Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge

TL;DR

This paper proves the global well-posedness of the Maxwell-Klein-Gordon system with finite-energy initial data in Lorenz gauge by uncovering a null structure similar to that in Coulomb gauge, leveraging Lorentz invariance.

Contribution

It demonstrates the presence of a null structure in Lorenz gauge for M-K-G, enabling finite-energy global well-posedness proof, previously known only in Coulomb gauge.

Findings

Null structure exists in Lorenz gauge for M-K-G.

Finite-energy global well-posedness established in Lorenz gauge.

Lorenz gauge's Lorentz invariance offers advantages over Coulomb gauge.

Abstract

It is known that the Maxwell-Klein-Gordon system (M-K-G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M-K-G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M-K-G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also a nonlinear elliptic equation.