Terence Tao

You splash around, there's some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller the velocity, the more calm it gets.

But potentially there is also a demon that keeps pushing

the energy of the fluid into a smaller and smaller scale.

And it will move faster and faster.

And at faster speeds, the effect of viscosity is relatively less.

And so it could happen that it creates some sort of what's called a self-similar blob scenario where, you know,

the energy of the fluid starts off at some large scale, and then it all sort of transfers its energy into a smaller region of the fluid, which then, at a much faster rate, moves into an even smaller region, and so forth.

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.