Smoothing Estimates of the Vlasov-Poisson-Landau System

Deng, Dingqun

doi:10.48550/arxiv.2103.04114

Cited by 1 publication

(1 citation statement)

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“…Nonlinear stability of global Maxwellians. In the ν = 1 case of (1.1a)-(1.1b) (or its twospecies analogue), the nonlinear asymptotic stability of global Maxwellians was first proven in Guo's [54] in a periodic box; see also [32,35]. The corresponding stability result on R 3 was proven in [89] (with alternative proofs in [57,66,99]).…”

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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