Local stability of perfect alignment for a spatially homogeneous kinetic model

Raoul, Gaël; Frouvelle, Amic; Degond, Pierre

Raoul, Gaël; Frouvelle, Amic; Degond, Pierre (2014), Local stability of perfect alignment for a spatially homogeneous kinetic model, Journal of Statistical Physics, 157, 1, p. 84-112. http://dx.doi.org/10.1007/s10955-014-1062-3

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00962234

Date

2014

Journal name

Journal of Statistical Physics

Volume

157

Number

Publisher

Springer

Pages

84-112

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.

Subjects / Keywords

comparison theorems; Riemannian manifold; unit sphere; Dirac mass; nonlinear stability; alignment of particles; kinetic equation