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

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  • We show that the Maxwell-Klein-Gordon equations in three dimensionsare globally well-posed in $H^s_x$ in the Coulomb gauge for all $s >\sqrt{3}/2 \approx 0.866$. This extends previous work ofKlainerman-Machedon [24] on finite energy data $s \geq1$, and Eardley-Moncrief [11] for still smoother data. Weuse the method of almost conservation laws, sometimes called the"I-method", to construct an almost conserved quantity based on theHamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$.One then uses Strichartz, null form, and commutator estimates tocontrol the development of this quantity. The main technicaldifficulty (compared with other applications of the method of almostconservation laws) is at low frequencies, because of the poorcontrol on the $L^2_x$ norm. In an appendix, we demonstrate theequations' relative lack of smoothing - a property that presentsserious difficulties for studying rough solutions using other knownmethods.
    Mathematics Subject Classification: 35J10, 42B25.


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