A commuting-vector-field approach to some dispersive estimates

Wong, Willie Wai Yeung

doi:10.1007/s00013-017-1114-4

Cited by 20 publications

(9 citation statements)

References 13 publications

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“…Recently, it has likewise found many applications for collisionless kinetic equations. In particular, the stability of vacuum has been established in many different settings [21,39,83,102], and the stability of the Minkowski spacetime for the Einstein-Vlasov system in general relativity has also been resolved [22,38,68,84,91]. (See also [5,4,44,41,98,97,96] for related works on stability of vacuum type results for collisionless models.)…”

Section: 21mentioning

confidence: 99%

The Vlasov--Poisson--Landau system in the weakly collisional regime

Chaturvedi¹,

Luk²,

Nguyen³

2021

Preprint

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Consider the Vlasov-Poisson-Landau system with Coulomb potential in the weakly collisional regime on a 3-torus, i.e.with ν ≪ 1. We prove that for ǫ > 0 sufficiently small (but independent of ν), initial data which are O(ǫν 1 3 )-Sobolev space perturbations from the global Maxwellians lead to globalin-time solutions which converge to the global Maxwellians as t → ∞. The solutions exhibit uniform-in-ν Landau damping and enhanced dissipation.Our main result is analogous to an earlier result of Bedrossian for the Vlasov-Poisson-Fokker-Planck equation with the same threshold. However, unlike in the Fokker-Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo's weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.

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Section: 21mentioning

confidence: 99%

The Vlasov--Poisson--Landau system in the weakly collisional regime

Chaturvedi¹,

Luk²,

Nguyen³

2021

Preprint

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“…Relations between these results and the stability of vacuum for the Boltzmann equation with angular cutoff is discussed in [10]. More recent results can be found in [13,14,15,16,17,24,50,64,65,63,66]. See also [23,46,56] for proof of the stability of the Minkowski spacetime for the Einstein-Vlasov system.…”

Section: Introductionmentioning

confidence: 99%

Stability of vacuum for the Landau equation with hard potentials

Chaturvedi

2020

Preprint

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We consider the spatially inhomogeneous Landau equation with Maxwellian and hard potentials (i.e with γ ∈ [0, 1)) on the whole space R 3 . We prove that if the initial data f in are close to the vacuum solution f vac = 0 in an appropriate weighted norm then the solution f exists globally in time. This work builds up on the author's earlier work on local existence of solutions to Landau equation with hard potentials.Our proof uses L 2 estimates and exploits the null-structure established by Luk [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11]. To be able to close our estimates, we have to couple the weighted energy estimates, which were established by the author in a previous paper [Local existence for the Landau equation with hard potentials, https://arxiv.org/abs/1910.11866], with the null-structure and devise new weighted norms that take this into account.

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“…As mentioned in Remark 1.5, our proof is based on a commutating vector field method. In the context of kinetic theory, the commutating vector field method has been most successful in capturing dispersion, see [6,7,18,19,28,32,33,36] for some results for collisionless models and [10,11,29] for some results on collisional models.…”

Section: Introduction Consider the Linear Transport Equation In 1dmentioning

confidence: 99%

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

Chaturvedi¹,

Luk²

2022

KRM

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Consider the linear transport equation in 1D under an external confining potential <inline-formula><tex-math id="M1">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula>:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $\end{document} </tex-math></disp-formula>For <inline-formula><tex-math id="M2">\begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon >0 $\end{document}</tex-math></inline-formula> small), we prove phase mixing and quantitative decay estimates for <inline-formula><tex-math id="M4">\begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}</tex-math></inline-formula>, with an inverse polynomial decay rate <inline-formula><tex-math id="M5">\begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}</tex-math></inline-formula>. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in <inline-formula><tex-math id="M6">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>D under the external potential <inline-formula><tex-math id="M7">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula>.

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A commuting-vector-field approach to some dispersive estimates

Cited by 20 publications

References 13 publications

The Vlasov--Poisson--Landau system in the weakly collisional regime

The Vlasov--Poisson--Landau system in the weakly collisional regime

Stability of vacuum for the Landau equation with hard potentials

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

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