Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up.

Vincent Calvez 1, 2 Thomas Gallouët 3
2 NUMED - Numerical Medicine
Inria Grenoble - Rhône-Alpes, UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : 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.
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  • ARXIV : 1404.0139

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Vincent Calvez, Thomas Gallouët. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up.. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015. ⟨hal-00968347⟩

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