Stationary states and stability analysis,, Differential Integral Equations, 25 (2012), 417.  doi: 10.1016/S0012-9593(00)00122-1.  Google Scholar

[35]

D. Ruelle, Statistical Mechanics: RIgorous Results,, W. A. Benjamin, (1969).   Google Scholar

[36]

E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity,, Ann. Sci. École Norm. Sup., 33 (2000), 561.  doi: 10.1016/S0012-9593(00)00122-1.  Google Scholar

[37]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model,, Progress in Nonlinear Differential Equations and their Applications, (2007).   Google Scholar

[38]

H. Sun, D. Uminsky and A. L. Bertozzi, Stability and clustering of self-similar solutions of aggregation equations,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.4745180.  Google Scholar

[39]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152.  doi: 10.1137/S0036139903437424.  Google Scholar

[40]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bulletin of Mathematical Biology, 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[41]

G. Toscani, One-dimensional kinetic models of granular flows,, M2AN Math. Model. Numer. Anal., 34 (2000), 1277.  doi: 10.1051/m2an:2000127.  Google Scholar

[42]

J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics,, J. Nonlinear Sci., 22 (2012), 935.  doi: 10.1007/s00332-012-9132-7.  Google Scholar

[43]

J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi, Predicting pattern formation in particle interactions,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202511400021.  Google Scholar

[44]

E. Weinan, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity,, Phys. D, 77 (1994), 383.  doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

show all references

References:
[1]

L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217.  doi: 10.1016/j.anihpc.2010.11.006.  Google Scholar

[2]

L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495.  doi: 10.1002/cpa.20223.  Google Scholar

[3]

L. A. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, (2005).   Google Scholar

[4]

D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials,, In Hyperbolic problems-theory, (2012), 136.   Google Scholar

[5]

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

[6]

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

[7]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615.   Google Scholar

[8]

A. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683.  doi: 10.1088/0951-7715/22/3/009.  Google Scholar

[9]

A. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pure Appl. Math., 64 (2011), 45.  doi: 10.1002/cpa.20334.  Google Scholar

[10]

A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202511400057.  Google Scholar

[11]

J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Confinement in nonlocal interaction equations,, Nonlinear Anal., 75 (2012), 550.  doi: 10.1016/j.na.2011.08.057.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Ration. Mech. Anal., 179 (2006), 217.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[16]

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

[17]

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

[18]

M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[19]

Q. Du and P. Zhang, Existence of weak solutions to some vortex density models,, SIAM J. Math. Anal., 34 (2003), 1279.  doi: 10.1137/S0036141002408009.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

E. Geigant, K. Ladizhansky and A. Mogilner, An integrodifferential model for orientational distributions of f-actin in cells,, SIAM Journal on Applied Mathematics, 59 (1998), 787.  doi: 10.1088/0951-7715/24/10/002.  Google Scholar

[25]

K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An integro-differential equation model for alignment and orientational aggregation,, J. Differential Equations, 264 (2009), 1387.  doi: 10.1016/j.jde.2008.11.006.  Google Scholar

[26]

T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential,, Phys. D, 260 (2013), 65.  doi: 10.1016/j.physd.2012.10.009.  Google Scholar

[27]

T. Kolokonikov, H. Sun, D. Uminsky, and A. Bertozzi., Stability of ring patterns arising from 2d particle interactions,, Phys. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.015203.  Google Scholar

[28]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows,, Arch. Ration. Mech. Anal., 172 (2004), 407.  doi: 10.1007/s00205-004-0307-8.  Google Scholar

[29]

F. Lin and P. Zhang, On the hydrodynamic limit of ginzburg-landau vortices,, Discrete Contin. Dynam. Systems, 6 (2000), 121.  doi: 10.3934/dcds.2000.6.121.  Google Scholar

[30]

E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity,, Boll. Unione Mat. Ital. (9), 2 (2009), 509.   Google Scholar

[31]

N. Masmoudi and P. Zhang, Global solutions to vortex density equations arising from sup-conductivity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 441.  doi: 10.1016/j.anihpc.2004.07.002.  Google Scholar

[32]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534.  doi: 10.1007/s002850050158.  Google Scholar

[33]

I. Primi, A. Stevens and J. J. L. Velázquez, Mass-selection in alignment models with non-deterministic effects,, Comm. Partial Differential Equations, 34 (2009), 419.  doi: 10.1080/03605300902797171.  Google Scholar

[34]

G. Raoul, Nonlocal interaction equations: Stationary states and stability analysis,, Differential Integral Equations, 25 (2012), 417.  doi: 10.1016/S0012-9593(00)00122-1.  Google Scholar

[35]

D. Ruelle, Statistical Mechanics: RIgorous Results,, W. A. Benjamin, (1969).   Google Scholar

[36]

E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity,, Ann. Sci. École Norm. Sup., 33 (2000), 561.  doi: 10.1016/S0012-9593(00)00122-1.  Google Scholar

[37]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model,, Progress in Nonlinear Differential Equations and their Applications, (2007).   Google Scholar

[38]

H. Sun, D. Uminsky and A. L. Bertozzi, Stability and clustering of self-similar solutions of aggregation equations,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.4745180.  Google Scholar

[39]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152.  doi: 10.1137/S0036139903437424.  Google Scholar

[40]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bulletin of Mathematical Biology, 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[41]

G. Toscani, One-dimensional kinetic models of granular flows,, M2AN Math. Model. Numer. Anal., 34 (2000), 1277.  doi: 10.1051/m2an:2000127.  Google Scholar

[42]

J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics,, J. Nonlinear Sci., 22 (2012), 935.  doi: 10.1007/s00332-012-9132-7.  Google Scholar

[43]

J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi, Predicting pattern formation in particle interactions,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202511400021.  Google Scholar

[44]

E. Weinan, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity,, Phys. D, 77 (1994), 383.  doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

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