Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation

Rong Yang; Li Chen

doi:10.3934/krm.2014.7.381

Article Contents

2014, Volume 7, Issue 2: 381-400. Doi: 10.3934/krm.2014.7.381

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Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation

Rong Yang^1, and
Li Chen^2,

1.
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
2.
Universität Mannheim, Lehrstuhl für Mathematik IV, 68131, Mannheim

Received: June 30, 2013

Revised: October 31, 2013

Published: May 2014

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Abstract

A Collision-Avoiding flocking particle system proposed in [8] is studied in this paper. The global wellposedness of its corresponding Vlasov-type kinetic equation is proved. As a corollary of the global stability result, the mean field limit of the particle system is obtained. Furthermore, the time-asymptotic flocking behavior of the solution to the kinetic equation is also derived. The technics used for local wellposedness and stability follow from similar ideas to those have been used in [3,14,22]. While in order to extend the local result globally, the main contribution here is to generate a series of new estimates for this Vlasov type equation, which imply that the growing of the characteristics can be controlled globally. Further estimates also show the long time flocking phenomena.

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Mathematics Subject Classification: Primary: 35A05, 35B40, 35D99; Secondary: 92D15.

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References

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