Dafermos and Rodnianski introduced an -weighted vector field method to obtain boundedness and decay of solutions to wave equations on a Lorentzian background, avoiding to use global vector field multipliers and commutators with weights in and thus proving to be more robust in dealing with black holes such as in Schwarzschild and Kerr background metrics. The approach has found numerous applications; see, for instance, Dafermos-Rodnianski, Moschidis, or Keir for many insightful discussions. In this blog post, I will give some basic details of this approach, based on lecture notes of S. Klainerman, focusing mostly on the flat Minkowski spacetime.

Continue reading →