TY - JOUR

T1 - Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

AU - Keel, Markus

AU - Roy, Tristan

AU - Tao, Terence

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/7

Y1 - 2011/7

N2 - 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.

AB - 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.

KW - Coulomb gauge

KW - Global well-posedness

KW - I-method

KW - Maxwell-Klein-Gordon equation

KW - X spaces

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U2 - 10.3934/dcds.2011.30.573

DO - 10.3934/dcds.2011.30.573

M3 - Article

AN - SCOPUS:79954479158

VL - 30

SP - 573

EP - 621

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 3

ER -