Mathematics – Analysis of PDEs
Scientific paper
2009-10-09
Mathematics
Analysis of PDEs
56 pages, no figures; to appear, DCDS-A. This is the final version, incorporating the referee comments
Scientific paper
We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of Klainerman-Machedon \cite{kl-mac:mkg} on finite energy data $s \geq 1$, and Eardley-Moncrief \cite{eardley} 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 $H^s_x$ rather than $H^1_x$. 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 $L^2_x$ 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.
Keel Markus
Roy Tristan
Tao Terence
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