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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

Vincent Calvez 1, and  Thomas O. Gallouët 2,

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.
Keywords: blow-up, particles methods, chemotaxis, Keller-Segel, critical mass, gradient flow..
Mathematics Subject Classification: Primary: 35K57, 35B44; Secondary: 35B40, 35Q92, 35K55, 65M99, 92C1.
Citation: Vincent Calvez, Thomas O. Gallouët. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up. Discrete and Continuous Dynamical Systems, 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, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[2]

A. Blanchet, On the parabolic-elliptic patlak-keller-segel system in dimension 2 and higher, Séminaire Équations aux Dérivées Partielles, (2011-2012), 26pp. doi: 10.5802/slsedp.6.

[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), 691-721. doi: 10.1137/070683337.

[4]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225.

[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), 32pp (electronic).

[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), 3515-3530. doi: 10.1090/S0002-9939-2012-11306-1.

[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, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007, 45-62. doi: 10.1090/conm/429/08229.

[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 (2009/10), 4305-4329. doi: 10.1137/080739574.

[9]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9.

[10]

A. Devys, Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie, Ph.D thesis, Université Lille 1, 2010.

[11]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121. doi: 10.3934/dcds.2009.25.109.

[12]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.

[13]

T. O. Gallouët, Optimal transport: Regularity and Applications, Ph.D thesis, Université de Lyon-Ecole Normale Supérieure de Lyon, 2012.

[14]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.

[15]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227 (electronic). doi: 10.1137/050628015.

[16]

J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151. doi: 10.1007/s10955-009-9717-1.

[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), 940-960. doi: 10.1080/03605302.2010.538783.

[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), 633-683 (1998).

[19]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[20]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[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 (2008/09), 1852-1881. doi: 10.1137/080722229.

[23]

E. F. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[24]

E. F. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[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), 31-80. doi: 10.1007/s00205-012-0549-9.

[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), 143-195. doi: 10.1215/S0012-7094-97-08605-1.

[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, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450. doi: 10.3934/dcds.2002.8.435.

[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), 13259-13263. doi: 10.1073/pnas.2233626100.

[29]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.

[30]

P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation, arXiv:1209.2517.

[31]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.

[32]

T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston Inc., Boston, MA, 2005. doi: 10.1007/0-8176-4436-9.

[33]

J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., 62 (2002), 1581-1633 (electronic). doi: 10.1137/S0036139900380049.

[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), 1198-1223 (electronic). doi: 10.1137/S0036139903433888.

[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), 1224-1248 (electronic). doi: 10.1137/S003613990343389X.

[36]

C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[2]

A. Blanchet, On the parabolic-elliptic patlak-keller-segel system in dimension 2 and higher, Séminaire Équations aux Dérivées Partielles, (2011-2012), 26pp. doi: 10.5802/slsedp.6.

[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), 691-721. doi: 10.1137/070683337.

[4]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225.

[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), 32pp (electronic).

[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), 3515-3530. doi: 10.1090/S0002-9939-2012-11306-1.

[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, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007, 45-62. doi: 10.1090/conm/429/08229.

[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 (2009/10), 4305-4329. doi: 10.1137/080739574.

[9]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9.

[10]

A. Devys, Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie, Ph.D thesis, Université Lille 1, 2010.

[11]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121. doi: 10.3934/dcds.2009.25.109.

[12]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.

[13]

T. O. Gallouët, Optimal transport: Regularity and Applications, Ph.D thesis, Université de Lyon-Ecole Normale Supérieure de Lyon, 2012.

[14]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.

[15]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227 (electronic). doi: 10.1137/050628015.

[16]

J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151. doi: 10.1007/s10955-009-9717-1.

[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), 940-960. doi: 10.1080/03605302.2010.538783.

[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), 633-683 (1998).

[19]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[20]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[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 (2008/09), 1852-1881. doi: 10.1137/080722229.

[23]

E. F. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[24]

E. F. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[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), 31-80. doi: 10.1007/s00205-012-0549-9.

[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), 143-195. doi: 10.1215/S0012-7094-97-08605-1.

[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, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450. doi: 10.3934/dcds.2002.8.435.

[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), 13259-13263. doi: 10.1073/pnas.2233626100.

[29]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.

[30]

P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation, arXiv:1209.2517.

[31]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.

[32]

T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston Inc., Boston, MA, 2005. doi: 10.1007/0-8176-4436-9.

[33]

J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., 62 (2002), 1581-1633 (electronic). doi: 10.1137/S0036139900380049.

[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), 1198-1223 (electronic). doi: 10.1137/S0036139903433888.

[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), 1224-1248 (electronic). doi: 10.1137/S003613990343389X.

[36]

C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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