Landau damping and survival threshold

TL;DR

This work provides a precise large-time description of the linearized Vlasov-Poisson dynamics near a broad class of spatially homogeneous equilibria with connected velocity support. Using a mode-by-mode Laplace-Fourier analysis, it derives a dielectric function $D(\lambda,k)$ and identifies a finite survival threshold $\kappa_0$ that delineates Langmuir oscillations from damping, with Langmuir modes persisting for $|k|\le\kappa_0$ and damping governed by Landau's law at the threshold; for $|k|>\kappa_0$ the dynamics is controlled by phase mixing of free transport. The paper proves the existence of two imaginary zeros $\lambda_\pm(k)=\pm i\tau_*(k)$ for $|k|\le\kappa_0$, with $\tau_*(k)$ obeying a Klein–Gordon–type dispersion, and provides explicit damping rates depending on the decay of $\mu$ at the maximal velocity $\Upsilon$, including sharp results for finite and infinite $\Upsilon$. A detailed Green function analysis yields a decomposition of the electric field into oscillatory and rapidly decaying remainder terms, with precise $L^p$ decay rates, and these results extend to the torus with a discrete spectrum, revealing time-periodic modes below threshold for large tori. Together, these results illuminate the transition between persistent Langmuir oscillations and Landau damping and furnish quantitative tools for subsequent nonlinear analyses of the VP system.

Abstract

In this paper, we establish the large time asymptotic behavior of solutions to the linearized Vlasov-Poisson system near general spatially homogenous equilibria $μ(\frac12|v|^2)$ with connected support on the torus $\mathbb{T}^3_x \times \mathbb{R}^3_v$ or on the whole space $\mathbb{R}^3_x \times \mathbb{R}^3_v$, including those that are non-monotone. The problem can be solved completely mode by mode for each spatial wave number, and their longtime dynamics is intimately tied to the ``survival threshold'' of wave numbers computed by $$κ_0^2 = 4π\int_0^Υ\frac{u^2μ(\frac12 u^2)}{Υ^2-u^2} \;du$$ where $Υ$ is the maximal speed of particle velocities. It is shown that purely oscillatory electric fields exist and obey a Klein-Gordon's type dispersion relation for wave numbers below { and up to} the threshold, thus rigorously confirming the existence of Langmuir's oscillatory waves { for a non-trivial range of spatial frequencies in this linearized setting}. At the threshold, the phase velocity of these oscillatory waves enters the range of admissible particle velocities, namely there are particles that move at the same propagation speed of the waves. It is this exact resonant interaction between particles and the oscillatory fields that causes the waves to be damped, classically known as Landau damping. Landau's law of decay is explicitly computed and is sensitive to the decaying rate of the background equilibria. The faster it decays at the maximal velocity, the weaker Landau damping is. Beyond the threshold, the electric fields are a perturbation of those generated by the free transport dynamics and thus decay rapidly fast due to the phase mixing mechanism.

Figures (2)

• Figure 1: Depicted are the solutions $\lambda_\pm(k)$ to the dispersion relation that start from $\lambda_\pm(0)=\pm i \tau_0$ at $k=0$, remain on the imaginary axis and obey a Klein-Gordon's dispersion relation $\tau_*(|k|)\sim \sqrt{\tau_0^2 + |k|^2}$ for all $0\le |k|\le \kappa_0$ up until the survival threshold $|k|=\kappa_0$ at which $\lambda_\pm(\kappa_0) = \pm i\sqrt{\tau_0^2 + \kappa_1^2}$, and then depart from the imaginary axis as soon as $|k|>\kappa_0$ due to Landau damping towards the phase mixing regime $\{ \Re \lambda \lesssim -|k|\}$. The group velocity $\tau_*'(k)$ is strictly increasing, while the phase velocity $\nu_*(k) = \tau_*(|k|)/|k|$ is strictly decreasing in $|k|$, with $\nu_*(0) = \infty$ and $\nu_*(\kappa_0) = \Upsilon$.
• Figure 2: Illustrated on the left is the contour of integration $\Gamma = \Gamma_1 \cup \Gamma_2 \cup \mathcal{C}_\pm$, while on the right is the contour $\Gamma^* = \Gamma_1 \cup \Gamma_2 \cup \mathcal{C}^*_\pm$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['lem-resolvent']}
• Lemma 2.3
• proof
• Remark 2.4