Terence Tao

So you're trying to make a blow-up.

Yeah.

So I basically engineer a blow-up by changing the rules of physics, which is one thing that mathematicians are allowed to do.

We can change the equation.

How does that help you get closer to the proof of something?

Right.

So it provides what's called an obstruction in mathematics.

So what I did was that basically if I turned off certain parts of the equation, which usually when you turn off certain interactions, make it less nonlinear, it makes it more regular and less likely to blow up.

But I found that by turning off a very well-designed set of interactions, I could force all the energy to blow in finite time.

So what that means is that if you wanted to prove

global regularity for Navier-Stokes, for the actual equation, you must use some feature of the true equation which my artificial equation does not satisfy.

So it rules out certain approaches.

So

The thing about math is it's not just about taking a technique that is going to work and applying it, but you need to not take the techniques that don't work.

And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem.

But it's only after a lot of experience that you realize there's no way that these methods are going to work.

So having these counterexamples for nearby problems kind of rules out, it saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work.

Right.

Yeah.

So the key phenomenon that my technique exploits is what's called supercriticality.