[diffeomorphism]

> but to me this sounds like the "proved" something by not being able to disprove Batchelor's law?

Not quite. The statement is more like, if Batchelor's law were to fail, then it has to be in one of the following specific ways. Then you show that these specific ways can't happen and get the result.

This is a common approach and needs a few ingredients:

- How could things go wrong?

- Show that these are all possibilities and that otherwise things work (hard problem).

- Isolate each scenario from the first step and show that things don't go wrong (hard problem again).

A classical example would be something like global existence for the two-dimensional Euler equations. If the solution were to fail after a finite time, then necessarily some quantity has to go to infinity, because otherwise we could find a solution for a small additional time (Beale-Kato-Majda criterion). We then show that this quantity does not go to infinity and we are done.

[whatitdobooboo]

I see, very helpful, thanks. But doesn't this leave room for what you potentially have not accounted for? I am just trying to wrap my head around how one can write a proof by saying something doesn't occur

[diffeomorphism]

Good question. That is part of the second point. You have to show that there is no room left.

For example say you want to show that a real valued solution stays bounded. Then you have to show that a solution always exists, starts at some small value and "it never occurs that the absolute value of the solution is bigger than 1000". Because you ruled out other scenarios this then implies that the solution is always bounded by 1000.