Terence Tao

So sometimes these forces are in balance at small scales, but not in balance at large scales or vice versa.

So Navier-Stokes is what's called supercritical.

So at smaller and smaller scales,

the transport terms are much stronger than the viscosity terms.

So the viscosity terms are the things that calm things down.

And so this is why the problem is hard.

In two dimensions, so the Soviet mathematician Ladislav Skaya, she, in the 60s, showed in two dimensions there was no blow-up.

And as you mentioned, the Navier-Stokes equation is what's called critical.

The effect of transport and the effect of viscosity are about the same strength, even at very, very small scales.

And we have a lot of technology to handle critical and also subcritical equations and improve regularity.

But for supercritical equations, it was not clear what was going on.

and i did a lot of work and then there's been a lot of follow-up showing that for many other types of super critical equations you can create all kinds of blow-up examples once the non-linear effects dominate the linear effects at small scales you can have all kinds of bad things happen so this is sort of one of the main insights of this this line of work is that super criticality versus criticality and sub-criticality this this makes a big difference i mean that's a key qualitative feature that distinguishes

Some equations were being sort of nice and predictable and, you know, like planetary motion.

I mean, there's certain equations that you can predict for millions of years or thousands at least.

Again, it's not really a problem, but there's a reason why we can't predict the weather past two weeks into the future because it's a super critical equation.

Lots of really strange things are going on at very fine scales.

Yeah, and if non-linearity is somehow more and more featured and interesting at small scales.

There's many equations that are non-linear, but in many equations you can approximate things by the bulk.

For example, planetary motion, if you want to understand the orbit of the Moon or Mars or something, you don't really need the microstructure of the seismology of the Moon or exactly how the mass is distributed.

You can almost approximate these planets by point masses.