Abstract
We review some recent results on the scattering structure of the linearized Vlasov-Poisson equation in d=1 space dimension. It started with [9] where the linearized Vlasov-Poisson equation is rewritten as a linear Vlasov-Ampère set of equations which makes the L 2 structure more visible. A consequence is that the linear Landau damping becomes an application of the scattering theory for Hamiltonian systems. Then we review the extension, firstly of the linearization around non homogeneous profiles which is treated with the theory of trace-class operators, secondly of the case with a forcing magnetic field which has the ability to eliminate the possibility of a linear Landau damping effect. Finally, we evoke some possibility for extension to space dimension x∈ℝ d with d>1.
Highlights
The presentation starts with a non linear Vlasov-Poisson model problem for negatively charged particules in a constant magnetic field B0 and in a bath of ions with non homogeneous density ρions∂tf + v · ∇xf − (E(t, x) + v × B0) · ∇vf = 0, f (t, x, v) ≥ 0, −E(∆t,φx=) =ρi−on∇s(xxφ) (−t, x)f.dv, (1) The long time dynamics of the linearized version of this system and of related linear systems presents many similarities with abstract scattering theory, some references in this direction are [1, 7, 12]
We review some recent results on the scattering structure of the linearized Vlasov-Poisson equation in d = 1 space dimension
We review the extension, firstly of the linearization around non homogeneous profiles which is treated with the theory of trace-class operators, secondly of the case with a forcing magnetic field which has the ability to eliminate the possibility of a linear Landau damping effect
Summary
Among the tools of abstract scattering theory, one can distinguish the LipmannSchwinger equation which calculates the generalized eigenvectors of H in function of those of H0, the construction of wave-operators W ±(H, H0) which establishes an isometric equivalence between the absolutely continuous part of the spectrum of H and the one of H0 and the trace-class method which justifies the existence of the wave operators provided the perturbation K satisfies the trace-class criterion. Going back to the physical problem attached to (9), square integrable initial data in the absolutely continuous part of the spectrum H converge weakly to zero in quadratic norm and this is essentially a generic proof of the linear Landau damping (the electric field tends to zero in time).
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