The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

TL;DR

The paper characterizes the short-time decay of propagators for linear time-invariant ODEs with a positive semi-definite Hermitian part via the hypocoercivity index $m_{HC}$. It proves a tight link: the leading short-time decay exponent is $a=2m_{HC}+1$, and the propagator norm satisfies $\|P(t)\|_2=1-c t^{a}+\mathcal{O}(t^{a+1})$, with $c>0$ determined by explicit kernel-based expressions involving the Hermitian and anti-Hermitian parts of the system matrix. Four equivalent Kalman-rank formulations underpin the HC-index, providing practical criteria to verify hypocoercivity and enabling a concise interpretation of finite-dimensional approximations of kinetic-type PDEs. The paper also supplies numerical demonstrations showing how the theory predicts both the exponent and the waiting time $t_0$ for decay to $1/e$, including behavior under $\epsilon$-dependent perturbations, highlighting its relevance for short-time dynamics in conservative-dissipative ODEs and kinetic theory applications.

Abstract

We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system.

Figures (3)

• Figure 1: Evolution of \ref{['ODE:B']} with matrix $B$ from \ref{['B:envelope']}: Comparison between the propagator norm (red line), its upper exponential envelope $1.25\exp(-t/10)$ (green line), and the norm of the solution with initial condition ${\mathbf x}(0)=\binom01$ (blue line) and that with initial condition ${\mathbf x}(0)=\binom{-0.1}{\sqrt{0.99}}$ (yellow line), all plotted on two time scales.
• Figure 2: The decay of $\|P_\epsilon(t)\|_2$ is given for six values of $\epsilon$. Left: For $t$ away from 0, this semigroup decays almost exponentially. With the logarithmic scale used here, the horizontal black line corresponds to $1/e$. The waiting times (defined as intersection with the line $1/e$) behave like ${\mathcal{O}}(\epsilon^{-2})$. We remark that the kink in the blue curve is not a numerical artifact. Right: This double logarithmic plot shows $1-\|P_\epsilon(t)\|_2\sim c_\epsilon\,t^3$ for small time, more precisely for $t\in[e^{-9},e^5]$. The curves have slope 3, and $c_\epsilon= \tilde{c}\,\epsilon^2$. The plot also shows the quite sharp transition from the initial algebraic behavior $1-\tilde{c}\,\epsilon^2\,t^3$ to the exponential behavior $c_\epsilon^* \, e^{-\tilde{\mu}\,\epsilon^2\,t}$.
• Figure 3: The decay of $\|P_\epsilon(t)\|_2$ is given for six values of $\epsilon$. Left: For $t$ away from 0, this semigroup decays almost exponentially. With the logarithmic scale used here, the horizontal black line corresponds to $1/e$. The waiting times (defined as intersection with the line $1/e$) behave like ${\mathcal{O}}(\epsilon^{-2})$. Right: This double logarithmic plot shows $1-\|P_\epsilon(t)\|_2\sim c_\epsilon\,t^7$ for small time, more precisely for $t\in[e^{-1.5},e^4]$. The curves have slope 7, and $c_\epsilon= \tilde{c}\ \epsilon^6$. The plot also shows the quite sharp transition from the initial algebraic behavior $1-\tilde{c}\ \epsilon^6\,t^7$ to the exponential behavior $c_{\epsilon}^*\ e^{-\tilde{\mu} \ \epsilon^2\ t}$.

Theorems & Definitions (29)

• Definition 1.1
• Definition 1.2
• Proposition 1.3: MMS16, AAC18
• Lemma 1.4
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof