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Quantitative decay
rates for dispersive solutions to the Einstein-scalar field
system in spherical symmetry
Jonathan Luk and Sung-Jin Oh |
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Vol. 8 (2015), No. 7, 1603–1674
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Abstract |
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We study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up. |
Keywords
Einstein-scalar field system, spherical symmetry,
quantitative decay rate
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Mathematical Subject Classification 2010
Primary: 35Q76
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Milestones
Received: 4 May 2014
Revised: 16 April 2015
Accepted: 24 June 2015
Published: 18 September 2015
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Authors |
| Jonathan Luk | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB United Kingdom |
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| Sung-Jin Oh | |
| Department of Mathematics UC Berkeley 970 Evans Hall Berkeley, CA 94720 United States |
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