On the asymptotic behavior of solutions to the Vlasov-Poisson system

Ionescu, Alexandru D.; Pausader, Benoît; Wang, Xuecheng; Widmayer, Klaus

doi:10.48550/arxiv.2005.03617

Cited by 8 publications

(27 citation statements)

References 18 publications

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“…We note that due to the faster decay rate of the electric field for d ≥ 4 in comparison with d = 3, modifications to the trajectories along which the distribution function scatters are not needed, which differs from the results of [18,23].…”

Section: Introductionmentioning

confidence: 70%

“…Partial results for the Cauchy problem are known in some situations, Date: January 25, 2022. The author was supported in part by NSF grants DMS-1911145 and DMS-2107938. including small data [1,16,18,30], monocharged and spherically-symmetric data [3,15,22], and lowerdimensional (d = 1, 2) settings [2,4,10,11,12,29]. These results all provide either time asymptotic growth estimates of characteristics or decay estimates of the electric field or charge density.…”

Section: Introductionmentioning

confidence: 94%

“…Unlike [23] we do not assume compactly-supported initial data, and instead propagate translated spatial moments of the distribution function in time. In addition to the spatial decay, the regularity assumptions on initial data could possibly be altered to arrive at similar convergence results in weaker topologies (see [18] for d = 3). Still, we will require only C 1 initial data rather than higher derivatives in L 1 or L 2 (as for vector field and harmonic methods) and can address all dimensions d ≥ 4 simultaneously.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.3. Theorems 1.1 and 1.2 can be extended to d = 3, but require a stronger decay assumption than (A) and a modification to the trajectories along which f scatters (see [18,23]). Furthermore, the results of [23] can be extended to d ≥ 4, but require…”

Section: Introductionmentioning

confidence: 99%

“…The particular novelty of these solutions is that, unlike previous small data solutions [1,18,28] for d = 3 and [30] for d ≥ 4, they allow f 0 ∞ to be arbitrarily large and further do not require a smallness assumption on derivatives of initial data.…”

Section: Introductionmentioning

confidence: 99%

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Scattering and Asymptotic Behavior of Solutions to the Vlasov-Poisson System in High Dimension

Pankavich¹

2022

Preprint

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We consider the repulsive Vlasov-Poisson system in dimension d ≥ 4. A sufficient condition on the decay rate of the associated electric field is presented that guarantees the scattering and determination of the complete asymptotic behavior of large data solutions as t → ∞. More specifically, we show that under this condition the spatial average of the particle distribution function converges, and we establish the precise asymptotic profiles of the electric field and macroscopic densities. An L ∞ scattering result for the particle distribution function along the associated trajectories of free transport is also proved. Finally, we construct small data solutions that display this asymptotic behavior. These solutions do not require smallness of f 0 ∞ or derivatives, as only a condition on integrated moments of the distribution function is imposed.

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Scattering and Asymptotic Behavior of Solutions to the Vlasov-Poisson System in High Dimension

Pankavich¹

2022

Preprint

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Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities

Lindblad,

Luhrmann,

Soffer

2020

Preprint

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We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. Our results and techniques are motivated by applications in the study of the asymptotic stability of kink solutions in classical nonlinear scalar field equations on the real line. Contents 1. Introduction 1 1.1. Motivation 2 1.2. Main results 3 1.3. Related works 5 1.4. Proof ideas 5 2. Preliminaries 8 2.1. Notation and conventions 8 2.2. Decay estimates 9 2.3. Harmonic analysis tools 10 3. Resonant Case 11 3.1. Local decay estimates 11 3.2. Asymptotics at the origin 15 3.3. Proof of Theorem 1.1 18 4. Non-Resonant Case 23 4.1. Normal form transformation 23 4.2. Energy estimates 25 4.3. L ∞ ξ control of the profile 32 4.4. Proof of Theorem 1.6 39 References 40

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

Bedrossian¹,

Masmoudi²,

Mouhot³

2020

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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.

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On the asymptotic behavior of solutions to the Vlasov-Poisson system

Abstract: We prove small data modified scattering for the Vlasov-Poisson system in dimension d = 3 using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamic related to the scattering mass.

Cited by 8 publications

References 18 publications

Scattering and Asymptotic Behavior of Solutions to the Vlasov-Poisson System in High Dimension

Scattering and Asymptotic Behavior of Solutions to the Vlasov-Poisson System in High Dimension

Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities

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

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