Imperial College London
Alternate

ProfessorBoguslawZegarlinski

Faculty of Natural Sciences, Department of Mathematics

Visiting Professor
 
 
 
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Contact

 

+44 (0)20 7594 8492b.zegarlinski Website

 
 
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Location

 

6M55Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Buttà:2019:10.3934/krm.2019031,
author = {Buttà, P and Flandoli, F and Ottobre, M and Zegarlinski, B},
doi = {10.3934/krm.2019031},
journal = {Kinetic and Related Models},
pages = {791--827},
title = {A non-linear kinetic model of self-propelled particles with multiple equilibria},
url = {http://dx.doi.org/10.3934/krm.2019031},
volume = {12},
year = {2019}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density ft, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only hypoelliptic) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate N-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as N→∞, to the solution of the PDE.
AU  - Buttà,P
AU  - Flandoli,F
AU  - Ottobre,M
AU  - Zegarlinski,B
DO  - 10.3934/krm.2019031
EP  - 827
PY  - 2019///
SN  - 1937-5093
SP  - 791
TI  - A non-linear kinetic model of self-propelled particles with multiple equilibria
T2  - Kinetic and Related Models
UR  - http://dx.doi.org/10.3934/krm.2019031
UR  - http://hdl.handle.net/10044/1/75023
VL  - 12
ER  -
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