Terence Tao

And I was interested in the global regularity problem again for this question.

Is it possible for all the energy here to collect at a point?

So the equation I considered was actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same.

And I was able barely to show that...

that you couldn't actually force a scenario where all the energy concentrated at one point.

The energy had to disperse a little bit, and the moment it dispersed a little bit, it would stay regular.

This was back in 2000.

That was part of why I got interested in Navier-Stokes afterwards.

I developed some techniques to solve that problem.

Part of it is that this problem is really non-linear because of the curvature of the sphere.

There was a certain nonlinear effect, which was a non-perturbative effect.

When you looked at it normally, it looked larger than the linear effects of the wave equation.

And so it was hard to keep things under control, even when the energy was small.

But I developed what's called a gauge transformation.

So the equation is kind of like an evolution of heaps of wheat, and they're all bending back and forth, and so there's a lot of motion.

But if you imagine stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion, and under this sort of stabilized flow, the flow becomes a lot more linear.

I discovered a way to transform the equation to reduce the amount of nonlinear effects, and then I was able to solve the equation.

I found this transformation while visiting my aunt in Australia.

And I was trying to understand the dynamics of all these fields, and I couldn't do it with a pen and paper.

And I had none of the facility of computers to do any computer simulations.