The Boson star equation with initial data of low regularity

TL;DR

This paper investigates the well-posedness of the boson star equation with low regularity initial data in three dimensions, establishing new results for both general and radial data, and demonstrating near-optimality through ill-posedness results.

Contribution

It proves local well-posedness for the boson star equation in low regularity Sobolev spaces, including the radial case, and shows these results are nearly optimal.

Findings

Well-posedness for s > 1/4 in H^s

Radial data well-posedness for s > 0

Ill-posedness results indicating near-optimal regularity thresholds

Abstract

The Cauchy problem for the L^2-critical boson star equation with initial data of low regularity in spatial dimension d=3 is studied. Local well-posedness in H^s for s > 1/4 is proved. Moreover, for radial initial data, local well-posedness is established in H^s for s > 0. Both results are shown to be almost optimal by providing complementary ill-posedness results.