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Rate of Convergence to Equilibrium for a Model describing Fibre Lay-down Processes on a Moving Conveyor Belt

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We study the hypocoercivity properties of a kinetic Fokker-Planck equation modelling the fibre lay-down process in the production of non-woven textiles. Under suitable assumptions on the potential describing the coiling properties of the fibres, we establish existence and uniqueness of a global Gibbs state and determine the rate of convergence of solutions in a suitably weighted L^2 space using only hypocoercivity techniques. The method is based on a micro / macro decomposition which is well adapted to the diffusion limit regime. Our result is an extension of the method used by Dolbeault et al. who assumed a stationary conveyor belt. For technical applications in the production process of non-wovens however, one is interested in a model including the movement of the conveyor belt, in which case the global Gibbs state is not known explicitly. This can be addressed by proving a stronger hypocoercivity estimate for solutions of the kinetic Fokker-Planck equation. This is a joint work with Clément Mouhot and Emeric Bouin.

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