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

Sanchit Chaturvedi; Jonathan Luk

doi:10.3934/krm.2022002

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2022, Volume 15, Issue 3: 403-416. Doi: 10.3934/krm.2022002

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

Sanchit Chaturvedi and
Jonathan Luk^,

Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg 380, Stanford, CA 94305, USA

^* Corresponding author: Jonathan Luk
^* Corresponding author: Jonathan Luk

Received: September 2021

Revised: December 2021

Early access: January 2022

Published: June 2022

The authors are supported by NSF grant DMS-2005435.

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Abstract

Consider the linear transport equation in 1D 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 { \varepsilon x^4}2 $ (with $ \varepsilon >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 $.

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Mathematics Subject Classification: Primary: 35B40, 82C40, 35Q83.

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