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

TL;DR

This paper proves phase mixing and decay estimates for solutions to a 1D linear transport equation with a confining potential, using a novel commuting vector field approach, with implications for nonlinear stability in Vlasov--Poisson systems.

Contribution

It introduces a new commuting vector field method to establish phase mixing and decay for a 1D transport equation with a confining potential, advancing understanding of stability in related nonlinear systems.

Findings

Established polynomial decay rate O(⟨t⟩^{-2}) for the solution derivative.

Developed a commuting vector field approach adapted to the confining potential setting.

Provided insights relevant to the nonlinear stability of the Vlasov--Poisson system in 1D.

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 \Phi \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 $\partial_t \varphi := - \Delta^{-1} \int_{\mathbb{R}} \partial_t f \, \mathrm{d} v$, with an inverse polynomial decay rate $O(\langle t\rangle^{-2})$. 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 $\Phi$.