The Maxwell equation is an intermediate linear model for
Einstein's equation lying between the scalar wave equation and the
linearised Einstein equation. This talk will present the 5 key
estimates necessary to prove a uniform bound on an energy and a
Morawetz integrated local energy decay estimate for the nonradiating
part.
The major obstacles, relative to the scalar wave equation are: that a
scalar equation must be found for at least one of the components,
since there is no known decay estimate directly at the tensor level;
that the scalar equation has a complex potential; and that there are
stationary solutions and, in the nonzero $a$ Kerr case, it is more
difficult to project away from these stationary solutions.
If time permits, some discussion of a geometric proof using the hidden
symmetries will be given.
This is joint work with L. Andersson and is arXiv:1310.2664.