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 $H_x^{\log }{L}_v^2(⟨v{⟩}^m)$, then the solution remains the same size in the same space. Moreover, a space-time type Landau damping holds, namely, $\mathrm{\parallel }E{\mathrm{\parallel }}_{L_t^2{L}_x^2}\lesssim {\nu }^{\frac{1}{2}}$; and a point-wise type Landau damping holds, namely, $\mathrm{\parallel }E(t){\mathrm{\parallel }}_{L^2}\lesssim {\nu }^{1\mathrm{/}2}⟨t{⟩}^{-N}$ for any $N>0$ for $t\ge {\nu }^{-1}$. We also prove that there exists initial perturbation in $H_x^1{L}_v^2(⟨v{⟩}^m)$ with size ${\nu }^{\frac{1}{2}-\frac{3}{2}{\epsilon }_0}$ with any ${\epsilon }_0>0$, such that the enhanced dissipation fails to hold in the following sense: there is $0<T\ll {\nu }^{-\frac{1}{3}}$ such that \begin{align*} |\langle v\rangle^m f_{\neq}(T)|{L^2_xL^2_v}\gtrsim \frac{1}{ν^{δ_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 μ]=(\partial_t +v\cdot \nabla_x)$\mathbf{D}$[g]. \end{align*}