Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

TL;DR

This paper proves global well-posedness of the 3D Maxwell-Klein-Gordon equations in a low regularity Sobolev space below the energy norm using the I-method, extending previous results to rougher initial data.

Contribution

It introduces an adaptation of the I-method to the Maxwell-Klein-Gordon equations at regularity below the energy norm, overcoming low-frequency control challenges.

Findings

Established global well-posedness for s > 0.866 in H^s_x

Developed almost conservation law approach for low regularity

Identified difficulties due to lack of smoothing in the equations

Abstract

We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of Klainerman-Machedon \cite{kl-mac:mkg} on finite energy data $s \geq 1$, and Eardley-Moncrief \cite{eardley} for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the $L^2_x$ norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.