Flowing towards a solution to the hardest math problems

January 7, 2018 8:52 PM

In 2000, the Clay Millennium Problems were posted - seven unsolved problems in mathematics. If you solve one, the Clay Institute will give you a million dollars. To date, only one problem has been solved, the Poincaré Conjecture, by the reclusive Grigory Perelman, who refused the prize, and who is famously described in this (controversial) New Yorker article [prev]. Now, as described in Wired, interesting questions are emerging over approaches to solve a second challenge, the Navier–Stokes Equations which predict how fluids flow.
posted by blahblahblah (13 comments total)

As a B.Sc. (Math) I am simultaneously appalled, mystified, and jealous of the lingo in the problem statements.
posted by Tad Naff at 10:10 PM on January 7, 2018 [1 favorite]

The original article was published in Quanta Magazine.
posted by J.K. Seazer at 11:14 PM on January 7, 2018 [1 favorite]

It would be more interesting if the Navier-Stokes equations did turn out to be inadequate.
posted by jamjam at 11:38 PM on January 7, 2018 [2 favorites]

Buckmaster and Vicol’s work shows that when you allow solutions to the Navier-Stokes equations to be very rough (like a sketch rather than a photograph), the equations start to output nonsense: They say that the same fluid, from the same starting conditions, could end up in two (or more) very different states. It could flow one way or a completely different way. If that were the case, then the equations don’t reliably reflect the physical world they were designed to describe.

So....don't do that ( allow solutions to the Navier-Stokes equations to be very rough)? Isn't it unsurprising that, if you tolerate initial ambiguity, you don't get results that satisfy an entirely deterministic metaphysics?

I'm sure the mathematics are profound but I don't see that there's a serious issue with understanding fluids here.
posted by thelonius at 4:48 AM on January 8, 2018 [1 favorite]

So....don't do that ( allow solutions to the Navier-Stokes equations to be very rough)?

I'm only a dumb engineer, but from my perspective this is a problem. We (engineers) solve these equations by coarsening the domain and approximating solutions. It's called finite element analysis, and the Leray approximation mentioned in the article basically does just that - we take an average in a small subdomain and assume the values in that "element" are uniform throughout the element. There simply isn't enough computing power in the world to do otherwise. If it turns out that doing this kind of approximation isn't valid in certain circumstances, that could have huge implications.
posted by backseatpilot at 5:31 AM on January 8, 2018 [11 favorites]

Supergenius Terence Tao has also published a few related papers on his excellent blog.

His hunch is that the equations aren't consistent and it may be possible to break the equations by mathematically constructing a fluid von Neumann machine that built smaller and smaller versions of itself until a singularity is reached.
posted by pomomo at 6:00 AM on January 8, 2018 [2 favorites]

If it turns out that doing this kind of approximation isn't valid in certain circumstances, that could have huge implications.

I don't know, their objections seemed to me to be more philosophical than practical.
posted by thelonius at 6:30 AM on January 8, 2018

I don't know, their objections seemed to me to be more philosophical than practical.

Pure Mathematicians look down on applied mathematicians.

Applied mathematicians look down on mathematicians whose work has been applied by engineers.

Pure posing.
posted by ocschwar at 6:32 AM on January 8, 2018

My memory from fluid mechanics lectures at university = 'And this big clanking inelegant factor/variable I'm going to add here... well we've no idea how this relates to the real world yet but we need it for the equations to work/balance so, moving on...' * several.
posted by fearfulsymmetry at 7:37 AM on January 8, 2018

So....don't do that ( allow solutions to the Navier-Stokes equations to be very rough)?

Let me attempt to answer this natural question.

Partial differential equations describe the evolution of a function in time. Intuitively given a function and some of its derivatives, you know how it changes after an infinitesimal time-step. Repeating this step over and over, you can integrate a solution and explore its future. For the Navier-Stokes equation the Clay problems are to
1. To find when smooth solutions exist given a starting point (boundary condition)
2. To find if and when smooth solutions "blow up" in the future eg. something goes to infinity unphysically

(for three dimensions. For some reason, 2D is much easier)

For weak solutions we don't demand that all derivatives exist, but a lesser condition (that they can be integrated against a smooth function). This might seem weird at first - how can you plug something into a differential equation if you can't differentiate it?! But it is actually much easier to work with.

(If you've met the delta function before, the concept is similar.)

For example, it encompasses certain limits of functions. Imagine a sequence of rounded corners that tend toward a sharp angle as their limit. If you restricted yourself to smooth functions you would have to artificially exclude the limit from consideration. But it might be perfectly well-defined as a (weak!) solution.

For other equations weak solutions are also physically natural eg. the Burger's equation can describe shock waves where the density of air jumps discontinuously. That is also a famous example of a smooth solution becoming unsmooth.

(but still physical in this case eg. energy is conserved)
posted by tirutiru at 8:15 AM on January 8, 2018 [7 favorites]

Libraries are filled with engineering graduate theses on various Navier-Stokes modelling approximations and the limits of their applicability, dating back to before World War II and the advent of computers. I got a Master's degree and a paper out of a correlation I developed between a constant in one of these approximations and the geometry of surface roughness, and it takes a few pages of math to drill down from the general Navier-Stokes equations to the limited case where this approximation can be applied (turbulent flow along a continuous surface).

The understanding when I was in school 20+ years ago was that the equations could only be "solved" (rather than modeled/approximated) if you knew the initial properties of the entire flow field down to the scale of molecular interactions--even if there was some way to measure this, there was no way to perform the calculations on a sufficiently large scale for practical application. If the equations can be solved, or even if the current models can be greatly simplified, you could realize large cost savings in aerodynamic design. This opens up their use in many fields where it is just not cost-effective today, where people are instead relying on trial and error and/or simplified models that constrain their design options considerably.
posted by cardboard at 8:27 AM on January 8, 2018 [4 favorites]

Pure Mathematicians look down on applied mathematicians.

Applied mathematicians look down on mathematicians whose work has been applied by engineers.

There are people like that, and famous historical examples, but they tend to be rare in my experience.
posted by Coventry at 10:31 AM on January 8, 2018

The answer is 42. I'll take my million in non-sequentially numbered 10s and 20s please.
posted by Chitownfats at 11:09 AM on January 8, 2018 [1 favorite]