Phase mixing for solutions to 1D transport equation in a confining potential

Abstract

<p style='text-indent:20px;'>Consider the linear transport equation in 1D under an external confining potential <inline-formula><tex-math id="M1">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M2">\begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon &gt;0 $\end{document}</tex-math></inline-formula> small), we prove phase mixing and quantitative decay estimates for <inline-formula><tex-math id="M4">\begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}</tex-math></inline-formula>, with an inverse polynomial decay rate <inline-formula><tex-math id="M5">\begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}</tex-math></inline-formula>. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in <inline-formula><tex-math id="M6">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>D under the external potential <inline-formula><tex-math id="M7">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula>.</p>