Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge

TL;DR

This paper proves local well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge at very low regularity levels, introducing new techniques to handle the elliptic component due to low regularity.

Contribution

It establishes well-posedness for low regularity initial data and develops a novel approach for the elliptic variable $A_0$ in this context.

Findings

Well-posedness for $s=1/4+\epsilon$ for gauge potentials

Well-posedness for $s=5/8+\epsilon$ for the Klein-Gordon field

Introduction of a new method to handle the elliptic variable $A_0$

Abstract

We consider the Maxwell-Klein-Gordon equation in 2D in the Coulomb gauge. We establish local well-posedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution $\phi$ of the gauged Klein-Gordon equation. The main tool for handling the wave equations is the product estimate established by D'Ancona, Foschi, and Selberg. Due to low regularity, we are unable to use the conventional approaches to handle the elliptic variable $A_0$, so we provide a new approach.