Optimal stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations

TL;DR

This paper determines the optimal stability threshold for the Vlasov-Poisson equation with weak Fokker-Planck collision in lower regularity spaces, proving Landau damping and identifying conditions where enhanced dissipation fails.

Contribution

It solves the open problem of the sharp stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations, introducing a precise wave operator to handle nonlocal terms.

Findings

Landau damping holds with initial perturbation size a0bd;a0a0and a0bd;a0a0in the critical weighted space.

Enhanced dissipation can fail for certain initial perturbations in lower regularity spaces.

The paper constructs a wave operator a0bd;a0a0to absorb nonlocal terms and prove stability thresholds.

Abstract

In this paper, we study the optimal stability threshold for the Vlasov-Poisson equation with weak Fokker-Planck collision. We prove that if the initial perturbation is of size $\nu^{\frac{1}{2}}$ in the critical weighted space $H_x^{\log}L^2_{v}(\langle v\rangle^m)$, then the solution remains the same size in the same space. Moreover, a space-time type Landau damping holds, namely, $\|E\|_{L^2_tL^2_x}\lesssim \nu^{\frac{1}{2}}$; and a point-wise type Landau damping holds, namely, $\|E(t)\|_{L^2}\lesssim \nu^{1/2}\langle t\rangle^{-N}$ for any $N>0$ for $t\geq \nu^{-1}$. We also prove that there exists initial perturbation in $H^{1}_xL^2_v(\langle v\rangle^m)$ with size $\nu^{\frac12-\frac32\epsilon_0}$ with any ${\epsilon_0>0}$, such that the enhanced dissipation fails to hold in the following sense: there is $0<T\ll \nu^{-\frac13}$ such that \begin{align*} \|\langle v\rangle^m f_{\neq}(T)\|_{L^2_xL^2_v}\gtrsim \frac{1}{\nu^{\delta_1}}\|\langle v\rangle^m f_{\neq}(0)\|_{ H^1_xL^2_v} \end{align*} with some $\delta_1>0$. The paper solves the open problem raised in [Bedrossian; arXiv: 2211.13707] about the sharp stability threshold in lower regularity spaces. The main idea is to construct a wave operator $\mathbf{D}$ with a very precise expression to absorb the nonlocal term, namely, \begin{align*} \mathbf{D}[\partial_tg+v\cdot \nabla_x g+E\cdot\nabla_v \mu]=(\partial_t +v\cdot \nabla_x)\mathbf{D}[g]. \end{align*}