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Hypocoercivity in Hilbert spaces

Franz Achleitner, Anton Arnold, Volker Mehrmann, Eduard A. Nigsch

TL;DR

This work extends the hypocoercivity framework from finite dimensions to separable Hilbert spaces by introducing the hypocoercivity index (HC-index) and a staircase form, enabling quantitative analysis of short-time and long-time decay for linear dissipative systems. It establishes equivalences and coercivity-type conditions that characterize hypocoercivity in the infinite-dimensional setting, and it derives precise short-time decay rates of the propagator in terms of the HC-index, including both upper and lower bounds and a converse. The staircase form clarifies structural properties, while the Lorentz kinetic equation serves as a central application, yielding uniform exponential decay for long times and cubic decay in the short-time regime, with the HC-index explicitly equal to 1 for the modal components and the full system under mass-zero constraint. The results provide a robust framework for analyzing infinite-dimensional hypocoercive dynamics and connect to control theory concepts, with implications for kinetic models and related dissipative systems.

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.

Hypocoercivity in Hilbert spaces

TL;DR

This work extends the hypocoercivity framework from finite dimensions to separable Hilbert spaces by introducing the hypocoercivity index (HC-index) and a staircase form, enabling quantitative analysis of short-time and long-time decay for linear dissipative systems. It establishes equivalences and coercivity-type conditions that characterize hypocoercivity in the infinite-dimensional setting, and it derives precise short-time decay rates of the propagator in terms of the HC-index, including both upper and lower bounds and a converse. The staircase form clarifies structural properties, while the Lorentz kinetic equation serves as a central application, yielding uniform exponential decay for long times and cubic decay in the short-time regime, with the HC-index explicitly equal to 1 for the modal components and the full system under mass-zero constraint. The results provide a robust framework for analyzing infinite-dimensional hypocoercive dynamics and connect to control theory concepts, with implications for kinetic models and related dissipative systems.

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.
Paper Structure (16 sections, 21 theorems, 173 equations, 1 figure)

This paper contains 16 sections, 21 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conditions for Hypocoercivity
  3. Hypocoercivity Index
  4. Short-time decay of the propagator norm
  5. Main results
  6. Upper bound of the propagator norm
  7. Lower bound of the propagator norm
  8. Reverse direction of Theorem \ref{['thm:short-t-decay:sandwich']}
  9. Staircase form
  10. Example: Lorentz kinetic equation
  11. Mode by mode analysis
  12. Analysis as infinite dimensional dynamical system
  13. Further examples
  14. Appendix
  15. Connection to linear systems and control theory
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let ${\mathbf{J}} \in \mathcal{B}(\mathcal{H})$ be skew-adjoint and ${\mathbf{R}} \in \mathcal{B}(\mathcal{H})$ be self-adjoint with ${\mathbf{R}} \ge 0$ and let $\dim \ker {\mathbf{R}} < \infty$. Then the following conditions are equivalent: If one of p1--p2 holds for a particular $m$, all of them hold for the same $m$. Moreover, these conditions then also hold for all $m' > m$, and the minimum

Figures (1)

  • Figure 1: To derive the uniform estimate $\|{\mathbf{P}}_{\mathbf n}(t)\|_{\mathcal{B}(L^2(\mathbb{S}^1))} \le 1-c t^3$ for $0\le t\le \tau$ in \ref{['Pn-decay']}, we combine a short-term decay estimate for the initial phase $[0,\frac{\tau}{|{\mathbf n}|}]$ (that shrinks w.r.t. $|{\mathbf n}|$) with the long-term decay estimate \ref{['unif-decay']} for the remaining time interval $[\frac{\tau}{|{\mathbf n}|},\tau]$.

Theorems & Definitions (50)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: AAM22Oseen
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 40 more