Modified scattering of small data solutions to the Vlasov-Poisson system with a trapping potential

TL;DR

This paper proves global existence and modified scattering for small data solutions to the Vlasov-Poisson system with a trapping potential, using hyperbolic flow properties without modified vector fields.

Contribution

It provides a new proof of global existence and establishes small data modified scattering for the Vlasov-Poisson system with a trapping potential, avoiding modified vector field techniques.

Findings

Global existence for small data solutions in 2D with trapping potential.

Distribution function converges to a modified state along characteristics.

Weak convergence to a Dirac mass on the unstable manifold.

Abstract

In this paper, we study small data solutions to the Vlasov-Poisson system with the simplest external potential, for which unstable trapping holds for the associated Hamiltonian flow. First, we provide a new proof of global existence for small data solutions to the Vlasov-Poisson system with the trapping potential $\frac{-|x|^2}{2}$ in dimension two. We exploit the uniform hyperbolicity of the Hamiltonian flow, by making use of the commuting vector fields contained in the stable and unstable invariant distributions of phase space for the linearized system. In contrast with the proof in \cite{VV23}, we do not use modified vector field techniques. Moreover, we obtain small data modified scattering for this non-linear system. We show that the distribution function converges to a new regular distribution function along modifications to the characteristics of the linearized problem. We define the linearly growing corrections to the characteristic system, in terms of a precise effective asymptotic force field. We make use of the scattering state to obtain the late-time asymptotic behavior of the spatial density. Finally, we prove that the distribution function (up to normalization) converges weakly to a Dirac mass on the unstable manifold of the origin.