Hypocoercivity in Hilbert spaces

TL;DR

This paper extends the concept of hypocoercivity from finite-dimensional systems to infinite-dimensional Hilbert spaces, providing new theoretical tools and decay estimates for linear evolution equations with dissipation.

Contribution

It introduces an infinite-dimensional staircase form and applies hypocoercivity analysis to the Lorentz kinetic equation, advancing the theoretical framework.

Findings

Derived quantitative decay estimates for hypocoercive systems.

Extended hypocoercivity characterizations to Hilbert spaces.

Applied results to the Lorentz kinetic equation.

Abstract

The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are extended to separable Hilbert spaces. Using the concept of a hypocoercivity index, quantitative estimates on the short-time and long-time decay behavior of a hypocoercive system are derived. As a useful tool for analyzing the structural properties, an infinite-dimensional staircase form is also derived and connections to linear systems and control theory are presented. Several examples illustrate the new concepts and the results are applied to the Lorentz kinetic equation.