### Math 712. Topics in Partial Differential Equations:Analysis of Black Holes

- Spring 19 -
Hans Lindblad

The lectures are MW 3.00-4.15 in Gillman 77.

The course will cover linear stability of black holes.
First we will study wave equations on the background of first and
Schwarzschild black hole and then a Kerr black hole.
We will prove local energy decay estaimtes and decay estiamtes.
Then we will study aolutions to the full linearization of Einstein's equations around
these black holes.
For the estimates of the wave equation of Schwarzschild black holes we will follow
J.Marzuola, J.Metcalfe, D.Tataru, M.Tohaneanu: Strichartz estimates on Schwarzschild black hole backgrounds
as well as
H. Lindblad, M.Tohaneanu: Global existence for quasilinear wave equations close to Schwarzschild
For the estimates of the wave equation of Schwarzschild black holes we will follow
D. Tataru and M. Tohaneanu: Local energy estimate on Kerr black hole backgrounds
(dowloadable here)i as well as work in preparation by myself and Mihai.

For the linear stability we will follow work by Dafermos, Holzegel adn Rodnianski as well as Johnson that I will post later.

However, we will start with an introduction to Black hole solutions to Einstein's equations from
the textbooks by Wald or Hawking&Ellis.
There are also some
M. Dafermos, I. Rodnianski: Lectures on black holes and linear waves