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Abstract
The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the 1D×1D and 2D×2D Vlasov-Poisson system illustrate the effectiveness of this approach.
Highlights
We consider potentials φ = φ(t, x) : R×Td → R and distribution functions f = f (t, x, v) : R × Td × Rd → R satisfying the Vlasov-Poisson system ∂tf + v · ∇xf − ∇xφ · ∇vf =0∆xφ = n(f ) − Rd f dv f (t = 0) = f0. (VP)Here periodic boundary conditions being used, Td is a d dimensional torus: there exist L1, . . . , Ld > 0 such that Td = (R/L1Z) × · · · × (R/LdZ)
In order to prove Propositions 1 and 2, as (VPL) and (VPL2) share the same structure, we focus on a general linearized Vlasov Poisson equation
Since space Fourier transform is injective on regular functions, we have proven that u, g is a solution of Poisson equation (18)
Summary
As in Proposition 1, we can first prove that space Fourier transform of μ is supported by (K + K) ∪ {0}, observe that h is a C∞ function and conclude that (h, μ) is a solution of (VPL) by Proposition 4. In Proposition 1, we have proved that it is enough to solve dispersion relation (10) to get a solution (g, ψ) to linearized Vlasov-Poisson equation (VPL). There exists rk,λ an analytic and bounded function on Σα such that the following expansion defines a solution of the dispersion relation (10). This proposition will be proven at the end of this section. Consider the following Theorem that is very useful to invert Laplace transforms and to control it
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