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.

UR - http://www.scopus.com/inward/record.url?scp=11144275780&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=11144275780&partnerID=8YFLogxK

U2 - 10.1007/s00220-004-1152-1

DO - 10.1007/s00220-004-1152-1

M3 - Article

AN - SCOPUS:11144275780

VL - 251

SP - 377

EP - 426

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -