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Linearized wave-damping structure of Vlasov-Poisson in \(\mathbb{R}^3\). (English) Zbl 1497.35465

Summary: In this paper we study the linearized Vlasov-Poisson equation for localized disturbances of an infinite, homogeneous Maxwellian background distribution in \(\mathbb{R}^3_x \times \mathbb{R}^3_v\). In contrast to the confined case \(\mathbb{T}^d_x \times \mathbb{R}_v^d\), or the unconfined case \(\mathbb{R}^d_x \times \mathbb{R}^d_v\) with screening, the dynamics of the disturbance are not scattering towards free transport as \(t \to \pm \infty\): 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.

MSC:

35Q83 Vlasov equations
35Q60 PDEs in connection with optics and electromagnetic theory
35Q49 Transport equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
78A25 Electromagnetic theory (general)
45D05 Volterra integral equations
44A10 Laplace transform
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

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