For centuries, mathematicians sought to understand and model fluid motion. Equations that describe how ripples create wrinkles on the lake’s surface have also helped researchers predict the weather, design aircraft better, and describe how blood flows through the circulatory system. These equations seem simple when written in the right mathematical language. However, their solutions are so complex that understanding even basic questions about them can be extremely difficult.
Perhaps the oldest and most striking of these equations, formulated by Leonhard Euler more than 250 years ago, describes the flow of an ideal, incompressible fluid: a fluid with no degree of density. viscous, or has no internal friction, and cannot be squeezed into a smaller volume. “Almost all nonlinear fluid equations are derived from Euler equations,” says Tarek Elgindi, a mathematician at Duke University. “You could say they were the first.”
However, much remains unknown about the Euler equations—including whether they will always be an accurate model of the ideal fluid flow. One of the central problems in fluid dynamics is figuring out if the equations will ever fail, giving meaningless values that make them impossible to predict the future state of the fluid.
Mathematicians have long suspected that there exist initial conditions that cause equations to break. But they couldn’t prove it.
In a preprint posted online in October, a pair of mathematicians have shown that a particular version of Euler’s equation sometimes actually fails. The proof marks a major breakthrough—and although it does not completely solve the problem of a more general version of the equations, it does give hope that such a solution will eventually be in the reach of hand. “It was a great result,” said Tristan Buckmaster, a mathematician at the University of Maryland who was not involved in the work. “There are no results of this type in the literature.”
Only one catch.
The 177-page evidence—the result of a decade-long research program—makes significant use of computers. This is said to make it difficult for other mathematicians to verify it. (In fact, they are still in the process of doing so, although many experts believe the new work will turn out right.) It also forces them to think about philosophical questions about “evidence.” what it is and how it will be. which means if the only possible way to solve such important questions in the future is with the help of computers.
Seeing the beast
In principle, if you know the position and velocity of each particle in the liquid, the Euler equations should be able to predict how the liquid will evolve at all times. But mathematicians want to know if that is indeed the case. Perhaps in some situations the equations will work out as expected, producing exact values for the state of the liquid at any given time, only if one of those values suddenly spikes. to infinity. At the time, the Euler equations were said to give rise to a “singularity” – or, more dramaticly, an “explosion”.
Once they reach that singularity, the equations will no longer be able to account for the fluid’s flow. But “a few years ago, what people could do fell very, very far from [proving blowup],” speak Charlie Feffermana mathematician at Princeton University.
It gets even more complicated if you’re trying to model a fluid with viscosity (like most real-world fluids). The multi-million dollar Millennium Prize from the Clay Mathematics Institute awaits anyone who can prove whether similar failures occur in the Navier-Stokes equations, a generalization of Euler’s equations take into account viscosity.
