Exponential Decay to Equilibrium for a Fiber Lay-Down Process on a Moving Conveyor Belt

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 micromacro 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]. Keywords— hypocoercivity, rate of convergence, fibre lay-down, existence and uniqueness of stationary state, perturbation, moving belt