Abstract

The wave equation, Maxwell’s equations, and the linearized Einstein’s equations have been intensively studied in order to understand the stability properties of “black hole” solutions to Einstein’s equations. Results for Maxwell’s equations on a black hole background were first obtained by Blue (2007), with further recent developments by Andersson, Baeckdahl and Blue (2015). In this talk I will present a different approach to proving decay properties of the Maxwell field on the Schwarzschild spacetime. The key ideas are: 1) find a quantity which satisfies a Regge-Wheeler Equation, admitting good estimates. Motivation for this approach lies in forthcoming work by Dafermos, Holzegel and Rodnianski on the linearized Einstein equations. 2) From such quantity, show that all the components of the field may be estimated. A key role is played by the r^p estimates of Dafermos and Rodnianski, i.e. energy estimates with a hierarchy of weights in the r variable. I will finally point out how this approach is suitable for application to a nonlinear problem involving the Maxwell-Born-Infeld equations.