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The Analysis of Stochastic Fiber Lay‐Down Models: Geometry and Convergence to Equilibrium of the Basic Model
Author(s) -
Grothaus Martin,
Klar Axel,
Maringer Johannes,
Stilgenbauer Patrik
Publication year - 2012
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201210294
Subject(s) - convergence (economics) , fokker–planck equation , stochastic differential equation , manifold (fluid mechanics) , fiber , degenerate energy levels , mathematics , process (computing) , differential equation , computer science , statistical physics , mathematical optimization , mathematical analysis , physics , engineering , mechanical engineering , economics , chemistry , organic chemistry , quantum mechanics , economic growth , operating system
The so‐called fiber lay‐down models arise in the production process of nonwovens. We introduce the generalized version of the basic fiber lay‐down model which can precisely be formulated in abstract form as some manifold‐valued stochastic differential equation. An important criterion for the quality of the nonwoven material is how the solution to the associated Fokker‐Planck equation converges towards its stationary state. Especially, one is interested in determining the speed of convergence. Here we present some results concerning the long‐time behavior by using classical stochastic methods as well as modern analytic methods from the theory of hypocoercivity. Demanding mathematical difficulties arising since the equation is degenerate. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)