For centuries mathematicians have I have been trying to understand and model the motion of fluids. The equation for how ripples crease the surface of a pond has helped researchers predict the weather, design better planes, and characterize how blood flows through the circulatory system. These equations are deceptively simple when written in a suitable mathematical language. But their solutions are so complex that even basic questions about them are very difficult to understand.
Perhaps the oldest and most prominent of these equations, formulated by Leonhard Euler over 250 years ago, describes ideal incompressible fluid flow. This fluid has no viscosity or internal friction and cannot be forced into a smaller volume. “Almost all nonlinear fluid equations are derived from Euler’s equations,” says Tarek He Elgindy, a mathematician at Duke University. “They are the first, you can tell.”
However, many questions remain about the Euler equation, such as whether it is always an accurate model of ideal fluid flow. One of the central problems in fluid dynamics is determining whether equations fail and output meaningless values, making the future state of the fluid unpredictable.
Mathematicians have long suspected the existence of initial conditions that cause the equations to collapse. But they haven’t been able to prove it.
In a preprint posted online in October, two mathematicians showed that certain versions of Euler’s equations do indeed sometimes fail. This proof represents a major breakthrough, and while it does not completely solve the problem of the more general version of the equation, it gives hope that such a solution is finally within reach. I will give it to you. “This is a surprising result,” said University of Maryland mathematician Tristan Buckmaster, who was not involved in the study. “There are no results of that kind in the literature.”
Only one catch.
The result of a ten-year research program, the 177-page proof makes great use of computers. This no doubt makes it difficult for other mathematicians to verify it. The only viable way to resolve such critical issues in the future is with the help of a computer.
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In principle, if the position and velocity of each particle in the fluid are known, the Euler equations should be able to predict how the fluid will change over time. But mathematicians want to know if that’s really the case. Perhaps in some circumstances the equations will go as expected and produce accurate values for the state of the fluid at any given time, but only one of those values will suddenly spike to infinity. At that point, the Euler equation is said to cause a “singularity”. More dramatically, it can also “blow up”.
Once the singularity is reached, the equations can no longer calculate the fluid flow. But “as of a few years ago, what people could do was [proving blowup]said Charlie Pfefferman, a mathematician at Princeton University.
Trying to model viscous fluids (as with almost all real-world fluids) gets even more complicated. The million-dollar Millennium Prize from the Clay Mathematics Institute awaits someone who can prove whether similar failures occur in the Navier-Stokes equations, a generalization of the Euler equations that describe viscosity.