This time, I won't comment on the cat (too much data in it).

2 . or you can regularize the governing equations themselves and then apply standard theory to show the existence of the desired solution.

Does this include changing the governing equations by possibly very different ones? For instance, we may have a primary system of equations that apply in a given domain or under certain circumstances. During the course of the evolution, we may however leave the domain of definition of the primary system, so that the whole set of equations need to be replaced by something else.

Of course, I have physics in mind. For instance, Newtonian mechanics needs to be upgraded to special relativity for fastly moving systems. In this case however, we can recover the former by taking the right limit of the latter.