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

M. Keel; Tristan Roy; Terence Tao

doi:10.3934/dcds.2011.30.573

Article Contents

2011, Volume 30, Issue 3: 573-621. Doi: 10.3934/dcds.2011.30.573 open access

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

1.
Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455
2.
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555
3.
Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095

Received: May 2010

Revised: December 2010

Published: September 2011

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: 35J10, 42B25.

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