Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

Abstract

Abstract. This article constitutes the final and main part of a three-paper sequence [24, 25], whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on R 1+4 for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers [24, 25], we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

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

“…The present paper is the first of a sequence of three papers [22,23], in which we give a complete proof of global well-posedness and scattering of (MKG) on R 1+4 for any finite energy data. This theorem is analogous to the threshold theorem for energy critical wave maps [18,29,30,[33][34][35][36][37].…”

“…Taking the contrapositive, we see that any finite time blow up of a solution to (MKG) must be accompanied by energy concentration at a point. In [22,23], following the scheme successfully developed by one of the authors (D. Tataru) and J. Sterbenz in the context of energy critical wave maps [29,30], we establish global well-posedness of (MKG) for finite energy data by showing that such a phenomenon cannot occur. We refer to the last and the main paper of the sequence [23] for an overview of the entire series.…”

“…We remark that we do not lose any generality by restricting to initial data sets in the global Coulomb gauge, as any finite energy initial data sets can be gauge transformed into this gauge; see Section 3. We formulate our local well-posedness theorem specifically in the global Coulomb gauge in view of the rest of the series [22,23], where we show global well-posedness and scattering in this gauge. An important feature of Theorem 1.1 is that it provides a lower bound on the lifespan in terms of the energy concentration scale r c of the data.…”

“…These techniques (in addition to Theorem 1.1 itself) are also crucially used in the last paper of the sequence [23], where we carry out a blow-up analysis of (MKG) to preclude concentration of energy and non-scattering.…”

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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-KleinGordon 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.

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