Inverse modified scattering and polyhomogeneous expansions for the Vlasov--Poisson system

TL;DR

This paper presents a new proof for the well-posedness of the inverse modified scattering problem in the Vlasov--Poisson system, demonstrating solutions that disperse and scatter according to a given profile, with explicit polyhomogeneous expansions.

Contribution

It introduces a novel proof technique that does not rely on full ellipticity, providing explicit coefficients for polyhomogeneous expansions of solutions.

Findings

Solutions exist for every suitable scattering profile.

Solutions admit explicit polyhomogeneous expansions.

The proof does not rely on full ellipticity of the Poisson equation.

Abstract

We give a new proof of well posedness of the inverse modified scattering problem for the Vlasov--Poisson system: for every suitable scattering profile there exists a solution of Vlasov--Poisson which disperses and scatters, in a modified sense, to this profile. Further, as a consequence of the proof, the solutions are shown to admit a polyhomogeneous expansion, to any finite but arbitrarily high order, with coefficients given explicitly in terms of the scattering profile. The proof does not exploit the full ellipticity of the Poisson equation.