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

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## Abstract

*not*decaying towards infinity. This gives the following consequences:

- 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.
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.
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.

## Notes

### Acknowledgements

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

## References

- 1.Beceanu, M., Soffer, A.: Large Outgoing Solutions to Supercritical Wave Equations, preprint , arXiV:1601.06335 (2016)
- 2.Christodoulou, D.: The problem of a self-gravitating scalar field. Comm. Math. Phys.
**105**(3), 337–361 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 3.Christodoulou, D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Comm. Pure Appl. Math.
**44**(3), 339–373 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Comm. Pure Appl. Math.
**46**(8), 1131–1220 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Christodoulou, D.: The Formation of Black Holes in General Relativity. Monographs in Mathematics. European Mathematical Soc, Oxford (2009)CrossRefzbMATHGoogle Scholar
- 6.Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series 41 (1993)Google Scholar
- 7.Krieger, J., Schlag, W.: Large Global Solutions for Energy Supercritical Nonlinear Wave Equations on \({\mathbb{R}}^{3+1}\), preprint , arXiV:1403.2913 (2014)
- 8.Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. (2)
**171**(3), 1401–1477 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Luk, J., Oh, S.-J.: Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry. Anal. PDE
**8**(7), 1603–1674 (2014). arXiv:1402.2984 MathSciNetCrossRefzbMATHGoogle Scholar - 10.Luk, J., Oh, S.-J.: Global Nonlinear Stability of Large Dispersive Solutions to the Einstein Equations (
**in preparation**) (2016)Google Scholar - 11.Wang, J., Yu, P.: A Large Data Regime for Non-Linear Wave Equations, preprint , arXiv:1210.2056 (2012)
- 12.Yang, S.: Global solutions of nonlinear wave equations with large data. Sel. Math. (
**online first**) (2014)Google Scholar