Low-regularity global well-posedness for the Klein-Gordon-Schr\"odinger system on $\mathbb R^{+}$

TL;DR

This paper proves local and global well-posedness for the Klein-Gordon-Schrödinger system on the half line with rough initial data, extending sharp results from the full line case.

Contribution

It establishes almost optimal well-posedness and regularity results for the system on the half line, including for initial data in negative Sobolev spaces.

Findings

Local-in-time well-posedness for rough initial data

Global well-posedness using conservation laws and iterative methods

Results are consistent with sharp full-line case results

Abstract

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schr\"odinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the $L^2$ conservation law of the Schr\"odinger part with a careful iteration of the rough wave part in lower order Sobolev norms.