Terence Tao

And each time it does this, it takes maybe half as long as the previous one, and then you could actually

converge to all the energy concentrating in one point in a finite amount of time.

That scenario is called finite-time blow-up.

In practice, this doesn't happen.

Water is what's called turbulent.

It is true that if you have a big eddy of water, it will tend to break up into smaller eddies.

But it won't transfer all the energy from one big eddy into one small eddy.

It will transfer into maybe three or four.

And then those ones split up into maybe three or four small eddies of their own.

And so the energy gets dispersed to the point where the viscosity can then keep the thing under control.

But if it can somehow concentrate all the energy

keep it all together and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blob can occur.

So there were papers who had claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just Navier-Stokes, but for many, many types of equations like this.

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.