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Local well-posedness of the \((4 + 1)\)-dimensional Maxwell-Klein-Gordon equation at energy regularity. (English) Zbl 1402.35273
Summary: This paper is the first part of a trilogy [the authors, Am. J. Math. 140, No. 1, 1–82 (2018; Zbl 1392.35309); Invent. Math. 205, No. 3, 781–877 (2016; Zbl 1364.35198)] 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.

35Q61 Maxwell equations
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