The Einstein equations in wave coordinates are an example of a system
which does not obey the "null condition". This leads to many
difficulties, most famously when attempting to prove global existence,
otherwise known as the "nonlinear stability of Minkowski space".
Previous approaches to overcoming these problems suffer from a lack of
generalisability - among other things, they make the a priori assumption
that the space is approximately scale-invariant. Given the current
interest in studying the stability of black holes and other related
problems, removing this assumption is of great importance.
The p-weighted energy method of Dafermos and Rodnianski promises to
overcome this difficulty by providing a flexible and robust tool to
prove decay. However, so far it has mainly been used to treat linear
equations. In this talk I will explain how to modify this method so that
it can be applied to nonlinear systems which only obey the "weak null
condition" - a large class of systems that includes, as a special case,
the Einstein equations. This involves combining the p-weighted energy
method with many of the geometric methods originally used by
Christodoulou and Klainerman. Among other things, this allows us to
enlarge the class of wave equations which are known to admit small-data
global solutions, it gives a new proof of the stability of Minkowski
space, and it also yields detailed asymptotics. In particular, in some
situations we can understand the geometric origin of the slow decay
towards null infinity exhibited by some of these systems: it is due to
the formation of "shocks at infinity".
Seminar series
Date
Thu, 16 May 2019
Time
12:00 -
13:00
Location
L4
Speaker
Joe Keir
Organisation
Cambridge DAMTP