Mathematics

A Fluid New Path in Grand Math Challenge

Submerged turbulent jet

C. Fukushima and J. Westerweel, Technical University of Delft

The Navier-Stokes equations of fluid flow, used to model ocean currents, weather patterns and other phenomena, have been dubbed one of the seven most important problems in modern mathematics.

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In Dr. Seuss’s book “The Cat in the Hat Comes Back,” the Cat makes a stain he can’t clean up, so he calls upon the help of Little Cat A, a smaller, perfect replica of the Cat who has been hiding under the Cat’s hat. Little Cat A then calls forth Little Cat B, an even smaller replica hidden under Little Cat A’s hat. Each cat in turn lifts his hat to reveal a smaller cat who possesses all the energy and good cheer of the original Cat, just crammed into a tinier package. Finally, Little Cat Z, who is too small to see, unleashes a VOOM like a giant explosion of energy, and the stain disappears.

A similar process lies at the heart of a speculative new approach to a problem that has bedeviled mathematicians for more than 150 years: understanding the solutions to the Navier-Stokes equations of fluid flow, which physicists use to model ocean currents, weather patterns and other phenomena. These equations are so complex that in most cases, no one knows whether the solution will be smooth and well-behaved, without any sudden shifts of direction or explosions of energy, for instance. And computer models of the solutions run aground, unable to accurately capture the behavior of small eddies.

Now, in a paper posted online on February 3, Terence Tao of the University of California, Los Angeles, a winner of the Fields Medal, mathematics’ highest honor, offers a possible way to break the impasse. He has shown that in an alternative abstract universe closely related to the one described by the Navier-Stokes equations, it is possible for a body of fluid to form a sort of computer, which can build a self-replicating fluid robot that, like the Cat in the Hat, keeps transferring its energy to smaller and smaller copies of itself until the fluid “blows up.” As strange as it sounds, it may be possible, Tao proposes, to construct the same kind of self-replicator in the case of the true Navier-Stokes equations. If so, this fluid computer would settle a question that the Clay Mathematics Institute in 2000 dubbed one of the seven most important problems in modern mathematics, and for which it offered a million-dollar prize. Is a fluid governed by the Navier-Stokes equations guaranteed to flow smoothly for all time, the problem asks, or could it eventually hit a “blowup” in which something physically impossible happens, such as a non-zero amount of energy concentrated into a single point in space?

Tao’s new program for tackling Navier-Stokes has changed his thinking about the problem.

Tao’s proposal is “a tall order,” said Charles Fefferman of Princeton University. “But it’s a very interesting way of thinking about the long-term future of the problem.”

The real ocean doesn’t spontaneously blow up, of course, and perhaps for that reason, most mathematicians have concentrated their energy on trying to prove that the solutions to the Navier-Stokes equations remain smooth and well-behaved forever, a property called global regularity. Purported proofs of global regularity surface every few months, but so far each one has had a fatal flaw. (The most recent attempt to garner serious attention, by Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, is still under review, but mathematicians have already uncovered significant problems with the proof, which Otelbaev is trying to solve.)

“Everyone in the research community would agree that the tools we have at the moment are not sufficient to prove global regularity,” said Susan Friedlander, of the University of Southern California in Los Angeles.

Tao originally set out with a fairly modest goal: simply to make rigorous the intuition that the existing tools are not good enough. Many would-be proofs of global regularity have tried to exploit a principle of conservation of energy, and Tao set out to show that this principle is not sufficient to establish global regularity. He constructed a counterexample, a sort of toy fluid-flow universe whose governing equations have many commonalities with the Navier-Stokes equations, including conservation of energy, but whose solutions can blow up.

A decade earlier, Nets Katz, now of the California Institute of Technology in Pasadena, and Natasa Pavlovic, now of the University of Texas, Austin, had established blowup for a toy version of a simpler fluid flow model by showing how to transfer a given amount of energy into smaller and smaller size scales until, after a finite amount of time, all the energy would be packed into a single point, and the fluid would blow up. But Katz and Pavlovic’s process distributed the energy across many different size scales at the same time, as if the Cat had lifted his hat to reveal not Little Cat A, but weak versions of many of the smaller cats. When Katz and Pavlovic tried to extend their process to a toy version of the Navier-Stokes equations, the fluid’s viscosity snuffed out this thinned-out energy and no blowup occurred.

Terence Tao

Kyle Alexander

Terence Tao of the University of California, Los Angeles, has proposed a new approach to understanding the solutions to the Navier-Stokes equations of fluid flow.

To make the transfer of energy more controllable, Tao tried to devise a system that would incorporate a delay at each step — a sort of timer that would push the energy cleanly from one size scale to the next at just the right moment. He found himself thinking like an electrical engineer in terms of circuits, capacitors, resistors and fuses, until he realized what he was really doing: trying to create a computer out of fluid.

Tao started designing fluid “logic gates,” the basic building blocks of all computers, which transform information or energy according to particular rules. One of his gates pumped energy from one region of a fluid to another; a second gate rapidly exchanged energy between two regions if a third region passed a certain energy threshold. Tao realized that he could string together five such gates to create a self-replicating fluid machine. He then designed a particular toy universe in which these five gates were all achievable.

While Tao’s construction applies only to this toy universe, “in principle, there’s nothing stopping the actual Navier-Stokes equations from doing this,” he said. “There’s no immediate mathematical objection to this working — just a huge practical objection.”

Fluid-based computers are already in use in some specialized settings such as in aircrafts, but these computers use pipes and valves made out of solid materials. Tao’s computer would have to be made entirely out of water.

“It’s a very fanciful idea, and I don’t expect this program to come to fruition any time in the next five years,” Tao said. Nevertheless, he said, it may be possible to harness fluid effects such as miniature vortex sheets to create water barriers that could serve as pipes.

The new proposal, however intriguing, is just speculation, cautioned Peter Constantin of Princeton University. “This is mathematics,” he said. “What’s proved is proved.”

Tao’s ideas certainly aren’t standard, Friedlander said. “But results about global regularity, whether positive or negative, will almost certainly require something that is nonstandard.” Tao’s ideas are quite likely to lead to some interesting mathematics, she said, whether or not they answer the Navier-Stokes question.

Meanwhile, Tao’s new program for tackling Navier-Stokes has changed his thinking about the problem. Previously, he had viewed both sides of the problem — proving global regularity or establishing that solutions can blow up — as equally remote goals (though in the end, only one can be true). Now he believes that it should be possible to concoct special scenarios in which a solution to the Navier-Stokes equations blows up. (This wouldn’t mean that the real ocean could blow up — instead, it would suggest that in these rare cases, the Navier-Stokes equations don’t fully capture the ocean’s physics.) “My mindset has certainly changed,” he said.

A central insight of computer science is that, whenever a physical phenomenon is complex enough, it should be possible to use it to build a universal computer — one capable of doing anything computers can do, including building self-replicating machines.

“Can you make fancy patterns of water that actually have some computation power?” Tao asked. “I’m betting that fluids are complex enough to do this.”

This article was reprinted on ScientificAmerican.com.

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  • If one can actually build self-replicating robots that build smaller robots in some fluid, then you could build a computer with exponentially many components. You could then use this computer to simulate a quantum mechanical system with a Hilbert space of exponentially many dimensions. Could quantum mechanics be a form of turbulence?

  • This is a fascinating model. While I don’t have the level of ability to understand the original paper (to my regret) I wonder if someone who does might be able to comment on a speculation that this approach could suggest a mechanism for inflation in our present universe? Viz:

    If the observed quark-gluon plasma was omnipresent in the early (-10E15 sec) universe, as is speculated, and behaves as a (supposedly perfect) superfluid then this approach to the Navier-Stokes equations may imply that a “blow-up” could have spontaneously occurred within some finite time. I presume that (if one were able to observe this phenomenon) this might appear much like a period of cosmological inflation?

    My thanks in anticipation of a response post (positive or not).

  • If Tao’s method can be successfully applied to the Navier-Stokes equations for fluid dynamics, is there any likelihood that plasma physicists will utilize it to describe plasmas as observed in stars and in galaxies?

    This opens up the question as to whether or not plasmas constitute “fluids” and can successfully be described thereby. Plasmas are by definition composed of some fraction of charged particles, so the inventor of magnetic hydrodynamics (MHD), Swedish physicist Hannes Alfvén, said no, plasmas are too complex for even his MHD model to describe successfully, even though MHD in the form of the Navier-Stokes equations are used in conjunction with Maxwell’s equations of electromagnetism in attempts to model large-scale plasmas. There are difficulties in this wicked problem, and simplifying assumptions have to be made at nearly every step for simulation to remain tractable.

    If Tao’s work could help, it might open up a welcome avenue of exploration in the topic of space and other forms of plasmas. Plasmas DO blow up occasionally, unlike the ocean, and simulations of arc discharges due to the carrying capacity of the conducting medium (plasma) sure needs to be better described if this is to work in the descriptions of stellar photospheres, supernovas, even the auroras or the impact of coronal mass discharges on the Earth’s magnetosphere and its connection with subterranean telluric currents.

  • Thanks for interesting information on work in progress. This also highlights the fact that parallel versions and modifications of famous problems tend to be oversimplified. An example of this is even the four colour theorem for surfaces other than the sphere, which becomes an n-colour problem, but turns out to be much simpler. So are certain exotic modifications of the Riemann hypothesis.

    Another issue is the mechanism by which turbulence is reached in this case, namely infinite energy density. After several iterations the system might reach the atomic level or the speed of light, where Navier-Stokes equations lose their meaning. It would be most interesting to explain natural turbulence without such infinities. Meanwhile, when we go swimming, we should avoid splash (as every good swimmer knows) to assure smooth and fast movement :-)

  • I agree that fluid matter doesn’t spontaneously blow up, rather energy is conserved as potential energy that exits as quantum many body fluctuation, whose distribution overall, can produce unique criticality.

    Hence emergence and complex patterns in systems is what presents itself. It’s the distribute of space and matter and potential versus kinetic energies and non-equilibrium thermodynamics (negative versus positive) entropic versus enthalpic gains that ultimately determines controlled blow up moments that in fact still within the fabric of enthalpy and not entirely chaotic. Hence controlled stochastics. The modeling of a domino effect, very simple yet extremely difficult in terms of its simplicity due to the number of factors it affects. Hence the number of atoms/things around an object and their interaction is what ultimately determines the change, fluctuation, pattern.

    Enthalpy versus entropy with reverse thermodynamic equilibrium. Non-equilibrium thermodynamics, is entropy(enthalpy((energetic((matter 1) – (matter2))).

    Dr. Rostro

  • Exotic model is not needed, particularly in hydrodynamics. Let Tao solves the Navier-Stokes equations.

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