Vol. 8, No. 7, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 9, 2997–3369
Issue 8, 2619–2996
Issue 7, 2247–2618
Issue 6, 1871–2245
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1948-206X (online)
ISSN 2157-5045 (print)
 
Author index
To appear
 
Other MSP journals
Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

Jonathan Luk and Sung-Jin Oh

Vol. 8 (2015), No. 7, 1603–1674
Abstract

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 ϕ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that ϕ 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
Mathematical Subject Classification 2010
Primary: 35Q76
Milestones
Received: 4 May 2014
Revised: 16 April 2015
Accepted: 24 June 2015
Published: 18 September 2015
Authors
Jonathan Luk
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Centre for Mathematical Sciences
Wilberforce Road
Cambridge
CB3 0WB
United Kingdom
Sung-Jin Oh
Department of Mathematics
UC Berkeley
970 Evans Hall
Berkeley, CA 94720
United States