Local stability of perfect alignment for a spatially homogeneous kinetic model
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
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00962234
Journal nameJournal of Statistical Physics
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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 / Keywordscomparison theorems; Riemannian manifold; unit sphere; Dirac mass; nonlinear stability; alignment of particles; kinetic equation
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