The Linear Theory of Landau Damping

Maslov, V. P.; Fedoryuk, M. V.

doi:10.1070/sm1986v055n02abeh003013

Cited by 30 publications

(19 citation statements)

References 13 publications

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“…1. A number of other works regarding the linearized Vlasov equations followed, providing mathematically rigorous treatments, clarifications, and generalizations [1,10,16,17,29,32,38]. The phenomenon is now known as Landau damping and is a cornerstone of plasma physics in approximately collisionless regimes; see, e.g., [7,34,37].…”

Section: Landau Damping and Existing Resultsmentioning

confidence: 99%

Landau Damping in Finite Regularity for Unconfined Systems with Screened Interactions

Bedrossian

Masmoudi

Mouhot

2017

Comm Pure Appl Math

112

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We prove Landau damping for the collisionless Vlasov equation with a class of L1 interaction potentials (including the physical case of screened Coulomb interactions) on ℝx3×ℝv3 for localized disturbances of an infinite, homogeneous background. Unlike the confined case double-struckTx3×ℝv3, results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from 0, the Landau damping of the density is similar to the confined case. The finite regularity is possible due to an additional dispersive mechanism available on ℝx3 that reduces the strength of the plasma echo resonance.© 2017 Wiley Periodicals, Inc.

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Section: Landau Damping and Existing Resultsmentioning

confidence: 99%

Landau Damping in Finite Regularity for Unconfined Systems with Screened Interactions

Bedrossian

Masmoudi

Mouhot

2017

Comm Pure Appl Math

112

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“…The mathematically rigorous theory of the linear damping was pioneered by Backus [4] and Penrose [73] as discussed above, and further clarified by many mathematicians, see e.g. [23,59]. Penrose seemed optimistic that Landau damping should occur in the nonlinear equations, and highlighted the fact that near equilibrium, the linear evolution would cause the nonlinear electric field to decay and that the nonlinear equations would be asymptotically linear.…”

Section: V N )mentioning

confidence: 99%

Landau Damping: Paraproducts and Gevrey Regularity

2016

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We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on T d in Gevrey− 1 s regularity (s > 1/3) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani [67]. Our proof combines in a novel way ideas from the original proof of Landau damping Mouhot and Villani [67] and the proof of inviscid damping in 2D Euler Bedrossian and Masmoudi [10]. As in Bedrossian and Masmoudi [10], we use paraproduct decompositions and controlled regularity loss along time to replace the Newton iteration scheme of Mouhot and Villani [67]. We perform time-response estimates adapted from Mouhot and Villani [67] to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work Bedrossian and Masmoudi [10]. We believe the work is an important step forward in developing a systematic theory of phase mixing in infinite dimensional Hamiltonian systems.

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“…[23][24][25][26][27][28][29][30] The present study is concerned with the behavior of solutions at intermediate times that are neither asymptotically large ͑t → ϱ͒ nor asymptotically small ͑t → 0 + ͒ and, in particular, on the range of timescales from roughly one plasma period ͑the inverse of the electron plasma frequency͒ to hundreds or even thousands of plasma periods. [23][24][25][26][27][28][29][30] The present study is concerned with the behavior of solutions at intermediate times that are neither asymptotically large ͑t → ϱ͒ nor asymptotically small ͑t → 0 + ͒ and, in particular, on the range of timescales from roughly one plasma period ͑the inverse of the electron plasma frequency͒ to hundreds or even thousands of plasma periods.…”

Section: Introductionmentioning

confidence: 99%

Transient growth in stable linearized Vlasov–Maxwell plasmas

Podesta

2010

Physics of Plasmas

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Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, ⍀ i t ӷ 1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov-Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov-Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov-Maxwell equations, namely, the one dimensional Vlasov-Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2 / pe , thus demonstrating the existence of transient growth of solutions for the linearized Vlasov-Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov-Maxwell system as well as for Landau fluid models.

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The Linear Theory of Landau Damping

Cited by 30 publications

References 13 publications

Landau Damping in Finite Regularity for Unconfined Systems with Screened Interactions

Landau Damping in Finite Regularity for Unconfined Systems with Screened Interactions

Landau Damping: Paraproducts and Gevrey Regularity

Transient growth in stable linearized Vlasov–Maxwell plasmas

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