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

Consider the linear transport equation in 1D under an external confining potential \begin{document}$ \Phi $\end{document}:

For \begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} (with \begin{document}$ \varepsilon >0 $\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}, with an inverse polynomial decay rate \begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $\end{document}D under the external potential \begin{document}$ \Phi $\end{document}.

 
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Award ID(s):
2005435
NSF-PAR ID:
10330775
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Kinetic and Related Models
Volume:
15
Issue:
3
ISSN:
1937-5093
Page Range / eLocation ID:
403
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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