Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 2029 - 2050, Volume 18, Issue 8, October 2013      doi:10.3934/dcdsb.2013.18.2029

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Vincent Calvez - Unité de Mathématiques Pures et Appliquées, CNRS UMR 5669, École Normale Supérieure de Lyon, 46 allée d'Italie, F-69364 Lyon cedex 07, France (email)
Lucilla Corrias - Laboratoire d'Analyse et Probabilité, Université d'Evry Val d'Essonne, 23 Bd. de France, F-91037 Evry Cedex, France (email)

Abstract: 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:  Chemotaxis, blow-up, particles methods, gradient flow.
Mathematics Subject Classification:  35B44, 35D30, 35Q92, 35K55, 65M99, 92C17, 92B05.

Received: January 2013;      Revised: May 2013;      Available Online: July 2013.