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

  • * Corresponding author: Jonathan Luk

    * Corresponding author: Jonathan Luk

The authors are supported by NSF grant DMS-2005435

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  • 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 $.

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


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