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Volumn 11, Issue 3, 2013, Pages 709-738

Stationary states of quadratic diffusion equations with long-range attraction

Author keywords

Krein Rutman theorem; Nonlinear diusion; Nonlocal aggregation; Population dynamics; Stationary states

Indexed keywords


EID: 84881632469     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2013.v11.n3.a3     Document Type: Article
Times cited : (46)

References (66)
  • 1
    • 73349096429 scopus 로고    scopus 로고
    • Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, ETH Zurich
    • L. Ambrosio, N. Gigli, and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, ETH Zurich. Basel: Birkhauser, Second Edition, 2008.
    • (2008) Basel: Birkhauser, Second Edition
    • Ambrosio, L.1    Gigli, N.2    Savarse, G.3
  • 2
    • 84881621189 scopus 로고    scopus 로고
    • Nonlocal interactions by repulsive- attractive potentials: Radial ins/stability
    • UAB, to appear in Physica D
    • D. Balague, J.A. Carrillo, T. Laurent, and G. Raoul, Nonlocal interactions by repulsive- attractive potentials: Radial ins/stability, UAB, to appear in Physica D, 2011.
    • (2011)
    • Balaguse, D.1    Carrillo, J.A.2    Laurent, T.3    Raoul, G.4
  • 3
    • 79959784958 scopus 로고    scopus 로고
    • Global minimizers for free energies of subcritical aggregation equations with degenerate diusion
    • J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diusion, Appl. Math. Letters, 24, 1927{1932, 2011.
    • (2011) Appl. Math. Letters , vol.24 , pp. 1927-1932
    • Bedrossian, J.1
  • 5
    • 65249090848 scopus 로고    scopus 로고
    • Blowup in multidimensional aggregation equa- tions with mildly singular interaction kernels
    • A. Bertozzi, J. Carrillo, and T. Laurent, Blowup in multidimensional aggregation equa- tions with mildly singular interaction kernels, Nonlin., 22, 683-710, 2009.
    • (2009) Nonlin. , vol.22 , pp. 683-710
    • Bertozzi, A.1    Carrillo, J.2    Laurent, T.3
  • 6
    • 34547507325 scopus 로고    scopus 로고
    • Finite-time blow-up of solutions of an aggregation equation in Rn
    • A. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Commun. Math. Phys., 274, 717-735, 2007.
    • (2007) Commun. Math. Phys , vol.274 , pp. 717-735
    • Bertozzi, A.1    Laurent, T.2
  • 7
    • 77951121715 scopus 로고    scopus 로고
    • Finite-time blow-up of L1-weak solutions of an aggre- gation equation
    • A.L. Bertozzi and J. Brandman, Finite-time blow-up of L1-weak solutions of an aggre- gation equation, Commun. Math. Sci., 8(1), 45-65, 2010.
    • (2010) Commun. Math. Sci , vol.8 , Issue.1 , pp. 45-65
    • Bertozzi, A.L.1    Brandman, J.2
  • 8
    • 78049410978 scopus 로고    scopus 로고
    • Lp theory for the multidimensional aggregation equation
    • A.L. Bertozzi, T. Laurent, and J. Rosado, Lp theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64(1), 45-83, 2011.
    • (2011) Commun. Pure Appl. Math , vol.64 , Issue.1 , pp. 45-83
    • Bertozzi, A.L.1    Laurent, T.2    Rosado, J.3
  • 9
    • 79953065830 scopus 로고    scopus 로고
    • Existence and uniqueness of solutions to an aggregation equation with degenerate diusion
    • A.L. Bertozzi and D. Slepcev, Existence and uniqueness of solutions to an aggregation equation with degenerate diusion, Commun. Pure Appl. Anal., 9, 1617-1637, 2010.
    • (2010) Commun. Pure Appl. Anal , vol.9 , pp. 1617-1637
    • Bertozzi, A.L.1    Slepcev, D.2
  • 10
    • 33645680010 scopus 로고    scopus 로고
    • Two-dimensional keller-segel model: Opti- mal critical mass and qualitative properties of the solutions
    • A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional keller-segel model: Opti- mal critical mass and qualitative properties of the solutions, Electron. J. Dier. Eqs., (44), 1-32, (electronic), 2006.
    • (2006) Electron. J. Dier. Eqs , Issue.44 , pp. 1-32
    • Blanchet, A.1    Dolbeault, J.2    Perthame, B.3
  • 11
    • 32644457070 scopus 로고    scopus 로고
    • An integro-dierential equation arising as a limit of indi- vidual cell-based models
    • M. Bodnar and J. Velazquez, An integro-dierential equation arising as a limit of indi- vidual cell-based models, J. Di. Equ., 222(2), 341-380, 2006.
    • (2006) J. Di. Equ , vol.222 , Issue.2 , pp. 341-380
    • Bodnar, M.1    Velazquez, J.2
  • 12
    • 0034148193 scopus 로고    scopus 로고
    • Modeling the aggregative behavior of ants of the species Polyergus rufescens
    • S. Boi, V. Capasso, and D. Morale, Modeling the aggregative behavior of ants of the species Polyergus rufescens, Nonlin. Anal., Real World Appl., 1(1), 163-176, 2000.
    • (2000) Nonlin. Anal. Real World Appl , vol.1 , Issue.1 , pp. 163-176
    • Boi, S.1    Capasso, V.2    Morale, D.3
  • 13
    • 33846217387 scopus 로고    scopus 로고
    • On an aggregation model with long and short range interactions
    • M. Burger, V. Capasso, and D. Morale, On an aggregation model with long and short range interactions, Nonlin. Anal., Real World Appl., 8(3), 939-958, 2007.
    • (2007) Nonlin. Anal. Real World Appl , vol.8 , Issue.3 , pp. 939-958
    • Burger, M.1    Capasso, V.2    Morale, D.3
  • 14
    • 57249084869 scopus 로고    scopus 로고
    • Large time behavior of nonlocal aggregation models with nonlinear diusion
    • M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diusion, Netw. Heterog. Media, 3(4), 749-785, 2008.
    • (2008) Netw. Heterog. Media , vol.3 , Issue.4 , pp. 749-785
    • Burger, M.1    Di Francesco, M.2
  • 15
    • 34248370060 scopus 로고    scopus 로고
    • The Keller-Segel model for chemotaxiswith prevention of overcrowding: Linear vs. nonlinear diusion
    • M. Burger, M. Di Francesco, and Y. Dolak-Stru, The Keller-Segel model for chemotaxiswith prevention of overcrowding: Linear vs. nonlinear diusion, SIAM J. Math. Anal., 38, 1288-1315, 2006.
    • (2006) SIAM J. Math. Anal , vol.38 , pp. 1288-1315
    • Burger, M.1    Di Francesco, M.2    Dolak-Stru, Y.3
  • 16
    • 33746377577 scopus 로고    scopus 로고
    • Volume eects in the Keller-Segel model: Energy estimates preventing blow-up
    • V. Calvez and J. Carrillo, Volume eects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86, 155-175, 2006.
    • (2006) J. Math. Pures Appl , vol.86 , pp. 155-175
    • Calvez, V.1    Carrillo, J.2
  • 17
    • 79551709781 scopus 로고    scopus 로고
    • Global-in-time weak measure solutions and nite-time aggregation for nonlocal interaction equations
    • J. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepcev, Global-in-time weak measure solutions, and nite-time aggregation for nonlocal interaction equations, Duke Math. J., 156, 229-271, 2011.
    • (2011) Duke Math. J , vol.156 , pp. 229-271
    • Carrillo, J.1    Di Francesco, M.2    Figalli, A.3    Laurent, T.4    Slepcev, D.5
  • 19
    • 1542399709 scopus 로고    scopus 로고
    • Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates
    • J. Carrillo, R. McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19(3), 971-1018, 2003.
    • (2003) Rev. Mat. Iberoam , vol.19 , Issue.3 , pp. 971-1018
    • Carrillo, J.1    McCann, R.2    Villani, C.3
  • 20
    • 32544445429 scopus 로고    scopus 로고
    • Contractions in the 2-Wasserstein length space and thermalization of granular media
    • J. Carrillo, R. McCann, and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179(2), 217-263, 2006.
    • (2006) Arch. Ration. Mech. Anal. , vol.179 , Issue.2 , pp. 217-263
    • Carrillo, J.1    McCann, R.2    Villani, C.3
  • 21
    • 0003277699 scopus 로고    scopus 로고
    • Asymptotic L1-decay of solutions of the porous medium equation to self-similarity
    • J.A. Carrillo and G. Toscani, Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49, 113-141, 2000.
    • (2000) Indiana Univ. Math. J , vol.49 , pp. 113-141
    • Carrillo, J.A.1    Toscani, G.2
  • 23
    • 84881650378 scopus 로고    scopus 로고
    • An aggregation equation with degenerate diusion: Qual- itative property of solutions preprint
    • L. Chayes, I. Kim, and Y. Yao, An aggregation equation with degenerate diusion: Qual- itative property of solutions, preprint.
    • Chayes, L.1    Kim, I.2    Yao, Y.3
  • 25
    • 26844499863 scopus 로고    scopus 로고
    • Global solutions of some chemotaxis and angio- genesis systems in high space dimensions
    • L. Corrias, B. Perthame, and H. Zaag, Global solutions of some chemotaxis and angio- genesis systems in high space dimensions, Milan J. Math., 72, 1-28, 2004.
    • (2004) Milan J. Math , vol.72 , pp. 1-28
    • Corrias, L.1    Perthame, B.2    Zaag, H.3
  • 26
    • 84881644217 scopus 로고    scopus 로고
    • Medical Image Segmentation Including Particle Methods
    • H. Dirks, Medical Image Segmentation Including Particle Methods, Diploma Thesis, WWU Munster, 2011.
    • (2011) Diploma Thes is WWU Munster
    • Dirks, H.1
  • 28
    • 33749561186 scopus 로고    scopus 로고
    • Order Structure and Topological Methods in Nonlinear Partial Dierential Equa- tions
    • Maximum Principles and Applications, Series in Partial Dierential Equations and Applications: Volume 2, Hackensack, New Jersey, World Scientic
    • Y. Du, Order Structure and Topological Methods in Nonlinear Partial Dierential Equa- tions, Vol. 1: Maximum Principles and Applications, Series in Partial Dierential Equations and Applications: Volume 2, Hackensack, New Jersey, World Scientic,2006.
    • (2006) , vol.1
    • Du, Y.1
  • 29
    • 79951580992 scopus 로고    scopus 로고
    • Stability of stationary states of non-local equations with singular interaction potentials
    • K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53(7-8), 1436-1450, 2011.
    • (2011) Math. Comput. Modelling , vol.53 , Issue.7-8 , pp. 1436-1450
    • Fellner, K.1    Raoul, G.2
  • 30
    • 78651233551 scopus 로고    scopus 로고
    • Stable stationary states of non-local interaction equations
    • K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20(12), 2267, 2010.
    • (2010) Math. Models Methods Appl. Sci. , vol.20 , Issue.12 , pp. 2267
    • Fellner, K.1    Raoul, G.2
  • 31
    • 80053259548 scopus 로고    scopus 로고
    • Swarm dynamics and equilibria for a nonlocal aggregation model
    • R.C. Fetecau, Y. Huang, and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlin., 24, 2681-2716, 2011.
    • (2011) Nonlin , vol.24 , pp. 2681-2716
    • Fetecau, R.C.1    Huang, Y.2    Kolokolnikov, T.3
  • 32
    • 84881631975 scopus 로고    scopus 로고
    • Consistency of variational continuous-domain quantization via kinetic theory
    • M. Fornasier, J. Haskovec, and G. Steidl, Consistency of variational continuous-domain quantization via kinetic theory, Appl. Anal., to appear, 2012.
    • (2012) Appl. Anal., to appear
    • Fornasier, M.1    Haskovec, J.2    Steidl, G.3
  • 33
    • 33846704688 scopus 로고    scopus 로고
    • The mean-eld limit for the dynamics of large particle systems
    • Proceedings of the Conference on Partial Dierential Equations Forges-les-Eaux France
    • F. Golse, The mean-eld limit for the dynamics of large particle systems, Proceedings of the Conference on Partial Dierential Equations, Forges-les-Eaux, France, June 2-6,2003.
    • (2003)
    • Golse, F.1
  • 34
    • 0003078083 scopus 로고
    • Nonlinear diusion limit for a sys- tem with nearest neighbor interactions
    • M.Z. Guo, G.C. Papanicolaou, and S.R.S. Varadhan, Nonlinear diusion limit for a sys- tem with nearest neighbor interactions, Commun. Math. Phys., 118, 31-59, 1988.
    • (1988) Commun. Math. Phys , vol.118 , pp. 31-59
    • Guo, M.Z.1    Papanicolaou, G.C.2    Varadhan, S.S.R.3
  • 35
    • 0039712703 scopus 로고    scopus 로고
    • Singularity patterns in a chemotaxis model
    • M.A. Herrero and J.J.L. Velazquez, Singularity patterns in a chemotaxis model, Math. Ann., 306(3), 583-623, 1996.
    • (1996) Math. Ann , vol.306 , Issue.3 , pp. 583-623
    • Herrero, M.A.1    Velazquez, J.J.L.2
  • 36
    • 0001404830 scopus 로고    scopus 로고
    • Global existence for a parabolic chemotaxis model with preven- tion of overcrowding
    • T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with preven- tion of overcrowding, Adv. Appl. Math., 26(4), 280-301, 2001.
    • (2001) Adv. Appl. Math. , vol.26 , Issue.4 , pp. 280-301
    • Hillen, T.1    Painter, K.2
  • 37
    • 77956257904 scopus 로고    scopus 로고
    • Self-similar blow-up solutions to an aggregation equation in Rn
    • Y. Huang and A.L. Bertozzi, Self-similar blow-up solutions to an aggregation equation in Rn, SIAM J. Appl. Math., 70(7), 2582-2603, 2010.
    • (2010) SIAM J. Appl. Math. , vol.70 , Issue.7 , pp. 2582-2603
    • Huang, Y.1    Bertozzi, A.L.2
  • 38
    • 84966208077 scopus 로고
    • On explosions of solutions to a system of partial dierential equations modelling chemotaxis
    • W. Jager and S. Luckhaus, On explosions of solutions to a system of partial dierential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329(2), 819-824, 1992.
    • (1992) Trans. Amer. Math. Soc. , vol.329 , Issue.2 , pp. 819-824
    • Jager, W.1    Luckhaus, S.2
  • 39
    • 58349119168 scopus 로고    scopus 로고
    • An integro-dierential equation model for alignment and orientational aggregation
    • K. Kang, B. Perthame, A. Stevens, and J. Velazquez, An integro-dierential equation model for alignment and orientational aggregation, J. Dier. Eqs., 246(4), 1387-1421, 2009.
    • (2009) J. Dier. Eqs , vol.246 , Issue.4 , pp. 1387-1421
    • Kang, K.1    Perthame, B.2    Stevens, A.3    Velazquez, J.4
  • 40
    • 0014748565 scopus 로고
    • Initiation of slide mold aggregation viewed as an instability
    • E.F. Keller and L.A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26, 399-415, 1970.
    • (1970) J. Theor. Biol , vol.26 , pp. 399-415
    • Keller, E.F.1    Segel, L.A.2
  • 41
    • 84861372019 scopus 로고    scopus 로고
    • The Patlak-Keller-Segel model and its variations: Properties of solu- tions via maximum principle
    • I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solu- tions via maximum principle, SIAM J. Math. Anal., 44(2), 568-602, 2012.
    • (2012) SIAM J. Math. Anal , vol.44 , Issue.2 , pp. 568-602
    • Kim, I.1    Yao, Y.2
  • 42
    • 15844396394 scopus 로고    scopus 로고
    • Preventing blow-up in a chemotaxis model
    • R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305(2), 566-588, 2005.
    • (2005) J. Math. Anal. Appl. , vol.305 , Issue.2 , pp. 566-588
    • Kowalczyk, R.1
  • 43
    • 85045529914 scopus 로고    scopus 로고
    • A chemotaxis model with threshold density and degenerate diusion
    • in Nonlinear Elliptic and Parabolic Problems, Progr. Nonlin. Di. Equ. Appl., Birkhauser, Basel
    • P. Laurencot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diusion, in Nonlinear Elliptic and Parabolic Problems, Progr. Nonlin. Di. Equ. Appl., Birkhauser, Basel, 64, 273-290, 2005.
    • (2005) , vol.64 , pp. 273-290
    • Laurencot, P.1    Wrzosek, D.2
  • 44
    • 36949038223 scopus 로고    scopus 로고
    • Local and global existence for an aggregation equation
    • T. Laurent, Local and global existence for an aggregation equation, Commun. Part. Dier. Eqs., 32(12), 1941-1964, 2007.
    • (2007) Commun. Part. Dier. Eqs. , vol.32 , Issue.12 , pp. 1941-1964
    • Laurent, T.1
  • 45
    • 77955386860 scopus 로고    scopus 로고
    • On a nonlocal aggregation model with nonlinear diusion
    • D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diusion, Discrete Contin. Dyn. Syst., 27(1), 301-323, 2010.
    • (2010) Discrete Contin. Dyn. Syst. , vol.27 , Issue.1 , pp. 301-323
    • Li, D.1    Zhang, X.2
  • 46
    • 3042676199 scopus 로고    scopus 로고
    • Long-time asymptotics of kinetic models of granular ows
    • H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular ows, Arch. Ration. Mech. Anal., 172(3), 407-428, 2004.
    • (2004) Arch. Ration. Mech. Anal. , vol.172 , Issue.3 , pp. 407-428
    • Li, H.1    Toscani, G.2
  • 47
    • 0000298073 scopus 로고
    • Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
    • E.H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118, 349-374, 1983.
    • (1983) Ann. Math , vol.118 , pp. 349-374
    • Lieb, E.H.1
  • 49
    • 85030707196 scopus 로고
    • The concentration-compactness principle in calculus of variations
    • in The Locally Compact Case Ann. Inst. Henri Poincare, Anal. Nonlin
    • P.L. Lions, The concentration-compactness principle in calculus of variations, in The Locally Compact Case, Ann. Inst. Henri Poincare, Anal. Nonlin., 1, 109-145, 1984.
    • (1984) , vol.1 , pp. 109-145
    • Lions, P.L.1
  • 50
    • 0031161793 scopus 로고    scopus 로고
    • A convexity principle for interacting gases
    • R.J. McCann, A convexity principle for interacting gases, Adv. Math., 128(1), 153-179, 1997.
    • (1997) Adv. Math , vol.128 , Issue.1 , pp. 153-179
    • McCann, R.J.1
  • 51
    • 49349113659 scopus 로고    scopus 로고
    • A simple model for biological aggregation with asymmetric sensing
    • P.A. Milewski and X. Yang, A simple model for biological aggregation with asymmetric sensing, Commun. Math. Sci., 6, 397-416, 2008.
    • (2008) Commun. Math. Sci , vol.6 , pp. 397-416
    • Milewski, P.A.1    Yang, X.2
  • 52
    • 0001508106 scopus 로고    scopus 로고
    • A non-local model for a swarm
    • A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38(6), 534-570, 1999.
    • (1999) J. Math. Biol. , vol.38 , Issue.6 , pp. 534-570
    • Mogilner, A.1    Edelstein-Keshet, L.2
  • 53
    • 13844298300 scopus 로고    scopus 로고
    • An interacting particle system modelling aggregation behavior: From individuals to populations
    • D. Morale, V. Capasso, and K. Oelschlaeger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50(1), 49-66, 2005.
    • (2005) J. Math. Biol. , vol.50 , Issue.1 , pp. 49-66
    • Morale, D.1    Capasso, V.2    Oelschlaeger, K.3
  • 54
    • 38249016648 scopus 로고
    • Large systems of interacting particles and porous medium equation
    • K. Olschlager, Large systems of interacting particles and porous medium equation, J. Di. Eqs., 88, 294-346, 1990.
    • (1990) J. Dik. Eqs , vol.88 , pp. 294-346
    • Olschlger, K.1
  • 55
    • 0001560970 scopus 로고    scopus 로고
    • The geometry of dissipative evolution equations: The porous medium equation
    • F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Commun. Part. Dier. Eqs., 26(1-2), 101-174, 2001.
    • (2001) Commun. Part. Dier. Eqs. , vol.26 , Issue.1-2 , pp. 101-174
    • Otto, F.1
  • 56
    • 4944248574 scopus 로고    scopus 로고
    • Volume-lling and quorum sensing in models for chemosensitive movement
    • K. Painter and T. Hillen, Volume-lling and quorum sensing in models for chemosensitive movement, Canadian Applied Mathematics Quaterly, 10(4), 280-301, 2003. [57] C. Patlak, Random walk with persi stence and external bias, Bull. Math. Biophys., 15, 311-338, 1953.
    • (2003) Canadian Applied Mathematics Quaterly , vol.10 , Issue.4 , pp. 280-301
    • Painter, K.1    Hillen, T.2
  • 57
    • 51249193541 scopus 로고
    • Random walk with persi stence and external bias
    • C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15, 311-338, 1953.
    • (1953) Bull. Math. Biophys. , vol.15 , pp. 311-338
    • Patlak, C.1
  • 59
    • 68949212910 scopus 로고    scopus 로고
    • Mass-selection in alignment models with non- deterministic eects
    • I. Primi, A. Stevens, and J. Velazquez, Mass-selection in alignment models with non- deterministic eects, Commun. Part. Dier. Eqs., 34(5), 419-456, 2009.
    • (2009) Commun. Part. Dier. Eqs , vol.34 , Issue.5 , pp. 419-456
    • Primi, I.1    Stevens, A.2    Velazquez, J.3
  • 60
    • 71849095800 scopus 로고    scopus 로고
    • Opinion evolution in closed community
    • K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, Int. J. Mod. Phys. C, 11, 1157-1165, 2000.
    • (2000) Int. J. Mod. Phys. C , vol.11 , pp. 1157-1165
    • Sznajd-Weron, K.1    Sznajd, J.2
  • 61
    • 16244371027 scopus 로고    scopus 로고
    • Swarming patterns in a two-dimensional kinematic model for biological groups
    • C.M. Topaz and A.L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65(1), 152-174, 2004.
    • (2004) SIAM J. Appl. Math. , vol.65 , Issue.1 , pp. 152-174
    • Topaz, C.M.1    Bertozzi, A.L.2
  • 62
    • 33748692474 scopus 로고    scopus 로고
    • A nonlocal continuum model for biological aggregation
    • C.M. Topaz, A.L. Bertozzi, and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68(7), 1601-1623, 2006.
    • (2006) Bull. Math. Bio. , vol.68 , Issue.7 , pp. 1601-1623
    • Topaz, C.M.1    Bertozzi, A.L.2    Lewis, M.3
  • 63
    • 0001070815 scopus 로고
    • Scaling limits for interacting diusions
    • S.R.S. Varadhan, Scaling limits for interacting diusions, Commun. Math. Phys., 135, 313-353, 1991.
    • (1991) Commun. Math. Phys. , vol.135 , pp. 313-353
    • Varadhan, S.R.S.1
  • 65
    • 73249119910 scopus 로고    scopus 로고
    • Topics in Optimal Transportation Graduate Studies in Mathematics
    • C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 58, 2003.
    • (2003) American Mathematical Society, Providence, RI , vol.58
    • Villani, C.1
  • 66
    • 37349070894 scopus 로고    scopus 로고
    • Optimal Transport. Old and New Grundlehren der Mathematischen Wissenschaften
    • Berlin: Springer
    • C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, Berlin: Springer, 338, 2008.
    • (2008) , vol.338
    • Villani, C.1


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