Terence Tao

So in partial differential equations, often these equations are like a tug of war between different forces.

So in Navier-Stokes, there's the dissipation force coming from viscosity, and it's very well understood.

It's linear.

It calms things down.

If viscosity was all there was, then nothing bad would ever happen.

But there's also transport, that energy in one location of space can get transported because the fluid is in motion to other locations.

And that's a nonlinear effect, and that causes all the problems.

So there are these two competing terms in the Navier-Stokes equation, the dissipation term and the transport term.

If the dissipation term dominates, if it's large, then basically you get regularity.

And if the transport term dominates, then we don't know what's going on.

It's a very nonlinear situation.

It's unpredictable.

It's turbulent.

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.