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Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions | | Igor Rodnianski
; Terence Tao
; | | Date: |
22 Sep 2003 | | Subject: | Analysis of PDEs MSC-class: 35Q60 | math.AP | | Abstract: | We show that in dimensions $n geq 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm $dot H^{n/2-1} imes dot H^{n/2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation 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. | | Source: | arXiv, math.AP/0309353 | | Services: | Forum | Review | PDF | Favorites | |
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