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

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*}