Landau damping and survival threshold

In this paper, we establish the large time asymptotic behavior of solutions to the linearized Vlasov-Poisson system near general spatially homogenous equilibria $\mu(\frac12|v|^2)$ 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 $$\kappa_0^2 = 4\pi \int_0^\Upsilon \frac{u^2\mu(\frac12 u^2)}{\Upsilon^2-u^2} \;du$$ where $\Upsilon$ 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.