March  2016, 36(3): 1175-1208. doi: 10.3934/dcds.2016.36.1175

Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

1. 

CNRS UMR 5669 "Unité de Mathématiques Pures et Appliquées", and project-team Inria NUMED, Ecole Normale Supérieure de Lyon, Lyon

2. 

Project-team MEPHYSTO, Inria Lille - Nord Europe, Villeneuve d'Ascq, France

Received  March 2014 Revised  May 2015 Published  August 2015

We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact.
Citation: Vincent Calvez, Thomas O. Gallouët. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1175-1208. doi: 10.3934/dcds.2016.36.1175
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space Of Probability Measures,, Second edition, (2008).   Google Scholar

[2]

A. Blanchet, On the parabolic-elliptic patlak-keller-segel system in dimension 2 and higher,, Séminaire Équations aux Dérivées Partielles, (): 2011.  doi: 10.5802/slsedp.6.  Google Scholar

[3]

A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model,, SIAM J. Numer. Anal., 46 (2008), 691.  doi: 10.1137/070683337.  Google Scholar

[4]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbbR^2$,, Comm. Pure Appl. Math., 61 (2008), 1449.  doi: 10.1002/cpa.20225.  Google Scholar

[5]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Differential Equations, 44 (2006).   Google Scholar

[6]

V. Calvez and J. A. Carrillo, Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities,, Proc. Amer. Math. Soc., 140 (2012), 3515.  doi: 10.1090/S0002-9939-2012-11306-1.  Google Scholar

[7]

V. Calvez, B. Perthame and M. Sharifi tabar, Modified Keller-Segel system and critical mass for the log interaction kernel,, in Stochastic Analysis and Partial Differential Equations, (2007), 45.  doi: 10.1090/conm/429/08229.  Google Scholar

[8]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms,, SIAM J. Sci. Comput., 31 (): 4305.  doi: 10.1137/080739574.  Google Scholar

[9]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[10]

A. Devys, Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie,, Ph.D thesis, (2010).   Google Scholar

[11]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up,, Discrete Contin. Dyn. Syst., 25 (2009), 109.  doi: 10.3934/dcds.2009.25.109.  Google Scholar

[12]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,, Numer. Math., 104 (2006), 457.  doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

T. O. Gallouët, Optimal transport: Regularity and Applications,, Ph.D thesis, (2012).   Google Scholar

[14]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations,, Comm. Pure Appl. Math., 38 (1985), 297.  doi: 10.1002/cpa.3160380304.  Google Scholar

[15]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations,, SIAM J. Sci. Comput., 28 (2006), 1203.  doi: 10.1137/050628015.  Google Scholar

[16]

J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system,, J. Stat. Phys., 135 (2009), 133.  doi: 10.1007/s10955-009-9717-1.  Google Scholar

[17]

J. Haškovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Partial Differential Equations, 36 (2011), 940.  doi: 10.1080/03605302.2010.538783.  Google Scholar

[18]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633.   Google Scholar

[19]

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

[20]

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

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[22]

N. I. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk,, SIAM J. Math. Anal., 40 (): 1852.  doi: 10.1137/080722229.  Google Scholar

[23]

E. F. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[24]

E. F. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[25]

S. Luckhaus, Y. Sugiyama and J. J. L. Velázquez, Measure valued solutions of the 2D Keller-Segel system,, Arch. Ration. Mech. Anal., 206 (2012), 31.  doi: 10.1007/s00205-012-0549-9.  Google Scholar

[26]

F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u+ |u|^{p-1}u$,, Duke Math. J., 86 (1997), 143.  doi: 10.1215/S0012-7094-97-08605-1.  Google Scholar

[27]

F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations,, Current developments in partial differential equations (Temuco, 8 (2002), 435.  doi: 10.3934/dcds.2002.8.435.  Google Scholar

[28]

N. Mittal, E. O. Budrene, M. P. Brenner and A. V. Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation,, Proc. Natl. Acad. Sci. USA, 100 (2003), 13259.  doi: 10.1073/pnas.2233626100.  Google Scholar

[29]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101.  doi: 10.1081/PDE-100002243.  Google Scholar

[30]

P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation,, , ().   Google Scholar

[31]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology,, Adv. Differential Equations, 6 (2001), 21.   Google Scholar

[32]

T. Suzuki, Free Energy and Self-Interacting Particles,, Progress in Nonlinear Differential Equations and their Applications, (2005).  doi: 10.1007/0-8176-4436-9.  Google Scholar

[33]

J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation,, SIAM J. Appl. Math., 62 (2002), 1581.  doi: 10.1137/S0036139900380049.  Google Scholar

[34]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1198.  doi: 10.1137/S0036139903433888.  Google Scholar

[35]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1224.  doi: 10.1137/S003613990343389X.  Google Scholar

[36]

C. Villani, Optimal Transport. Old and New,, Grundlehren der Mathematischen Wissenschaften, (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space Of Probability Measures,, Second edition, (2008).   Google Scholar

[2]

A. Blanchet, On the parabolic-elliptic patlak-keller-segel system in dimension 2 and higher,, Séminaire Équations aux Dérivées Partielles, (): 2011.  doi: 10.5802/slsedp.6.  Google Scholar

[3]

A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model,, SIAM J. Numer. Anal., 46 (2008), 691.  doi: 10.1137/070683337.  Google Scholar

[4]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbbR^2$,, Comm. Pure Appl. Math., 61 (2008), 1449.  doi: 10.1002/cpa.20225.  Google Scholar

[5]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Differential Equations, 44 (2006).   Google Scholar

[6]

V. Calvez and J. A. Carrillo, Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities,, Proc. Amer. Math. Soc., 140 (2012), 3515.  doi: 10.1090/S0002-9939-2012-11306-1.  Google Scholar

[7]

V. Calvez, B. Perthame and M. Sharifi tabar, Modified Keller-Segel system and critical mass for the log interaction kernel,, in Stochastic Analysis and Partial Differential Equations, (2007), 45.  doi: 10.1090/conm/429/08229.  Google Scholar

[8]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms,, SIAM J. Sci. Comput., 31 (): 4305.  doi: 10.1137/080739574.  Google Scholar

[9]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[10]

A. Devys, Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie,, Ph.D thesis, (2010).   Google Scholar

[11]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up,, Discrete Contin. Dyn. Syst., 25 (2009), 109.  doi: 10.3934/dcds.2009.25.109.  Google Scholar

[12]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,, Numer. Math., 104 (2006), 457.  doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

T. O. Gallouët, Optimal transport: Regularity and Applications,, Ph.D thesis, (2012).   Google Scholar

[14]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations,, Comm. Pure Appl. Math., 38 (1985), 297.  doi: 10.1002/cpa.3160380304.  Google Scholar

[15]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations,, SIAM J. Sci. Comput., 28 (2006), 1203.  doi: 10.1137/050628015.  Google Scholar

[16]

J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system,, J. Stat. Phys., 135 (2009), 133.  doi: 10.1007/s10955-009-9717-1.  Google Scholar

[17]

J. Haškovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Partial Differential Equations, 36 (2011), 940.  doi: 10.1080/03605302.2010.538783.  Google Scholar

[18]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633.   Google Scholar

[19]

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

[20]

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

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[22]

N. I. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk,, SIAM J. Math. Anal., 40 (): 1852.  doi: 10.1137/080722229.  Google Scholar

[23]

E. F. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[24]

E. F. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[25]

S. Luckhaus, Y. Sugiyama and J. J. L. Velázquez, Measure valued solutions of the 2D Keller-Segel system,, Arch. Ration. Mech. Anal., 206 (2012), 31.  doi: 10.1007/s00205-012-0549-9.  Google Scholar

[26]

F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u+ |u|^{p-1}u$,, Duke Math. J., 86 (1997), 143.  doi: 10.1215/S0012-7094-97-08605-1.  Google Scholar

[27]

F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations,, Current developments in partial differential equations (Temuco, 8 (2002), 435.  doi: 10.3934/dcds.2002.8.435.  Google Scholar

[28]

N. Mittal, E. O. Budrene, M. P. Brenner and A. V. Oudenaarden, Motility of escherichia coli cells in clusters formed by chemotactic aggregation,, Proc. Natl. Acad. Sci. USA, 100 (2003), 13259.  doi: 10.1073/pnas.2233626100.  Google Scholar

[29]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101.  doi: 10.1081/PDE-100002243.  Google Scholar

[30]

P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation,, , ().   Google Scholar

[31]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology,, Adv. Differential Equations, 6 (2001), 21.   Google Scholar

[32]

T. Suzuki, Free Energy and Self-Interacting Particles,, Progress in Nonlinear Differential Equations and their Applications, (2005).  doi: 10.1007/0-8176-4436-9.  Google Scholar

[33]

J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation,, SIAM J. Appl. Math., 62 (2002), 1581.  doi: 10.1137/S0036139900380049.  Google Scholar

[34]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1198.  doi: 10.1137/S0036139903433888.  Google Scholar

[35]

J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1224.  doi: 10.1137/S003613990343389X.  Google Scholar

[36]

C. Villani, Optimal Transport. Old and New,, Grundlehren der Mathematischen Wissenschaften, (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

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