I see a number of people commenting on the size of the proof (roughly 900 pages) which is not uncommon in this particular sub-field of PDEs. For context, I had the distinct privilege of studying under Sergiu Klainerman for my PhD on this topic. My own dissertation was about 600 pages. From my personal experience, I have come to understand a few factors that contribute to large proof sizes. 1. A lot of work is on inequalities involving integrals with many terms. These are difficult to express without taking up substantial space on the page. Some inequality derivations themselves might take multiple pages if you want to go step-by-step to illustrate how they are done. 2. Writing a proof of this size is not unlike building a medium-to-large size codebase. You have a lot of Theorems/Classes that need to fit together, and by employing some form of separation of concerns you can end up with something quite large and complex. 3. Verifying this kind of proof isn't usually done all at once. A lot of verification happens on the individual lemmas before they're pieced together. Once the entire paper is written, verification is more of a process where you rely on intuition for what the "hard parts" of the proof are and drilling down on those. But when writing the paper, you must of course account for all the details regardless of whether they are "easy" or "hard", and there can be many.

Having said all this, I have not read their paper and it has been 5 years since I was in this space. This is a truly remarkable accomplishment and the result of decades of hard work!

I'll end with an amusing anecdote. A fellow grad student, when deciding between U of Chicago and Princeton for his PhD program was pitched by a U of Chicago professor who once said something like "Of course you could go to Princeton and write 700 page papers that nobody reads." When this story was shared during a conversation over tea at Princeton, another professor retorted, "Or you could have gone to U Chicago to work with him and write 70 page papers that nobody reads!"