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Rough solutions of Einstein vacuum equations in CMCSH gauge. (English) Zbl 1294.83009

Summary: In this paper, we consider very rough solutions to the Cauchy problem for the Einstein vacuum equations in CMC spatial harmonic gauge, and obtain the local well-posedness result in \(H^s\), \(s > 2\). The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric \(\mathbf g\), we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation \(\square_{\mathbf g} \phi = 0\) directly.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C15 Exact solutions to problems in general relativity and gravitational theory
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
35L05 Wave equation
53Z05 Applications of differential geometry to physics
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