On the Linearized Vlasov–Poisson System on the Whole Space Around Stable Homogeneous Equilibria

Han-Kwan, Daniel; Nguyen, Toan T.; Rousset, Frédéric

doi:10.1007/s00220-021-04228-2

Cited by 20 publications

(23 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See Subsection 1.3 for the notation x , x, y , ∇ , and W k+0,p w . We remark that another preprint proving similar results with somewhat different methods [18] has recently been completed as well. These works have been completed totally independently.…”

Section: Resultsmentioning

confidence: 66%

Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

Bedrossian¹,

Masmoudi²,

Mouhot³

2020

Preprint

View full text Add to dashboard Cite

In this paper we study the linearized Vlasov-Poisson equation for localized disturbances of an infinite, homogeneous Maxwellian background distribution in R 3x × R 3 v . In contrast with the confined case T dx × R d v , or the unconfined case R d x × R d v with screening, the dynamics of the disturbance are not scattering towards free transport as t → ±∞: we show that the electric field decomposes into a very weakly-damped Klein-Gordon-type evolution for long waves and a Landau-damped evolution. The Klein-Gordon-type waves solve, to leading order, the compressible Euler-Poisson equations linearized about a constant density state, despite the fact that our model is collisionless, i.e. there is no trend to local or global thermalization of the distribution function in strong topologies. We prove dispersive estimates on the Klein-Gordon part of the dynamics. The Landau damping part of the electric field decays faster than free transport at low frequencies and damps as in the confined case at high frequencies; in fact, it decays at the same rate as in the screened case. As such, neither contribution to the electric field behaves as in the vacuum case.

show abstract

Section: Resultsmentioning

confidence: 66%

Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

Bedrossian¹,

Masmoudi²,

Mouhot³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In particular, the key lower bound in the Penrose criterion (1.8) cannot hold. This critical difficulty has been observed in [13,14,5,23] in the study of the linearized system. In the nonlinear setting the failure of the Penrose condition leads to the presence of small denominators and resonances in the system.…”

mentioning

confidence: 83%

Nonlinear Landau damping for the Vlasov-Poisson system in $\R^3$: the Poisson equilibrium

Ionescu¹,

Pausader²,

Wang³

et al. 2022

Preprint

View full text Add to dashboard Cite

We prove asymptotic stability of the Poisson homogeneous equilibrium among solutions of the Vlassov-Poisson system in the Euclidean space R 3 . More precisely, we show that small, smooth, and localized perturbations of the Poisson equilibrium lead to global solutions of the Vlasov-Poisson system, which scatter to linear solutions at a polynomial rate as t → ∞.The Euclidean problem we consider here differs significantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a "Penrose condition". As a result, our system contains resonances (small divisors) and the electric field is a superposition of an electrostatic component and a larger oscillatory component, both with polynomially decaying rates. Contents 1. Introduction 1 2. Dynamics of the density and bootstrap setup 7 3. Preliminary estimates 12 4. The contributions of the initial data 21 5. Bounds on the static terms and the type-I reaction term 24 6. Bounds on the type-II reaction term 40 7.

show abstract

“…For the Vlasov-HMF a similar result can be proved under Sobolev regularity in the homogeneous case, see [22]. The question of Landau damping in Sobolev regularity for the Vlasov-Poisson system has been recently addressed, for instance in [7,36], where Landau damping results are proved in a weakly collisional regime, or in [9,27,28,10] in the case of unconfined systems.…”

Section: Introductionmentioning

confidence: 89%

On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model

Faou¹,

Horsin²,

Rousset³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion (Penrose criterion). We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear Landau damping effect with an algebraic rate of damping.

show abstract

On the Linearized Vlasov–Poisson System on the Whole Space Around Stable Homogeneous Equilibria

Abstract: We study the linearized Vlasov-Poisson system around suitably stable homogeneous equilibria on R d × R d (for any d ≥ 1) and establish dispersive L ∞ decay estimates in the physical space.

Cited by 20 publications

References 16 publications

Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

Nonlinear Landau damping for the Vlasov-Poisson system in $\R^3$: the Poisson equilibrium

On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model

Contact Info

Product

Resources

About