Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes

TL;DR

This paper proves exponential convergence to equilibrium for a fiber lay-down model using a micro/macro decomposition approach, extending previous methods to a Fokker-Planck equation relevant in fiber manufacturing processes.

Contribution

It adapts and applies a micro/macro decomposition method to establish exponential convergence for a specific Fokker-Planck model of fiber lay-down.

Findings

Proves exponential convergence to a unique stationary state.

Uses a weighted L^2 norm for analysis.

Extends existing methods to a fiber lay-down Fokker-Planck equation.

Abstract

This paper is devoted to the adaptation of the method developed in [4,3] to a Fokker-Planck equation for fiber lay-down which has been studied in [1,5]. Exponential convergence towards a unique stationary state is proved in a norm which is equivalent to a weighted $L^2$ norm. The method is based on a micro / macro decomposition which is well adapted to the diffusion limit regime.