Scattering and Asymptotic Behavior of Solutions to the Vlasov-Poisson System in High Dimension

TL;DR

This paper analyzes the long-term behavior of solutions to the high-dimensional Vlasov-Poisson system, establishing conditions for scattering, asymptotic profiles, and constructing small data solutions without smallness constraints on initial data.

Contribution

It provides new sufficient conditions on electric field decay for scattering and asymptotic analysis in high dimensions, including explicit profiles and small data solutions.

Findings

Convergence of spatial average of distribution function

Asymptotic profiles of electric field and densities

Scattering results for particle distribution function

Abstract

We consider the repulsive Vlasov-Poisson system in dimension $d \geq 4$. A sufficient condition on the decay rate of the associated electric field is presented that guarantees the scattering and determination of the complete asymptotic behavior of large data solutions as $t \to \infty$. More specifically, we show that under this condition the spatial average of the particle distribution function converges, and we establish the precise asymptotic profiles of the electric field and macroscopic densities. An $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport is also proved. Finally, we construct small data solutions that display this asymptotic behavior. These solutions do not require smallness of $\|f_0\|_\infty$ or derivatives, as only a condition on integrated moments of the distribution function is imposed.