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Local stability of perfect alignment for a spatially homogeneous kinetic model

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Date
2014
Link to item file
http://hal.archives-ouvertes.fr/hal-00962234
Dewey
Probabilités et mathématiques appliquées
Sujet
comparison theorems; Riemannian manifold; unit sphere; Dirac mass; nonlinear stability; alignment of particles; kinetic equation
Journal issue
Journal of Statistical Physics
Volume
157
Number
1
Publication date
2014
Article pages
84-112
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s10955-014-1062-3
URI
https://basepub.dauphine.fr/handle/123456789/13033
Collections
  • CEREMADE : Publications
Metadata
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Author
Raoul, Gaël
Frouvelle, Amic
Degond, Pierre
Type
Article accepté pour publication ou publié
Abstract (EN)
We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the absence of an energy that would decrease for any initial condition. We overcome this difficulty thanks to a functional which is decreasing in time in a neighborhood of any Dirac mass (in the sense of the Wasserstein distance). The results are then extended to the case where the unit sphere is replaced by a general Riemannian manifold.

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