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We show existence and uniqueness of a stationary state for a kineticFokker-Planck equation modelling the fibre lay-down process in the productionof non-woven textiles. Following a micro-macro decomposition, we usehypocoercivity techniques to show exponential convergence to equilibrium withan explicit rate assuming the conveyor belt moves slow enough. This work is anextension of (Dolbeault et al., 2013), where the authors consider the case of astationary conveyor belt. Adding the movement of the belt, the global Gibbsstate is not known explicitly. We thus derive a more general hypocoercivityestimate from which existence, uniqueness and exponential convergence can bederived. To treat the same class of potentials as in (Dolbeault et al., 2013),we make use of an additional weight function following the Lyapunov functionalapproach in (Kolb et al., 2013).

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