, Volume 251, Issue 2, pp 377-426

Global Regularity for the Maxwell-Klein-Gordon Equation with Small Critical Sobolev Norm in High Dimensions

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Abstract.

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 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.

Communicated by P. Constantin
Acknowledgements The authors are deeply indebted to Sergiu Klainerman, without whose encouragement and insight this project would not even have been initiated. We have tremendously benefited from numerous discussions with him, in particular on the issue of the Cronstrom gauge. We would also like to thank Joachim Krieger and the anonymous referee for their helpful comments and suggestions.
I.R. is a Clay Prize Fellow and supported in part by the NSF grant DMS-01007791
T.T. is a Clay Prize Fellow and supported in part by a grant from the Packard Foundation