Researchers have long been numerically solving the partial differential equations that govern important fluid phenomena such as weather, fusion plasmas, and aerodynamics. Of course, the accuracy of the results is always limited by the finite precision and spatial resolution of computer representations of the equations.
Computers have also become a powerful tool for exact, rigorous mathematics. Proof assistants, for example, instill confidence a logical argument is sound and all cases have been considered. Programs can tirelessly examine superhumanly large libraries of combinations, such as those underlying the four-color map theorem proof in 1976.
What seems surprising, however, is that researchers have used numerical computations to rigorously prove contentious statements about the solutions to fluid equations. In particular, researchers recently proved equations developed by Leonhard Euler to describe fluid flow have solutions that “blow up,” meaning some quantities become infinite at a finite time.
These “singularities” have only been proven to occur with carefully selected initial conditions within highly symmetric boundaries. Nonetheless, knowing they exist could change the way researchers think about less idealized situations. “If you don’t exactly hit a singularity, but you get close, maybe that means the behavior of the system is unpredictable,” said Charles Fefferman, Herbert E. Jones, Jr. ’43 University Professor of Mathematics at Princeton University.
Moreover, the recent results may be extended to clarify the impact of fluid friction, or viscosity, which is left out of the Euler equations but included in the more realistic and important Navier-Stokes equations. Proving the existence—or impossibility—of singularities when there is viscosity (and no boundaries) would win one of the million-dollar prizes associated with solving any of the seven well-known complex math problems classified as Millennium Problems by the Clay Mathematics Institute.
Although the results for the Euler equations do not yet meet this challenge, they could provide important clues about how to do it. They could also inspire mathematicians to demonstrate the singularities using traditional analytical methods, which some find more satisfying.
Searching for Singularities
Fluid flow patterns emerge from simple behavior at a local level, in which each parcel of fluid responds to forces such as pressure gradients. A critical term in the equations describes how internal rotation of the fluid, or vorticity, is carried along with the moving fluid. Because this “vortex-stretching” term includes both speed and rotation, it is intrinsically nonlinear, greatly complicating the resulting behavior. Although the equations are straightforward to write down (Euler did so in 1757), finding their solutions is not.
The goal is to predict how fluids will evolve from any particular initial conditions, which specify the velocity at every point in space at a starting time. The complexities of fluid flow quickly overwhelm researchers’ ability to describe it analytically, but computer simulations have long been an important complement to experiments.
An important outstanding question, however, is whether there is always a unique “globally regular” solution that behaves well, for any initial conditions. A few years ago, Tarek Elgindi, a mathematician at Duke University, found a singularity in a simplified model. “He produced a singular solution of the Euler equation,” Fetterman said, “but it isn’t quite as smooth as one would like.”
In fact, more than 20 years ago, some researchers had found numerical solutions that seemed to be blowing up. Using more precise computations, however, others, including Thomas Hou of the California Institute of Technology, showed that what looked like singularities actually stopped blowing up without developing infinities.
A decade ago, however, Hou and his colleague Luo Guo (now at the Hang Seng University of Hong Kong) showed more convincing numerical signs of a singularity for the Euler equations in a carefully chosen geometry. They studied a fluid bounded by a perfectly cylindrical container with alternating bands of fluid initially flowing clockwise and counterclockwise, which meet at a circular line at the boundary.
The goal is to predict how fluids will evolve from any particular initial conditions, which specify the velocity at every point in space at a starting time.
Around this line, the opposite flows induce a rapidly growing secondary rotational flow. Importantly, the cylindrical symmetry of the container and the mirror symmetry of the flow ensure that the nonlinearities that enhance the vorticity remain concentrated at the singularity. The vorticity appears to become infinite at a finite time, although the fluid velocity remains finite.
Computer Proof
The recent work by Hou and his student Jiajie Chen (now at the Courant Institute at New York University) is regarded as a proof that true singularities occur in this situation. Interestingly, the proof starts with numerical profiles, although Hou notes “a computer can never get infinite resolution.”
Nonetheless, as Fefferman explained, although most numbers cannot be represented exactly in a computer, calculations can guarantee arbitrarily tight upper and lower bounds on their values. More sophisticated techniques can also ensure that a computer representation of a function is as close as necessary, in some chosen sense. “You can deal not only with numbers but also with functions on the computer, and make statements that have a precise logical meaning and make manipulations that are guaranteed to be provably correct.”
The proof required that the singularity occurs despite small deviations. “It’s very important to have a computer to give you a candidate profile,” Hou said, “Based on that, you can do analysis,” systematically examining functions that are very “close” to the candidate.
Otherwise, “The blow-up may have some unstable mode, meaning that you can get close to it but may not be able to get to the singularity itself. It may find a way to escape,” he said. “There’s some unstable direction that a small perturbation will drift away from the singularity.” The researchers repeatedly enlisted the computer to confirm all escape routes were blocked if the deviations are small enough.
Importantly, the researchers did the stability analysis in a rescaled version of the problem. They assume that the spatial variation is nearly the same, except that the length scale changes as a power of the time remaining until the singularity. “We scale the finite time simulation to infinite time in the rescaled time domain, and we rescale in space, so we have a smooth profile,” Hou said, even though the coordinates themselves are singular. “The equivalent problem is much easier, because the profile becomes smooth.”
Nonetheless, although many people expected a singularity for the Euler equations, “It’s very hard to find such a candidate,” Hou said. “If you randomly choose some smooth initial data, almost surely that will not blow up. You have to find a very special condition to generate this self-similar, sustainable, stable blow-up.”
Beyond the Boundaries
“You can think of these self-similar coordinates as sort of de-singularizing the singularities,” said Tristan Buck-master of the University of Maryland, whose team has also advanced the search for smooth solutions. “You want to prove something horrible happens by proving something non-horrible happens.”
The rescaling depends on determining the precise exponent for the transformation. Buckmaster and his colleagues have been using “physics-informed neural networks” to find the discrete values that allow a smooth solution.
Unlike neural networks that iteratively suggest parameters to a separate, traditional solver, however, in this case “The neural network itself is just a nonlinear representation of the function,” Buckmaster said. Moreover, “We know a lot about [how] the solution should look, so we can build that in directly to the neural network,” he said, including symmetries, conservation laws, and asymptotic behavior. “That helps a lot.”
Buckmaster hopes his methods will help identify singularities for the boundary-free problem. “Traditional numerical methods are really not very useful in finding these solutions.”
Hou and Chen’s solution “is a fantastic result. They solved it,” Buck-master said. Nonetheless, he said, “the name of the game is not to have a boundary,” because the flow will always have a discontinuity there, which is where the singularity forms. Buck-master hopes his methods will help identify singularities for the boundary-free problem. “Traditional numerical methods are really not very useful in finding these solutions.”
“I think there is a serious chance that one team or the other or both will … find a singularity for the Euler equations without a wall,” Fefferman said. “I personally think that is at least as interesting as the Clay [Navier-Stokes] problem.”
Including Viscosity
The implications for the Navier-Stokes solutions are surprisingly murky, however. Having solved the viscosity-free Euler case, one might “hope that if the coefficient of friction is actually very, very small but not zero, that the solutions are approximately the same,” Fefferman said, and there is even a proved theorem that this should be so. “That’s very, very plausible, but it turns out that in the real world it’s absolutely false—or so it seems.”
“There’s unreasonable effectiveness of a tiny amount of friction in the flow of fluids,” Fefferman noted. Singularities could help resolve this seeming paradox, he said, since after the singularity there may actually be no solution of the Euler equations—although there are other possible resolutions.
Buckmaster said singularities are even more dependent on boundaries when viscosity is present. Whether they persist or escape depends on the precise value of the self-scaling exponent with viscosity, which he hopes his neural networks will be particularly good at finding.
Viscosity is also critical in the important phenomenon of turbulence, which results in a cascade of energy to whorls at smaller and smaller scales, until viscosity finally turns it to heat. While this process has a geometric self-similarity, it tends to spread vorticity throughout the fluid, rather than concentrating it, Hou said. “In fact, turbulence tends to destroy singularity.” Nonetheless, he recently described a scenario leading to somewhat different singularities in the Navier-Stokes equations.
“If the singularities exist,” Fefferman said, “it is possible that there is a zoo of different singularities and that the current exciting work is merely discovering the ones that are easiest to describe.”
Further Reading
Chen, J. and Hou, T.Y.
Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data, https://arxiv.org/abs/2210.07191 (2022).
Cepelewicz, J.
Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations, Quanta, Nov. 16, 2022, https://bit.ly/3ZAiXvU
Cepelewicz, J.
Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations, Quanta, April 12, 2022, https://bit.ly/3YBmJns
The Navier-Stokes Millennium Problem, with an official problem description by Charles Fefferman, Claymath.org, http://bit.ly/3l63PXV.
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