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

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 [F. Pasqualotto, The spin \(\pm 1\) Teukolsky equations and the Maxwell system on Schwarzschild, Preprint 2016, arXiv:1612.07244] 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 setup a bootstrap argument. We use the \(r^p\) method of Dafermos and Rodnianski in [M. Dafermos and I. 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, Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arXiv:0910.4957] 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.