Terence Tao: evaluation

Transcript context

…How deeply connected is it to that specific problem of fluid dynamics or is this some more general intuition you build up about mathematics? Right. Yeah. So, the key phenomenon that my technique exploits is what’s called super-criticality. So, in partial [inaudible 00:15:46] 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 from… 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. 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 things that calm things down. And so this is why the problem is hard. In two dimensions, so the Soviet mathematician Ladyzhenskaya, she in the ’60s shows in two dimensions there was no blowup. And in two dimensions, the Navier-Stokes Equation is what’s called critical, the effect of transport and the effect of viscosity 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 prove 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 supercritical equations, you can create all kinds of blowup examples. Once the nonlinear 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 line of work is that super-criticality versus criticality and subcriticality, this makes a big difference. That’s a key qualitative feature that distinguishes some equations for being sort of nice and predictable and… Like planetary motion, 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 supercritical equation. Lots of really strange things are going on at very fine scales. So, whenever there is some huge source of nonlinearity, that can create a huge problem for predicting what’s going to happen?…

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