Frouvelle, Amic and Liu, Jian-Guo
(2012)
*Dynamics in a Kinetic Model of Oriented Particles with Phase Transition.*
SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 4 (2).
pp. 791-826.
ISSN 0036-1410

Text
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## Abstract

Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing more than the Smoluchowski equation on the sphere with dipolar potential. In this self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, the other equilibria are the Fisher–von Mises distributions with arbitrary direction and a concentration parameter determined by the intensity of the noise. For any initial condition, we give a rigorous proof of convergence of the solution to a steady state as time goes to infinity. In particular, when the noise is under the threshold and with nonzero initial mean velocity, the solution converges exponentially fast to a unique Fisher–von Mises distribution. We also found a new conservation relation, which can be viewed as a convex quadratic entropy when the noise is above the threshold. This provides a uniform exponential rate of convergence to the uniform distribution. At the threshold, we show algebraic decay to the uniform distribution.

Item Type: | Article |
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Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Engineering, Science and Mathematics > School of Mathematics > Department of Applied Mathematics |

Depositing User: | Amic Frouvelle |

Date Deposited: | 06 Oct 2012 09:08 |

Last Modified: | 07 May 2017 19:33 |

URI: | http://preprints.acmac.uoc.gr/id/eprint/147 |

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