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 $\mathbb{R}^{1+4}$ 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

A heat flow approach to Onsager's conjecture for the Euler equations on manifolds

We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to $\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, our approach yields an alternative proof of the sharp result of the latter authors. Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.Comment: 15 pages. Improved exposition, corrected typos. Added a criterion for energy conservation in terms of the H\"older norm in Theorem 1.