Flocking : Phase transition and asymptotic behaviour
Li, Xingyu (2019-06), Flocking : Phase transition and asymptotic behaviour. https://basepub.dauphine.fr/handle/123456789/19390
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-02143985
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)This paper is devoted to a continuous Cucker-Smale model with noise, which has isotropic and polarized stationary solutions depending on the intensity of the noise. The first result establishes the threshold value of the noise parameter which drives the phase transition. This threshold value is used to classify all stationary solutions and their linear stability properties. Using an entropy, these stability properties are extended to the non-linear regime. The second result is concerned with the asymptotic behaviour of the solutions of the evolution problem. In several cases, we prove that stable solutions attract the other solutions with an optimal exponential rate of convergence determined by the spectral gap of the linearized problem around the stable solutions. The spectral gap has to be computed in a norm adapted to the non-local term.
Subjects / Keywordssymmetry breaking; free energy; large time asymp- totics; Flocking model; phase transition; asymptotic rate of convergence AMS subject classifications; asymptotic rate of convergence; spectral gap; stability
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