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Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $\begin{array}{c}{\mathrm{\partial <\#comment/>}}_{t}F(t,x,v)+v_i{\mathrm{\partial <\#comment/>}}_{x_i}F(t,x,v)+E_i(t,x){\mathrm{\partial <\#comment/>}}_{v_i}F(t,x,v)=\mathrm{\nu <\#comment/>}Q(F,F)(t,x,v),\\ E(t,x)=\mathrm{\nabla <\#comment/>}{\mathrm{\Delta <\#comment/>}}^{-<\#comment/>1}\left({\int <\#comment/>}_{{\mathbb{R}}^3}F\right(t,x,v)\mathrm{d}v-<\#comment/>{\int <\#comment/>-<\#comment/>}_{{\mathbb{T}}^3}{\int <\#comment/>}_{{\mathbb{R}}^3}F(t,x,v\left)\mathrm{d}v\mathrm{d}x\right),\end{array}$with $\mathrm{\nu <\#comment/>}\ll <\#comment/>1$. We prove that for $\mathrm{ϵ<\#comment/>}>0$sufficiently small (but independent of $\mathrm{\nu <\#comment/>}$), initial data which are $O\left(\mathrm{ϵ<\#comment/>}{\mathrm{\nu <\#comment/>}}^{1/3}\right)$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to <\#comment/>\mathrm{\infty <\#comment/>}$. The solutions exhibit uniform-in- $\mathrm{\nu <\#comment/>}$Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. 

Consider the linear transport equation in 1D under an external confining potential \begin{document}$$ \Phi $$\end{document}: \begin{document}$$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $$\end{document} For \begin{document}$$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $$\end{document} (with \begin{document}$$ \varepsilon >0 $$\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $$\end{document}, with an inverse polynomial decay rate \begin{document}$$ O({\langle} t{\rangle}^{-2}) $$\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $$\end{document}D under the external potential \begin{document}$$ \Phi $$\end{document}$.