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1. Consider the linear transport equation in 1D under an external confining potential \begin{document}$\Phi$\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 Vlasovâ€“Poisson system in \begin{document}$1$\end{document}D under the external potential \begin{document}$\Phi$\end{document}.

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