Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity

TL;DR

This paper establishes local well-posedness for the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity, forming a foundational step towards proving global well-posedness and scattering for finite energy data.

Contribution

It introduces a large energy local well-posedness theorem in the Coulomb gauge and develops novel initial data excision and gluing techniques at critical regularity.

Findings

Lifespan bounded by energy concentration scale

Reduction of global well-posedness to energy non-concentration

Development of new excision and gluing methods

Abstract

This paper is the first part of a trilogy dedicated to a proof of global well-posedness and scattering of the (4+1)-dimensional mass-less Maxwell-Klein-Gordon equation (MKG) for any finite energy initial data. The main result of the present paper is a large energy local well-posedness theorem for MKG in the global Coulomb gauge, where the lifespan is bounded from below by the energy concentration scale of the data. Hence the proof of global well-posedness is reduced to establishing non-concentration of energy. To deal with non-local features of MKG we develop initial data excision and gluing techniques at critical regularity, which might be of independent interest.