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The New York Times Magazine JULY 24, 2015 article The Singular Mind of Terry Tao starts off with:

This April, as undergraduates strolled along the street outside his modest office on the campus of the University of California, Los Angeles, the mathematician Terence Tao mused about the possibility that water could spontaneously explode. A widely used set of equations describes the behavior of fluids like water, but there seems to be nothing in those equations, he told me, that prevents a wayward eddy from suddenly turning in on itself, tightening into an angry gyre, until the density of the energy at its core becomes infinite: a catastrophic ‘‘singularity.’’ Someone tossing a penny into the fountain by the faculty center or skipping a stone at the Santa Monica beach could apparently set off a chain reaction that would take out Southern California.

Is there any way to state the problem described in terms of undergraduate level physics and math? I don't mean why doesn't real water really explode necessarily, I mean what is the mathematical issue.

Is this related to the Navier–Stokes Existence and Smoothness Millennium Prize problem?

Edit: Based on the helpful comments, I went back and looked at the NYTimes Magazine article again and found this, which is illustative:

Imagine, he said, that someone awfully clever could construct a machine out of pure water. It would be built not of rods and gears but from a pattern of interacting currents. As he talked, Tao carved shapes in the air with his hands, like a magician. Now imagine, he went on, that this machine were able to make a smaller, faster copy of itself, which could then make another, and so on, until one ‘‘has infinite speed in a tiny space and blows up.’’ Tao was not proposing constructing such a machine — ‘‘I don’t know how!’’ he said, laughing. It was merely a thought experiment, of the sort that Einstein used to develop the theory of special relativity. But, Tao explained, if he can show mathematically that there is nothing, in principle, preventing such a fiendish contraption from operating, then it would mean that water can, in fact, explode.

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    $\begingroup$ There was nice development recently where $\int_{a}^{b} f = (b-a)av_f(a,b) $ would provide category theory version of integration via simple average-function and mulltiplication. This would easily extend to navier-stokes, but dunno if it produces anything better than numerical solution -- and the millennium prize problem description says that numerical solution exhibits blowup.... $\endgroup$
    – tp1
    Commented May 16, 2017 at 14:18
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    $\begingroup$ I did not read all the details in the paper, but roughly speaking, Terry Tao considers a modifed version of Navier-Stokes, and proves that for these equations, there exist initial data which leads to a finite time blow-up, i.e. to the formation of a singularity in finite time. This is related to the Navier-Stokes Millenium problem indeed (albeit it is a modifed version of Navier-Stokes). That being said, his approach in the paper is highly original! $\endgroup$
    – Malkoun
    Commented May 16, 2017 at 14:23
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    $\begingroup$ This is all about the millennium problem, yes. It is just a question of starting with "nice" initial data, do you get "bad" data later? And the problem is that none of the techniques we have to show that "niceness" is preserved over time (which are applicable in many evolution PDE) apply to Navier-Stokes in particular. Moreover, Tao recently showed that Navier-Stokes is at best "right on the cusp" of developing singularities, since a relatively modest modification of it does develop singularities. $\endgroup$
    – Ian
    Commented May 16, 2017 at 14:25
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    $\begingroup$ As I recall they are carefully set up to vorticial motion to repeatedly move energy from long length scales to successively shorter length scales. $\endgroup$
    – Ian
    Commented May 16, 2017 at 23:31
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    $\begingroup$ Have you read Tao's blog post on finite time blowup for an averaged version of NSE? There's a more mathematical and still accessible explanation (especially at the end). $\endgroup$
    – Michał Miśkiewicz
    Commented May 17, 2017 at 20:18

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Tao's idea is to set up initial data that successively moves energy from longer length scales to shorter length scales by constructing smaller, faster vortices. As in the NYT quote, it is a bit like a machine which constructs a small copy of itself and then dissipates, and then the copy creates an even smaller copy of itself and then dissipates, etc. This process then accelerates fast enough that, roughly speaking, after only a finite amount of time the machine already has no size. In the modified Navier-Stokes used in his recent paper, this construction actually goes through and leads to finite time blowup.

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