American Institute of Mathematical Sciences

Advanced
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 & 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.  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), 691-721. 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), 1449-1481. 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), 32pp (electronic).  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), 3515-3530. 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, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007, 45-62. 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), 217-237. 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, Université Lille 1, 2010. Google Scholar

[11]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121. 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), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

T. O. Gallouët, Optimal transport: Regularity and Applications, Ph.D thesis, Université de Lyon-Ecole Normale Supérieure de Lyon, 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), 297-319. 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), 1203-1227 (electronic). 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), 133-151. 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), 940-960. 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), 633-683 (1998).  Google Scholar

[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.  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), 103-165.  Google Scholar

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. 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), 399-415. 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), 235-248. 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), 31-80. 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), 143-195. 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, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450. 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), 13259-13263. 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), 101-174. 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), 21-50.  Google Scholar

[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.  Google Scholar

[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.  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), 1198-1223 (electronic). 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), 1224-1248 (electronic). doi: 10.1137/S003613990343389X.  Google Scholar

[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.  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, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 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), 691-721. 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), 1449-1481. 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), 32pp (electronic).  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), 3515-3530. 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, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007, 45-62. 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), 217-237. 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, Université Lille 1, 2010. Google Scholar

[11]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121. 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), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[13]

T. O. Gallouët, Optimal transport: Regularity and Applications, Ph.D thesis, Université de Lyon-Ecole Normale Supérieure de Lyon, 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), 297-319. 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), 1203-1227 (electronic). 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), 133-151. 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), 940-960. 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), 633-683 (1998).  Google Scholar

[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.  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), 103-165.  Google Scholar

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. 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), 399-415. 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), 235-248. 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), 31-80. 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), 143-195. 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, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450. 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), 13259-13263. 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), 101-174. 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), 21-50.  Google Scholar

[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.  Google Scholar

[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.  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), 1198-1223 (electronic). 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), 1224-1248 (electronic). doi: 10.1137/S003613990343389X.  Google Scholar

[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.  Google Scholar

[1]

Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809
[2]

Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109
[3]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[4]

Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029
[5]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013
[6]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[7]

Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
[8]

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
[9]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[10]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[11]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[12]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
[13]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[14]

Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537
[15]

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
[16]

Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79
[17]

Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2453-2464. doi: 10.3934/cpaa.2015.14.2453
[18]

Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489
[19]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[20]

Maria Antonietta Farina, Monica Marras, Giuseppe Viglialoro. On explicit lower bounds and blow-up times in a model of chemotaxis. Conference Publications, 2015, 2015 (special) : 409-417. doi: 10.3934/proc.2015.0409

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (120)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy