For centuries, mathematicians Fluid motion has been studied and modeled by researchers. Scientists have been able to use equations to explain how water ripples create a pond’s surface. This has allowed them better predict weather patterns, make better planes and understand how blood flows through the circulation system. If written in the proper mathematical language, they can appear very simple. The solutions to these equations are complex enough that it can be difficult for even the most basic questions to be understood.
Leonhard Euler’s equations are the most well-known and oldest of all these equations. They describe the flow and pressure of an ideal incompressible fluid. A fluid that has no viscosity and internal friction is impossible to compress into smaller volumes. “Almost all nonlinear fluid equations are kind of derived from the Euler equations,” said Tarek Elgindi, a mathematician at Duke University. “They’re the first ones, you could say.”
Yet much remains unknown about the Euler equations—including whether they’re always an accurate model of ideal fluid flow. One of the central problems in fluid dynamics is to figure out if the equations ever fail, outputting nonsensical values that render them unable to predict a fluid’s future states.
For decades mathematicians believed that equations could be broken down if they had the right initial conditions. But they haven’t been able to prove it.
Two mathematicians from the University of Michigan have demonstrated that certain Euler equations fail sometimes in an online preprint published October. The proof marks a major breakthrough—and while it doesn’t completely solve the problem for the more general version of the equations, it offers hope that such a solution is finally within reach. “It’s an amazing result,” said Tristan Buckmaster, a mathematician at the University of Maryland who was not involved in the work. “There are no results of its kind in the literature.”
There’s just one catch.
The 177-page proof—the result of a decade-long research program—makes significant use of computers. It is therefore difficult to prove it by other mathematicians. Although they’re still trying to verify it, experts are confident that the work will prove correct. It also forces them to reckon with philosophical questions about what a “proof” is, and what it will mean if the only viable way to solve such important questions going forward is with the help of computers.
Looking for the Beast
The Euler equations can be used to forecast how fluids will change over time if we know where and at what velocity each particle is located in fluid. But mathematicians want to know if that’s actually the case. It is possible that in certain situations the equations can produce exact fluid state values at every moment. But one of these values could suddenly rise to infinity. At that point, the Euler equations are said to give rise to a “singularity”—or, more dramatically, to “blow up.”
Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow. But “as of a few years ago, what people were able to do fell very, very far short of [proving blowup],” said Charlie Fefferman, a mathematician at Princeton University.
It gets even more complicated if you’re trying to model a fluid that has viscosity (as almost all real-world fluids do). The Clay Mathematics Institute is offering a million-dollar Millennium Prize to anyone who proves similar failures in the Navier–Stokes equations. This generalization of Euler equations accounts for viscosity.
