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

Vincent Calvez | Zendy; Thomas Gallouët | Zendy

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

Author(s) -

Vincent Calvez,

Thomas Gallouët

Publication year - 2015

Publication title -

discrete and continuous dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.289

H-Index - 70

eISSN - 1553-5231

pISSN - 1078-0947

DOI - 10.3934/dcds.2016.36.1175

Subject(s) - rigidity (electromagnetism) , logarithm , balanced flow , work (physics) , particle system , homogeneous , physics , particle (ecology) , stability (learning theory) , flow (mathematics) , limiting , mathematics , classical mechanics , statistical physics , mathematical analysis , mechanics , thermodynamics , computer science , mechanical engineering , oceanography , quantum mechanics , machine learning , engineering , geology , operating system

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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