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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, Birkhäuser Verlag, Basel, 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), 979-990. 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), 615-641.  Google Scholar [4] P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math., 68 (1995), 229-239.  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), 319-334.  Google Scholar [6] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. 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), 691-721. 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), 1-33.  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), 3515-3530. 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), 561-584. 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 "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 [12] E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbbS^n$, Geom. Funct. Anal., 2 (1992), 90-104. 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), 217-263. 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 "New Trends in Mathematical Physics,'' World Sci. Publ., Hackensack, NJ, (2004), 234-244.  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), 1-28. 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, CRC Press, Boca Raton, FL, 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), 819-824. 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), 1-17. 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), 399-415. 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), 225-234. 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), 17-51. 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), 066109. 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), 061111, 22 pp. doi: 10.1103/PhysRevE.78.061111.  Google Scholar [24] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 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, Birkhäuser Verlag, Basel, 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), 979-990. 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), 615-641.  Google Scholar [4] P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math., 68 (1995), 229-239.  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), 319-334.  Google Scholar [6] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. 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), 691-721. 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), 1-33.  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), 3515-3530. 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), 561-584. 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 "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 [12] E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbbS^n$, Geom. Funct. Anal., 2 (1992), 90-104. 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), 217-263. 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 "New Trends in Mathematical Physics,'' World Sci. Publ., Hackensack, NJ, (2004), 234-244.  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), 1-28. 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, CRC Press, Boca Raton, FL, 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), 819-824. 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), 1-17. 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), 399-415. 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), 225-234. 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), 17-51. 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), 066109. 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), 061111, 22 pp. doi: 10.1103/PhysRevE.78.061111.  Google Scholar [24] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

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