Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk, Jonathan

doi:10.1007/s40818-019-0067-2

Cited by 27 publications

(34 citation statements)

References 71 publications

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“…We mention [12,13] for the best regularity results available for the Boltzmann equation without cut-off. For the stability of vacuum, see [11,34,47] for the Landau, cutoff and non-cutoff Boltzmann equation with moderate soft potential respectively.…”

Section: 5mentioning

confidence: 99%

The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

Cao¹,

Deng²,

Li³

2021

Preprint

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In this work, we consider the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb potential near a global Maxwellian µ. We establish the global existence, uniqueness and large time behavior for solutions in a polynomial-weighted Sobolev space H 2x,v ( v k ) for some constant k > 0. The proof is based on extra dissipation generated from semigroup method and energy estimates on electrostatic field.

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Section: 5mentioning

confidence: 99%

The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

Cao¹,

Deng²,

Li³

2021

Preprint

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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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“…This lets one prove transport bounds in the presence of collision, and in fact to take advantage of the coercivity given by collision. Such a commutating vector field method is inspired by related techniques for treating dispersion in nonlinear wave equations and other kinetic models [28,29,63,69,83]. See Sections 1.1.4 and 1.2.6.…”

Section: Introductionmentioning

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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Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Cited by 27 publications

References 71 publications

The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

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

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

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