TY - JOUR
T1 - Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions
AU - Rodnianski, Igor
AU - Tao, Terence
N1 - Copyright:
Copyright 2005 Elsevier B.V., All rights reserved.
PY - 2004/11
Y1 - 2004/11
N2 - We show that in dimensions n ≥ 6 one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm Ḣn/2-1 × Ḣn/2-2 of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [17, 7, 14, 12] but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.
AB - We show that in dimensions n ≥ 6 one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm Ḣn/2-1 × Ḣn/2-2 of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [17, 7, 14, 12] but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.
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U2 - 10.1007/s00220-004-1152-1
DO - 10.1007/s00220-004-1152-1
M3 - Article
AN - SCOPUS:11144275780
SN - 0010-3616
VL - 251
SP - 377
EP - 426
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 2
ER -