Hypocoercivity for Kolmogorov backward evolution equations and applications

TL;DR

This paper extends hypocoercivity methods to analyze the long-time behavior of degenerate Kolmogorov backward operators, with applications to the spherical velocity Langevin equation relevant in industrial fiber production.

Contribution

It develops a new framework for hypocoercivity applicable to degenerate operators, including domain issues, and applies it to a model with industrial relevance.

Findings

Proves exponential decay to equilibrium for the spherical velocity Langevin equation.

Provides conditions for hypocoercivity verification on fixed operator cores.

Constructs contraction semigroups using hypoellipticity and perturbation theory.

Abstract

In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. The latter can be seen as some kind of an analogue to the classical Langevin equation in case spherical velocities are required. This model is of important industrial relevance and describes the fiber lay-down in the production process of nonwovens. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and pertubation theory.