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October  2013, 18(8): 2029-2050. doi: 10.3934/dcdsb.2013.18.2029

Blow-up dynamics of self-attracting diffusive particles driven by competing convexities

Vincent Calvez 1, and  Lucilla Corrias 2,

1. 

Unité de Mathématiques Pures et Appliquées, CNRS UMR 5669, École Normale Supérieure de Lyon, 46 allée d'Italie, F-69364 Lyon cedex 07, France
2. 

Laboratoire d'Analyse et Probabilité, Université d'Evry Val d'Essonne, 23 Bd. de France, F-91037 Evry Cedex, France

Received  January 2013 Revised  May 2013 Published  July 2013

In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is an approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel system in high dimensions.
Keywords: blow-up, gradient flow., particles methods, Chemotaxis.
Mathematics Subject Classification: 35B44, 35D30, 35Q92, 35K55, 65M99, 92C17, 92B0.
Citation: Vincent Calvez, Lucilla Corrias. Blow-up dynamics of self-attracting diffusive particles driven by competing convexities. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2029-2050. doi: 10.3934/dcdsb.2013.18.2029
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Lectures in Mathematics ETH Zürich, (2005).   Google Scholar

[2]

D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media,, J. Stat. Phys., 91 (1998), 979.  doi: 10.1023/A:1023032000560.  Google Scholar

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615.   Google Scholar

[4]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III,, Colloq. Math., 68 (1995), 229.   Google Scholar

[5]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I,, Colloq. Math., 66 (1994), 319.   Google Scholar

[6]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1999), 845.  doi: 10.1137/S0036139996313447.  Google Scholar

[7]

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

[8]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Diff. Eqns., 2006 (2006), 1.   Google Scholar

[9]

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

[10]

V. Calvez, L. Corrias and A. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension,, Comm. Partial Differential Equations, 37 (2012), 561.  doi: 10.1080/03605302.2012.655824.  Google Scholar

[11]

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

[12]

E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbbS^n$,, Geom. Funct. Anal., 2 (1992), 90.  doi: 10.1007/BF01895706.  Google Scholar

[13]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Rat. Mech. Anal., 179 (2006), 217.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[14]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations,, in, (2004), 234.   Google Scholar

[15]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math., 72 (2004), 1.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[16]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Property of Functions,'', Studies in Advanced Mathematics, (1992).   Google Scholar

[17]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.  doi: 10.2307/2153966.  Google Scholar

[18]

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

[19]

E. F. Keller and L. A. 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

[20]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[21]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis,, J. Funct. Anal., 191 (2002), 17.  doi: 10.1006/jfan.2001.3802.  Google Scholar

[22]

C. Sire and P.-H. Chavanis, Post-collapse dynamics of self-gravitating Brownian particles and bacterial populations,, Phys. Rev. E, 69 (2004).   Google Scholar

[23]

C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations,, Phys. Rev. E (3), 78 (2008).  doi: 10.1103/PhysRevE.78.061111.  Google Scholar

[24]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, (2003).  doi: 10.1007/b12016.  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,", Lectures in Mathematics ETH Zürich, (2005).   Google Scholar

[2]

D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media,, J. Stat. Phys., 91 (1998), 979.  doi: 10.1023/A:1023032000560.  Google Scholar

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615.   Google Scholar

[4]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III,, Colloq. Math., 68 (1995), 229.   Google Scholar

[5]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I,, Colloq. Math., 66 (1994), 319.   Google Scholar

[6]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1999), 845.  doi: 10.1137/S0036139996313447.  Google Scholar

[7]

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

[8]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Diff. Eqns., 2006 (2006), 1.   Google Scholar

[9]

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

[10]

V. Calvez, L. Corrias and A. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension,, Comm. Partial Differential Equations, 37 (2012), 561.  doi: 10.1080/03605302.2012.655824.  Google Scholar

[11]

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

[12]

E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbbS^n$,, Geom. Funct. Anal., 2 (1992), 90.  doi: 10.1007/BF01895706.  Google Scholar

[13]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Rat. Mech. Anal., 179 (2006), 217.  doi: 10.1007/s00205-005-0386-1.  Google Scholar

[14]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations,, in, (2004), 234.   Google Scholar

[15]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math., 72 (2004), 1.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[16]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Property of Functions,'', Studies in Advanced Mathematics, (1992).   Google Scholar

[17]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.  doi: 10.2307/2153966.  Google Scholar

[18]

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

[19]

E. F. Keller and L. A. 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

[20]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[21]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis,, J. Funct. Anal., 191 (2002), 17.  doi: 10.1006/jfan.2001.3802.  Google Scholar

[22]

C. Sire and P.-H. Chavanis, Post-collapse dynamics of self-gravitating Brownian particles and bacterial populations,, Phys. Rev. E, 69 (2004).   Google Scholar

[23]

C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations,, Phys. Rev. E (3), 78 (2008).  doi: 10.1103/PhysRevE.78.061111.  Google Scholar

[24]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, (2003).  doi: 10.1007/b12016.  Google Scholar

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