Decay for the Maxwell field outside a slowly rotating Kerr black hole

14 January 2014
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.
  • Partial Differential Equations Seminar