Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes

Jean Dolbeault (1) , Axel Klar (2) , Clément Mouhot (3) , Christian Schmeiser (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Departement of Mathematics, TU Kaiserslautern
3 DPMMS - Department of Pure Mathematics and Mathematical Statistics
4 Fakultät für Mathematik [Wien]

This paper is devoted to the adaptation of the method developed in [4,3] to a Fokker-Planck equation for fiber lay-down which has been studied in [1,5]. Exponential convergence towards a unique stationary state is proved in a norm which is equivalent to a weighted $L^2$ norm. The method is based on a micro / macro decomposition which is well adapted to the diffusion limit regime.

Mots clés: kinetic equations; stochastic differential equations; Fokker-Planck equation; fiber dynamics; hypocoercivity; spectral gap; Poincaré inequality; hypoelliptic operators; degenerate diffusion; transport operator; large time behavior; convergence to equilibrium; exponential rate of convergence

Domaines: Equations aux dérivées partielles [math.AP]; Génie des procédés

Jean Dolbeault, Axel Klar, Clément Mouhot, Christian Schmeiser. Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes. Applied Mathematics Research eXpress, 2013, 2013, pp.165-175. ⟨10.1093/amrx/abs015⟩. ⟨hal-00658343⟩