Non-linear wave equations (in 3+1 dimensions) are ubiquitous in physics, playing a vital role in many subfields of it. There are many methods available to show short-time existence of solutions for such equations, but global-in-time existence is a much more subtle problem.

In the 1980s Klainermann introduced a so-called *null condition* on the nonlinearities of a nonlinear wave equation which was shown to guarantee global-in-time existence of solutions for all sufficiently small initial data.

It is known that the null condition is only sufficient, but not necessary for the existence of global solutions. Fifteen years ago the *weak null condition* was introduced by Lindblad and Rodnianski, and all systems of non-linear wave equations, where we know that they have global existence of solutions for small initial data (i.e., also the ones which do not satisfy the null condition), satisfy the weak null condition.

A few months ago Joseph Keir posted a paper on the arXiv ( https://arxiv.org/abs/1808.09982 ) in which he shows that under an extra condition nonlinear wave equations in 3+1 dimensions satisfying the weak null condition do indeed have global solutions for small initial data. This extra condition that he imposes is satisfied by all currently known equations which are known to have global existence.

Since Joseph Keir’s paper is an astounding 372 pages long, it will probably take some time for the community to verify all his arguments. For the more interested reader, his paper actually also has a long and extensive introduction, so it might be worthwhile to read it.