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

Abstract

We show the existence and uniqueness of a stationary state for a kinetic Fokker--Planck equation modeling the fiber lay-down process in the production of nonwoven 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 slowly enough. This work is an extension of [Dolbeault et al., Appl. Math. Res. Express. AMRX, 2013 (2013), pp. 165-175], 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., Appl. Math. Res. Express. AMRX, (2013), pp. 165-175] we make use of an additional weight function following the Lyapunov functional approach in [M. Kolb, M. Savov, and A. Wübker, SIAM J. Math. Anal., 45 (2013), pp. 1-13].

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