Terence Tao

And so in the past, there have been many attempts to try to obtain what's called global regularity for Navier-Stokes, which is the opposite of finite time blow-up, that velocity stays smooth.

And it all failed.

There was always some sign error or some subtle mistake, and it couldn't be salvaged.

So what I was interested in doing was trying to explain why we were not able to disprove finite time blow-up.

I couldn't do it for the actual equations of fluids, which were too complicated.

But if I could average the equations of motion of the Navier-Stokes, basically, if I could turn off certain types of ways in which water interacts and only keep the ones that I want.

So in particular, if there's a fluid and it could transfer its energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller eddy while still preserving the law of conservation of energy.

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