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Annals of PDE

, 5:19 | Cite as

Nonlinear Stability for the Maxwell–Born–Infeld System on a Schwarzschild Background

  • Federico PasqualottoEmail author
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Abstract

In this paper we prove small data global existence for solutions to the Maxwell–Born–Infeld (MBI) system on a fixed Schwarzschild background. This system has appeared in the context of string theory and can be seen as a nonlinear model problem for the stability of the background metric itself, due to its tensorial and quasilinear nature. The MBI system models nonlinear electromagnetism and does not display birefringence. The key element in our proof lies in the observation that there exists a first-order differential transformation which brings solutions of the spin \(\pm 1\) Teukolsky Equations, satisfied by the extreme components of the field, into solutions of a “good” equation (the Fackerell–Ipser Equation). This strategy was established in Pasqualotto (Ann Henri Poincaré 20(4):1263–1323, 2019) for the linear Maxwell field on Schwarzschild. We show that analogous Fackerell–Ipser equations hold for the MBI system on a fixed Schwarzschild background, which are however nonlinearly coupled. To essentially decouple these right hand sides, we set up a bootstrap argument. We use the \(r^p\) method of Dafermos and Rodnianski (A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: XVIth international congress on mathematical physics. World Science Publication, Hackensack, pp 421–432, 2010) in order to deduce decay of some null components, and we infer decay for the remaining quantities by integrating the MBI system as transport equations.

Notes

Acknowledgements

I would like to thank Prof. Jonathan Luk for his patience and guidance, for suggesting the problem to me, and for inviting me to Stanford University to finish the project. I would also like to thank my advisor, Prof. Mihalis Dafermos, for his patience, his encouragement and his comments on preliminary versions of the manuscript. Moreover, I thank Jan Sbierski for the suggestion to look at Fritz John’s approach to local existence. Moreover, I thank John Anderson and Yakov Shlapentokh-Rothman for very valuable discussions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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