Abstract
We prove local existence of solutions to the Einstein–null dust system under polarized \({\mathbb{U}}\)(1) symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a reductive structure in the system of equations.
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References
Andersson L., Moncrief V.: Elliptic-hyperbolic systems and the Einstein equations. Ann. Henri Poincare 4, 1–34 (2003)
Choquet-Bruhat Y.: General Relativity and the Einstein Equations, Oxford Mathematical Monographs. Oxford University Press, Oxford (2009)
Choquet-Bruhat Y., Friedrich H.: Motion of isolated bodies. Class. Quantum Gravity 23(20), 5941–5949 (2006)
Huneau C.: Constraint equations for 3+1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case. Ann. Henri Poincaré 17(2), 271–299 (2016)
Huneau, C., Luk, J.: High-frequency backreaction for the Einstein equations under polarized \({\mathbb{U}}\)(1) symmetry (2017). arXiv:1706.09501
McOwen R.C.: The behavior of the Laplacian on weighted Sobolev spaces. Commun. Pure Appl. Math. 32(6), 783–795 (1979)
Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society, Providence (2006)
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Communicated by P. Chrusciel
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Huneau, C., Luk, J. Einstein Equations Under Polarized \({\mathbb{U}}\)(1) Symmetry in an Elliptic Gauge. Commun. Math. Phys. 361, 873–949 (2018). https://doi.org/10.1007/s00220-018-3167-z
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DOI: https://doi.org/10.1007/s00220-018-3167-z