PARTICLE APPROXIMATION OF THE ONE DIMENSIONAL KELLER-SEGEL EQUATION, STABILITY AND RIGIDITY OF THE BLOW-UP

被引：10

作者：

Calvez, Vincent ^{[1
,2
]}

Gallouet, Thomas O. ^{[3
]}

机构：

[1] Ecole Normale Super Lyon, CNRS, Unite Math Pures & Appl, UMR 5669, F-69364 Lyon, France

[2] Project Team Inria NUMED, Lyon, France

[3] Inria Lille Nord Europe, Project Team MEPHYSTO, Villeneuve Dascq, France

来源：

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS | 2016年 / 36卷 / 03期

关键词：

Keller-Segel; chemotaxis; blow-up; critical mass; particles methods; gradient flow; REFINED ASYMPTOTICS; POINT DYNAMICS; SINGULAR LIMIT; MODEL; AGGREGATION; CONVERGENCE; SCHEME;

D O I：

10.3934/dcds.2016.36.1175

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：1175 / 1208

页数：34

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