Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt

Emeric Bouin (1) , Franca Hoffmann (2, 3) , Clément Mouhot (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics [Imperial College London]
3 Cambridge Center for Analysis
4 DPMMS - Department of Pure Mathematics and Mathematical Statistics

We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modeling the fibre lay-down process in the production of non-woven textiles. Following a micro-macro decomposition, we use hypocoercivity techniques to show exponential convergence to equilibrium with an explicit rate assuming the conveyor belt moves slow enough. This work is an extention of [2], where the authors consider the case of a stationary conveyor belt. Adding the movement of the belt, the global Gibbs state is not known explicitly. We thus derive a more general hypocoercivity estimate from which existence, uniqueness and exponential convergence can be derived. To treat the same class of potentials as in [2], we make use of a an additional weight function following the Lyapunov functional approach in [8].

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Emeric Bouin, Franca Hoffmann, Clément Mouhot. Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt. 2016. ⟨hal-01315222v1⟩