Rodnianski, Igor; Tao, Terence Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions. (English) Zbl 1106.35073 Commun. Math. Phys. 251, No. 2, 377-426 (2004). Summary: We show that in dimensions \(n\geq 6\) one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm \(\dot H^{n/2-1}\times\dot H^{n/2-2}\) of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [T. Tao, Int. Math. Res. Not. 2001, No. 6, 299–328 (2001; Zbl 0983.35080), S. Klainerman and I. Rodnianski, Int. Math. Res. Not. 2001, No. 13, 656–677 (2001), J. Shatah and M. Struwe, Int. Math. Res. Not. 2002, No. 11, 555–571 (2002; Zbl 1024.58014), A. Nahmod, A. Stefanov and K. Uhlenbeck, Commun. Anal. Geom. 11, No. 1, 49-83 (2003; Zbl 1085.58022)] but unlike the wave map equation, the Coulomb gauge nonlinearity cannot be iterated away directly. We use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn is achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge. Cited in 2 ReviewsCited in 29 Documents MSC: 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81T13 Yang-Mills and other gauge theories in quantum field theory 35L15 Initial value problems for second-order hyperbolic equations Keywords:Strichartz estimates; covariant wave equation; Littlewood-Paley multipliers; parametrix Citations:Zbl 0983.35080; Zbl 1024.58014; Zbl 1085.58022 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bahouri, H., Chemin, J-Y.: Équations d?ondes quasilinéaires et les inegalites de Strichartz. Am. J. Math. 121, 1337-1377 (1999) · Zbl 0952.35073 · doi:10.1353/ajm.1999.0038 [2] Cuccagna, S.: On the local existence for the Maxwell-Klein-Gordon system in R3+1. Comm. Part. Diff. 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