Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions

In this paper, we study the Vlasov-Poisson-Fokker-Planck (VPFP) equation with a small collision frequency $0<\nu \ll 1$, exploring the interplay between the regularity and size of perturbations in the context of the asymptotic stability of the global Maxwellian. Our main result establishes the Landau damping and enhanced dissipation phenomena under the condition that the perturbation of the global Maxwellian falls within the Gevrey-$\frac{1}{s}$ class and obtain that the stability threshold for the Gevrey-$\frac{1}{s}$ class with $s>s_{\mathrm{k}}$ can not be larger than $\gamma =\frac{1-3s_{\mathrm{k}}}{3-3s_{\mathrm{k}}}$ for $s_{\mathrm{k}}\in [0,\frac{1}{3}]$. Moreover, we show that for Gevrey-$\frac{1}{s}$ with $s>3$, and for $t\ll {\nu }^{\frac{1}{3}}$, the solution to VPFP converges to the solution to Vlasov-Poisson equation without collision.