For centuries, mathematicians have attempted to understand and model the movement of liquids. The equations that describe how waves ripple the surface of a pond have also helped researchers predict the weather, design better airplanes and characterize how blood flows through the circulatory system. These equations are deceptively simple when written in the correct mathematical language. However, their solutions are so complex that understanding even basic questions about them can be prohibitively difficult.
Perhaps the oldest and most prominent of these equations, formulated by Leonhard Euler more than 250 years ago, describes the flow of an ideal, incompressible fluid: a fluid with no viscosity or internal friction that cannot be forced into a smaller volume. “Almost all nonlinear fluid equations are somehow derived from Euler’s equations,” said Tarek Elgindi, a mathematician at Duke University. “They are the first, you could say.”
Still, much remains unknown about Euler’s equations – including whether they are always an accurate model of ideal fluid flow. One of the central problems in fluid dynamics is figuring out if the equations will ever fail, giving out nonsensical values that make them incapable of predicting the future states of a fluid.
Mathematicians have long suspected that there are initial conditions that cause the equations to collapse. But they couldn’t prove it.
In a preprint posted online in October, two mathematicians showed that a particular version of Euler’s equations does sometimes fail. The proof represents a major breakthrough – and while it doesn’t completely solve the problem for the more general version of the equations, it gives 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 this kind in the literature.”
There’s only one catch.
The 177-page proof – the result of a decades-long research program – makes heavy use of computers. This makes it arguably difficult for other mathematicians to verify. (In fact, they’re still at it, though many pundits believe the new work will turn out to be accurate.) It also forces them to anticipate philosophical questions about what “proof” is and what it becomes, if the only viable path to solve such important issues in the future is the help of computers.
sighting of the beast
In principle, if you know the location and velocity of each particle in a liquid, the Euler equations should be able to predict how the liquid will evolve for all time. But mathematicians want to know if this is really the case. Perhaps in some situations the equations play out as expected, giving precise values for the state of the fluid at any point in time, only for one of those values to suddenly shoot to infinity. At this point, Euler’s equations are said to “singularize”—or, more dramatically, “explode.”
Once they reach this singularity, the equations can no longer calculate the fluid’s flow. But “just a few years ago what people could do was very, very behind [proving blowup]’ said Charlie Fefferman, a mathematician at Princeton University.
It gets even more complicated when you try to model a liquid with viscosity (like almost all real liquids do). A million-dollar Millennium Prize from the Clay Mathematics Institute awaits anyone who can prove whether similar errors occur in the Navier-Stokes equations, a generalization of the Euler equations that explain viscosity.