Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

Abstract

We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in Hsx in the Coulomb gauge for all s > √3/2 ≈ 0.866. This extends previous work of Klainerman-Machedon [24] on finite energy data s ≥ 1, and Eardley-Moncrief [11] for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of Hsx rather than H1x. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the L2x norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.