Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

Abstract

In this article we review a powerful and fairly simple hypocoercivity method and its application to qualitative analysis of industrially relevant fibre lay‐down models. The hypocoercivity strategy is formulated in an abstract Hilbert space setting with all necessary conditions; we provide some interpretation of these conditions and briefly explain why they are needed to make the machinery working. Once the conditions are fulfilled we gain exponential decay of the strongly continuous semigroup associated to the fibre lay‐down process at an explicitly known rate. We interpret this result as describing for instance homogeneity of nonwoven fibre webs. Primarily, this article portrays the concepts, however it is based on original results by Grothaus and Stilgenbauer, which can be found in the previous studies.

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincaré Inequalities

“…Hypocoercivity methods go back to Villani ( [1]), were further developed by Dolbeault, Mouhot and Schmeiser ( [2]) as well as Grothaus and Stilgenbauer ([3]), and generalized by Grothaus and Wang ([4]) to the case with weak Poincaré inequalities. Since then, the latter have been successfully used to derive concrete convergence rates for solutions to degenerate Fokker-Planck partial differential equations, see for example [5,6,7], where last mentioned reference allows for velocity-dependent second-order coefficients. In order to obtain the rate of convergence to their stationary solution and to their equilibrium measure for solutions to degenerate stochastic differential equations, respectively, it is assumed that a Poincaré inequality exists for the measure induced by the potential giving the force.…”