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.

[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.

[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).

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[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.

[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.

[35]

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

[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.

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

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.

[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.

[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).

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[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.

[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.

[35]

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

[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.

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[1]

Vincent Calvez, Gaël Raoul, Christian Schmeiser. Confinement by biased velocity jumps: Aggregation of escherichia coli. Kinetic & Related Models, 2015, 8 (4) : 651-666. doi: 10.3934/krm.2015.8.651

[2]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309

[3]

María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255

[4]

David Cowan. Rigid particle systems and their billiard models. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1/2) : 111-130. doi: 10.3934/dcds.2008.22.111

[5]

Alan Mackey, Theodore Kolokolnikov, Andrea L. Bertozzi. Two-species particle aggregation and stability of co-dimension one solutions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1411-1436. doi: 10.3934/dcdsb.2014.19.1411

[6]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617

[7]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469

[8]

Shin-Ichiro Ei, Hirofumi Izuhara, Masayasu Mimura. Infinite dimensional relaxation oscillation in aggregation-growth systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1859-1887. doi: 10.3934/dcdsb.2012.17.1859

[9]

Jitendra Kumar, Gurmeet Kaur, Evangelos Tsotsas. An accurate and efficient discrete formulation of aggregation population balance equation. Kinetic & Related Models, 2016, 9 (2) : 373-391. doi: 10.3934/krm.2016.9.373

[10]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455

[11]

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

[12]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785

[13]

Doron Levy, Tiago Requeijo. Modeling group dynamics of phototaxis: From particle systems to PDEs. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 103-128. doi: 10.3934/dcdsb.2008.9.103

[14]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205

[15]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

[16]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395

[17]

Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339

[18]

Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic & Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008

[19]

François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 163-180. doi: 10.3934/nhm.2016.11.163

[20]

Benjamin Jourdain, Julien Reygner. Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]