There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following:

Solve (easier) approximate problems, show some form of compactness for the approximate solutions to get a limit, then identify the limit as a solution to the original problem.

Show local well-posedness for the problem (that is, for short times and without changing the initial value too much), establish a priori bounds for the solution, conclude global well-posedness by a blowup criterion (that is, local well-posedness contradicts a finite blow-up time in virtue of the a priori bounds).

I'm interested in heuristically comparing both approaches and identifying advantageous of one over the other. A first advantage of the blowup approach over the compactness approach is that it applies to problems that lack compactness properties, for instance when working in the full Euclidean space instead of a bounded domain. What are other general observations when comparing both approaches?