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

TL;DR

This paper proves the existence, uniqueness, and exponential convergence to equilibrium of a kinetic Fokker-Planck model for fibre lay-down on a moving conveyor belt, extending previous stationary belt results with explicit decay rates.

Contribution

It extends hypocoercivity methods to a conveyor belt scenario, deriving explicit convergence rates for a more general fiber lay-down model.

Findings

Existence and uniqueness of stationary state

Exponential convergence to equilibrium with explicit rate

Extension of hypocoercivity techniques to moving conveyor belt

Abstract

We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling 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 extension of (Dolbeault et al., 2013), 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 (Dolbeault et al., 2013), we make use of an additional weight function following the Lyapunov functional approach in (Kolb et al., 2013).