Propagation Estimates for the Boson Star Equation

TL;DR

This paper establishes that solutions to the boson star equation with general two-body interactions do not propagate faster than light, providing sharp estimates and analyzing asymptotic behavior under certain conditions.

Contribution

It introduces new propagation and phase-space estimates for the boson star equation with broad classes of interaction potentials, including short-range cases.

Findings

Solutions cannot propagate faster than the speed of light, up to exponentially small errors.

Asymptotic phase-space propagation estimates are derived for small, regular initial data.

Minimal velocity estimates depend on the scattering state's momentum.

Abstract

We consider the boson star equation with a general two-body interaction potential $w$ and initial data $\psi_0$ in a Sobolev space. Under general assumptions on $w$, namely that $w$ decomposes as a sum of a finite, signed measure and an essentially bounded function, we prove that the (local in time) solution cannot propagate faster than the speed of light, up to a sharp exponentially small remainder term. If $w$ is short-range and $\psi_0$ is regular and small enough, we prove in addition asymptotic phase-space propagation estimates and minimal velocity estimates for the (global in time) solution, depending on the momentum of the scattering state associated to $\psi_0$.