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

Abstract

Consider the linear transport equation in $1$D under an external confining potential $\Phi$: \begin{equation*} \partial_t f + v \partial_x f - \partial_x Φ\partial_v f = 0. \end{equation*} For $\Phi =\frac{x^2}{2}+\frac{\epsilon x^4}{2}$ (with $\epsilon >0$ small), we prove phase mixing and quantitative decay estimates for ${\partial }_t\phi :=-{\Delta }^{-1}{\int }_{\mathbb{R}}{\partial }_{t}f\mathrm{d}v$, with an inverse polynomial decay rate $O(⟨t{⟩}^{-2})$. 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 $1$D under the external potential $\Phi$.

Dedicated to the memory of Robert Glassey