June  2013, 18(4): 847-863. doi: 10.3934/dcdsb.2013.18.847

Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles

1. 

Department of Mathematica Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

2. 

University Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Safi

3. 

Departamento de Matemática Aplicada, Universidad de Granada, Spain, Spain

Received  March 2012 Revised  May 2012 Published  February 2013

This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations modeling binary mixtures of multi-cellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative-destructive events. The analysis refers to a suitable hyperbolic approximation to show how the macroscopic tissue behavior can be described from the underlying cellular description. The approach is specifically focused on the modeling of chemotaxis phenomena by the Keller--Segel approximation.
Citation: Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 847-863. doi: 10.3934/dcdsb.2013.18.847
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[2]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modelling of multicellular growing systems,, Math. Comput. Modelling., 51 (2010), 441. doi: 10.1016/j.mcm.2009.12.002.

[3]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1. doi: 10.1142/S0218202510004568.

[4]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512005885.

[5]

N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives,, Physics of Life Reviews, 8 (2011), 1.

[6]

A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations,, Math. Models Methods Appl. Sci., 14 (2004), 853. doi: 10.1142/S0218202504003477.

[7]

A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models,, Nonlinear Analysis: Real World. Appl., 12 (2011), 1111. doi: 10.1016/j.nonrwa.2010.09.005.

[8]

A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,, Math. Models Methods Appl. Sci., 23 (2013). doi: 10.1142/S0218202512500650.

[9]

A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system,, Z. Agnew. Math. Phys., 55 (2004), 295. doi: 10.1007/s00033-003-3057-9.

[10]

A. Bellouquid and M. Delitala, Mathematical models and tools of kinetic theory towards modelling complex biological systems,, Math. Models Methods Appl. Sci., 15 (2005), 1639. doi: 10.1142/S0218202505000923.

[11]

A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Birkäuser, (2006).

[12]

C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition,, Comp. Math. Meth. in Medicine, (2012).

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F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and V. Vauchelet, Waves for the hyperbolic Keller-Segel model,, Math. Models Methods Appl. Sci., 21 (2011), 825.

[14]

F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7.

[15]

F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis,, Math. Models Methods Appl. Sci., 16 (2006), 1173. doi: 10.1142/S0218202506001509.

[16]

M. A. J. Chaplain, M. Lachowicz, Z. Szyman'ska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719. doi: 10.1142/S0218202511005192.

[17]

K. C. Chen, R. M. Ford and P. T. Cummings, Perturbation expansion of Alt's cell balance equations reduces to Segel's 1d equation for shallow chemoattractant gradients,, SIAM J. Appl. Math., 59 (1999), 35. doi: 10.1137/S0036139996301283.

[18]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Models., 5 (2012), 51. doi: 10.3934/krm.2012.5.51.

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Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 2 (2003), 153. doi: 10.1007/s00285-002-0173-7.

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A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'', SIMAI-Springer Series, (2012).

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H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming,, Math. Models Methods Appl. Sci., 21 (2011), 939.

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F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2.

[23]

J. Folkman, Role of angiogenesis in tumor growth and methastasis,, Seminar Oncology, 29 (2002), 15.

[24]

H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology,, Nature, 402 (1999).

[25]

J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Part. Diff. Eqs., 36 (2011), 940. doi: 10.1080/03605302.2010.538783.

[26]

M. A. Herrero, A. Köhn and J. M. Pérez-Pomares, Modelling vascular morphogenesis: Current views on blood vessels development,, Math. Models Methods Appl. Sci., 19 (2009), 1483. doi: 10.1142/S021820250900384X.

[27]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10 (1997), 1739. doi: 10.1088/0951-7715/10/6/016.

[28]

T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation,, Disc. Cont. Dyn. Syst. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961.

[29]

T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3.

[30]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167.

[31]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I,, Deutsch. Math.-Verein, 105 (2003), 103.

[32]

E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.

[33]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.

[34]

E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis,, J. Theor. Biol., 30 (1971), 235.

[35]

E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread,, Proc. Conf. Math. Biol. Heidelberg, (1980), 379.

[36]

P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis,, Math. Models Methods Appl. Sci., 21 (2011), 955.

[37]

R. Kowalczyck, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. doi: 10.1016/j.jmaa.2004.12.009.

[38]

M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors,, Math. Models Methods Appl. Sci., 15 (2005), 1667. doi: 10.1142/S0218202505000935.

[39]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Probab. Eng. Mechanics, 26 (2011), 54.

[40]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Analysis: Real World Applications, 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014.

[41]

C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552.

[42]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM J. Math. Analysis, 40 (2008), 215. doi: 10.1137/050645269.

[43]

A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model,, Math. Models Methods Appl. Sci., 20 (2010), 440. doi: 10.1142/S0218202510004301.

[44]

I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", Springer, (1998). doi: 10.1007/978-1-4612-2210-1.

[45]

J. D. Murray, "Mathematical Biology,", Springer, (1989).

[46]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392.

[47]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772.

[48]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Appl. Math. Quart., 10 (2002), 501.

[49]

C. S. Patlak, Random walk with persistant and external bias,, Bull. Math. Biol., 15 (1953), 311.

[50]

L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model,, Studies Appl. Math, 123 (2009), 337. doi: 10.1111/j.1467-9590.2009.00457.x.

[51]

C. Ringhofer, C. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains,, SIAM J. Num. Anal., 39 (2001), 1078. doi: 10.1137/S0036142998335984.

[52]

L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.

[53]

M. A. Stolarska, K. I. M. Yangjin and H. G. Othmer, Multi-scale models of cell and tissue dynamics,, Phil. Trans. Royal Society A: Math. Phys. Eng. Sci., 367 (2009), 3525. doi: 10.1098/rsta.2009.0095.

[54]

Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling,, Nonlinear Analysis: RWA, 12 (2011), 418. doi: 10.1016/j.nonrwa.2010.06.027.

[55]

R. A. Weinberg, "The Biology of Cancer,", Garland Sciences - Taylor and Francis, (2007).

show all references

References:
[1]

A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews - Cancer, 8 (2008), 227.

[2]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modelling of multicellular growing systems,, Math. Comput. Modelling., 51 (2010), 441. doi: 10.1016/j.mcm.2009.12.002.

[3]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1. doi: 10.1142/S0218202510004568.

[4]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512005885.

[5]

N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives,, Physics of Life Reviews, 8 (2011), 1.

[6]

A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations,, Math. Models Methods Appl. Sci., 14 (2004), 853. doi: 10.1142/S0218202504003477.

[7]

A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models,, Nonlinear Analysis: Real World. Appl., 12 (2011), 1111. doi: 10.1016/j.nonrwa.2010.09.005.

[8]

A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,, Math. Models Methods Appl. Sci., 23 (2013). doi: 10.1142/S0218202512500650.

[9]

A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system,, Z. Agnew. Math. Phys., 55 (2004), 295. doi: 10.1007/s00033-003-3057-9.

[10]

A. Bellouquid and M. Delitala, Mathematical models and tools of kinetic theory towards modelling complex biological systems,, Math. Models Methods Appl. Sci., 15 (2005), 1639. doi: 10.1142/S0218202505000923.

[11]

A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Birkäuser, (2006).

[12]

C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition,, Comp. Math. Meth. in Medicine, (2012).

[13]

F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and V. Vauchelet, Waves for the hyperbolic Keller-Segel model,, Math. Models Methods Appl. Sci., 21 (2011), 825.

[14]

F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7.

[15]

F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis,, Math. Models Methods Appl. Sci., 16 (2006), 1173. doi: 10.1142/S0218202506001509.

[16]

M. A. J. Chaplain, M. Lachowicz, Z. Szyman'ska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719. doi: 10.1142/S0218202511005192.

[17]

K. C. Chen, R. M. Ford and P. T. Cummings, Perturbation expansion of Alt's cell balance equations reduces to Segel's 1d equation for shallow chemoattractant gradients,, SIAM J. Appl. Math., 59 (1999), 35. doi: 10.1137/S0036139996301283.

[18]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Models., 5 (2012), 51. doi: 10.3934/krm.2012.5.51.

[19]

Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 2 (2003), 153. doi: 10.1007/s00285-002-0173-7.

[20]

A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'', SIMAI-Springer Series, (2012).

[21]

H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming,, Math. Models Methods Appl. Sci., 21 (2011), 939.

[22]

F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2.

[23]

J. Folkman, Role of angiogenesis in tumor growth and methastasis,, Seminar Oncology, 29 (2002), 15.

[24]

H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology,, Nature, 402 (1999).

[25]

J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Part. Diff. Eqs., 36 (2011), 940. doi: 10.1080/03605302.2010.538783.

[26]

M. A. Herrero, A. Köhn and J. M. Pérez-Pomares, Modelling vascular morphogenesis: Current views on blood vessels development,, Math. Models Methods Appl. Sci., 19 (2009), 1483. doi: 10.1142/S021820250900384X.

[27]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10 (1997), 1739. doi: 10.1088/0951-7715/10/6/016.

[28]

T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation,, Disc. Cont. Dyn. Syst. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961.

[29]

T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3.

[30]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167.

[31]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I,, Deutsch. Math.-Verein, 105 (2003), 103.

[32]

E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.

[33]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.

[34]

E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis,, J. Theor. Biol., 30 (1971), 235.

[35]

E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread,, Proc. Conf. Math. Biol. Heidelberg, (1980), 379.

[36]

P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis,, Math. Models Methods Appl. Sci., 21 (2011), 955.

[37]

R. Kowalczyck, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. doi: 10.1016/j.jmaa.2004.12.009.

[38]

M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors,, Math. Models Methods Appl. Sci., 15 (2005), 1667. doi: 10.1142/S0218202505000935.

[39]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Probab. Eng. Mechanics, 26 (2011), 54.

[40]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Analysis: Real World Applications, 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014.

[41]

C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552.

[42]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM J. Math. Analysis, 40 (2008), 215. doi: 10.1137/050645269.

[43]

A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model,, Math. Models Methods Appl. Sci., 20 (2010), 440. doi: 10.1142/S0218202510004301.

[44]

I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", Springer, (1998). doi: 10.1007/978-1-4612-2210-1.

[45]

J. D. Murray, "Mathematical Biology,", Springer, (1989).

[46]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392.

[47]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772.

[48]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Appl. Math. Quart., 10 (2002), 501.

[49]

C. S. Patlak, Random walk with persistant and external bias,, Bull. Math. Biol., 15 (1953), 311.

[50]

L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model,, Studies Appl. Math, 123 (2009), 337. doi: 10.1111/j.1467-9590.2009.00457.x.

[51]

C. Ringhofer, C. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains,, SIAM J. Num. Anal., 39 (2001), 1078. doi: 10.1137/S0036142998335984.

[52]

L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.

[53]

M. A. Stolarska, K. I. M. Yangjin and H. G. Othmer, Multi-scale models of cell and tissue dynamics,, Phil. Trans. Royal Society A: Math. Phys. Eng. Sci., 367 (2009), 3525. doi: 10.1098/rsta.2009.0095.

[54]

Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling,, Nonlinear Analysis: RWA, 12 (2011), 418. doi: 10.1016/j.nonrwa.2010.06.027.

[55]

R. A. Weinberg, "The Biology of Cancer,", Garland Sciences - Taylor and Francis, (2007).

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Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339

[19]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

[20]

Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013

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