VECTOR FIELDS, MORAWETZ AND STRICHARTZ METHODS IN BLACK BACKGROUND

Abstract

Many physical phenomena are described by linear and nonlinear wave equations. The mathematical understanding of these equations is a major field in mathematical analysis. The purpose of this thesis is to survey and to provide details of three methods that can be used to analyse wave equations in the most general frames. Given a spacetime equipped with a metric tensor $\textbf{g}$ \--- which, in this thesis, is either Minkowski or Schwarzschild \--- the task is to investigate the well-posedness theorems of wave equations involving the general wave operator $\Box_\textbf{g}=g^{\mu\nu}\partial_\mu\partial_\nu$. These different analysis methods are provided in this thesis. The first one is purely geometric, and is the classic vector fields theory. The second one is based on the derivation of a Morawetz estimate, which can be adapted to study problems in general relativity. The third method is the Strichartz estimate, which is based on Fourier and Littlewood-Paley theories. These estimates will be first established for Minkowski spacetime and then generalised to the Schwarzschild spacetime.