Annals of PDE

, 4:3 | Cite as

Solutions to the Einstein-Scalar-Field System in Spherical Symmetry with Large Bounded Variation Norms

  • Jonathan Luk
  • Sung-Jin OhEmail author
  • Shiwu Yang


It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are forward-in-time global (in the sense of future causal geodesic completeness). We construct a class of spherically symmetric solutions which are global but the initial norms are consistent with initial data not decaying towards infinity. This gives the following consequences:
  1. 1.

    We prove that there exist forward-in-time global solutions with arbitrarily large (and in fact infinite) initial bounded variation (BV) norms and initial Bondi masses.

  2. 2.

    While general solutions with non-decaying data do not approach Minkowski spacetime, we show using the results of Luk and Oh (Anal PDE 8(7):1603–1674, 2014. arXiv:1402.2984) that if a sufficiently strong asymptotic flatness condition is imposed on the initial data, then the solutions we construct (with large BV norms) approach Minkowski spacetime with a sharp inverse polynomial rate.

  3. 3.

    Our construction can be easily extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future.

Finally, we discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical norms.



S.-J. Oh was a Miller Research Fellow, and thanks the Miller Institute for support.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Stanford UniversityPalo AltoUSA
  2. 2.KIASSeoulKorea
  3. 3.BICMRPeking UniversityBeijingChina

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