Global wellposedness of the MaxwellKleinGordon equation below the energy norm
Abstract
We show that the MaxwellKleinGordon equations in three dimensions are globally wellposed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of KlainermanMachedon \cite{klmac:mkg} on finite energy data $s \geq 1$, and EardleyMoncrief \cite{eardley} for still smoother data. We use the method of almost conservation laws, sometimes called the "Imethod", 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.1850
 Bibcode:
 2009arXiv0910.1850K
 Keywords:

 Mathematics  Analysis of PDEs;
 35L10
 EPrint:
 56 pages, no figures