The weak null condition and the p-weighted energy method

The weak null condition and the p-weighted energy method

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Joe Keir, University of Cambridge
Fine Hall 314

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 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 a detailed description of null infinity. 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".