Mathematicians Second-Guess Centuries-Old Fluid Equations

THE NAVIER-STOKES EQUATIONS capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820s, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart.

While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work—or to provide an example where they don’t—stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward.

Mathematicians have developed many ways of trying to solve the problem. New work posted online in September raises serious questions about whether one of the main approaches pursued over the years will succeed. The paper, by Tristan Buckmaster and Vlad Vicol of Princeton University, is the first result to find that under certain assumptions, the Navier-Stokes equations provide inconsistent descriptions of the physical world.

“We’re figuring out some of the inherent issues with these equations and why it’s quite possible [that] people have to rethink them,” said Buckmaster.

Buckmaster and Vicol’s work shows that when you allow solutions to the Navier-Stokes equations to be very rough (like a sketch rather than a photograph), the equations start to output nonsense: They say that the same fluid, from the same starting conditions, could end up in two (or more) very different states. It could flow one way or a completely different way. If that were the case, then the equations don’t reliably reflect the physical world they were designed to describe.

Blowing Up the Equations
To see how the equations can break down, first imagine the flow of an ocean current. Within it there may be a multitude of crosscurrents, with some parts moving in one direction at one speed and other areas moving in other directions at other speeds. These crosscurrents interact with one another in a continually evolving interplay of friction and water pressure that determines how the fluid flows.

Mathematicians model that interplay using a map that tells you the direction and magnitude of the current at every position in the fluid. This map, which is called a vector field, is a snapshot of the internal dynamics of a fluid. The Navier-Stokes equations take that snapshot and play it forward, telling you exactly what the vector field will look like at every subsequent moment in time.

The equations work. They describe fluid flows as reliably as Newton’s equations predict the future positions of the planets; physicists employ them all the time, and they’ve consistently matched experimental results. Mathematicians, however, want more than anecdotal confirmation—they want proof that the equations are inviolate, that no matter what vector field you start with, and no matter how far into the future you play it, the equations always give you a unique new vector field.

This is the subject of the Millennium Prize problem, which asks whether the Navier-Stokes equations have solutions (where solutions are in essence a vector field) for all starting points for all moments in time. These solutions have to provide the exact direction and magnitude of the current at every point in the fluid. Solutions that provide information at such infinitely fine resolution are called “smooth” solutions. With a smooth solution, every point in the field has an associated vector that allows you to travel “smoothly” over the field without ever getting stuck at a point that has no vector—a point from which you don’t know where to move next.

Smooth solutions are a complete representation of the physical world, but mathematically speaking, they may not always exist. Mathematicians who work on equations like Navier-Stokes worry about this kind of scenario: You’re running the Navier-Stokes equations and observing how a vector field changes. After some finite amount of time, the equations tell you a particle in the fluid is moving infinitely fast. That would be a problem. The equations involve measuring changes in properties like pressure, friction, and velocity in the fluid — in the jargon, they take “derivatives” of these quantities — but you can’t take the derivative of an infinite value any more than you can divide by zero. So if the equations produce an infinite value, you can say they’ve broken down, or “blown up.” They can no longer describe subsequent states of your fluid.

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