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978-3-8439-1479-6, Reihe Mathematik

Patrik Stilgenbauer
The Stochastic Analysis of Fiber Lay-Down Models: An Interplay between Pure and Applied Mathematics involving Langevin Processes on Manifolds, Ergodicity for Degenerate Kolmogorov Equations and Hypocoercivity

266 Seiten, Dissertation Technische Universität Kaiserslautern (2014), Softcover, A5

Zusammenfassung / Abstract

This thesis is motivated by the mathematical analysis, especially the study of the longtime behavior, of stochastic fiber lay-down models arising in the production process of nonwovens. In these models, the lay-down of single fibers is described with the help of stochastic differential equations on manifolds that allow the computer simulation of whole fiber webs with regard to validation and optimization of final fleece products. We will get to know a fascinating interaction between pure mathematics on the one hand, and applied mathematics on the other hand. Our approaches are based on Functional Analysis, in particular on the theory of operator semigroups, and on Stochastic Analysis.

In the first part of this thesis new industrial relevant fiber lay-down models are developed. This also requires the development of new Langevin type equations; Langevin processes appear in modeling the dynamics of molecular systems and interacting particle systems in Statistical Mechanics and Mathematical Physics. We show their deep geometric connection to the fiber lay-down models and especially, we are interested in higher dimensional manifold-valued versions of the Langevin diffusions.

The convergence to equilibrium of the fiber lay-down process is of essential interest for the fabrication of qualitatively valuable nonwoven fleece products. However, the study of the longtime behavior of the fiber lay-down models provide demanding scientific challenges. Due to the degeneracy of the models, new modern mathematical concepts lying in between Functional Analysis and Stochastic Analysis need to be explored first or refined.

In the second part of the thesis a new class of such analytical and stochastic methods for showing hypocoercivity, ergodicity or mixing properties are provided. We develop abstract concepts that imply the relaxation to equilibrium and an explicit rate of convergence for a huge class of degenerate (partly singularly distorted) kinetic Kolmogorov evolution equation. In particular, we discuss the application to the classical Langevin equation. The developed hypocoercivity and ergodicity methods represent the main part of the underlying thesis.

In the last part of the thesis these methods are applied to the basic version of the fiber lay-down model. We investigate the longtime behavior, prove the desired mixing properties and show ergodicity as well as hypocoercivity with rate of convergence for the industrial application of interest.