Local Well-Posedness of the (4 + 1)-Dimensional Maxwell–Klein–Gordon Equation at Energy Regularity

Abstract

This paper is the first part of a trilogy [22, 23] 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.