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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

 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.
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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