The linear Vlasov-Poisson-Ampère equation from the viewpoint of abstract scattering theory

Després, Bruno

Séminaire Laurent Schwartz — EDP et applications (2019-2020), Exposé no. 1, 12 p.

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 $\mathbf{x}\in {ℝ}^d$ with $d>1$.

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