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Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation
This article constitutes the final and main part of a three-paper sequence,
whose goal is to prove global well-posedness and scattering of the energy
critical Maxwell-Klein-Gordon equation (MKG) on for
arbitrary finite energy initial data. Using the successively stronger
continuation/scattering criteria established in the previous two papers, 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.Comment: 64 page
Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity
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.Comment: 59 page
Energy dispersed solutions for the (4+1)-dimensional Maxwell-Klein-Gordon equation
This article is devoted to the mass-less energy critical Maxwell-Klein-Gordon
system in 4+1 dimensions. In earlier work of the second author, joint with
Krieger and Sterbenz, we have proved that this problem has global
well-posedness and scattering in the Coulomb gauge for small initial data. This
article is the second of a sequence of three papers of the authors, whose goal
is to show that the same result holds for data with arbitrarily large energy.
Our aim here is to show that large data solutions persist for as long as one
has small energy dispersion; hence failure of global well-posedness must be
accompanied with a non-trivial energy dispersion.Comment: 63 page
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