Nonlinear stability for the Maxwell-Born-Infeld system on a\n Schwarzschild background

2017/06/23 by Federico Pasqualotto, Pasqualotto, Federico · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)

paper · pdf · doi:10.48550/arxiv.1706.07764

openalex publication_date 2017/06/23 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28

Abstract

In this paper we prove small data global existence for solutions to the\nMaxwell-Born-Infeld (MBI) system on a fixed Schwarzschild background. This\nsystem has appeared in the context of string theory and can be seen as a\nnonlinear model problem for the stability of the background metric itself, due\nto its tensorial and quasilinear nature. The MBI system models nonlinear\nelectromagnetism and does not display birefringence. The key element in our\nproof lies in the observation that there exists a first-order differential\ntransformation which brings solutions of the spin \± 1 Teukolsky equations,\nsatisfied by the extreme components of the field, into solutions of a "good"\nequation (the Fackerell-Ipser Equation). This strategy was established in [F.\nPasqualotto, The spin \± 1 Teukolsky equations and the Maxwell system on\nSchwarzschild, Annales Henri Poincar 'e, 20(4):1263-1323, 2019,\narXiv:1612.07244] for the linear Maxwell field on Schwarzschild. We show that\nanalogous Fackerell-Ipser equations hold for the MBI system on a fixed\nSchwarzschild background, which are however nonlinearly coupled. To essentially\ndecouple these right hand sides, we set up a bootstrap argument. We use the\nr^p method of Dafermos and Rodnianski in [M. Dafermos and I. Rodnianski, A\nnew physical-space approach to decay for the wave equation with applications to\nblack hole spacetimes, in XVIth International Congress on Mathematical Physics,\nPavel Exner ed., Prague 2009 pp. 421-433, 2009, arXiv:0910.4957] in order to\ndeduce decay of some null components, and we infer decay for the remaining\nquantities by integrating the MBI system as transport equations.\n

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