A New Take on the Navier–Stokes Equations

Benjamin Skuse

The Navier–Stokes equations have governed our understanding of how fluids such as water and air flow for nearly 200 years. Today, they are ubiquitous in science and society, used for a host of applications from modelling weather, ocean currents and blood flow, to designing aircraft, vehicles and power stations. Developed by French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes between 1822 and 1850, these partial differential equations very accurately describe the motion of viscous fluids.

But despite their widespread success and use, mathematically these equations have one glaring flaw. From smooth initial conditions in three dimensions, it is not clear whether they converge to sensible solutions, converge to gibberish, or even converge at all. Do they always adhere to reality or is there a discrepancy between the Navier–Stokes equations and the real physical world?

A Millennium Prize Problem

This head-scratcher is known as the Navier–Stokes existence and smoothness problem, and is such an important challenge that it has been recognised as one of Clay Mathematics Institute’s Millennium Prize Problems, the seven most important open problems in mathematics. The mathematician who provides a solution to any of these problems will be offered $1 million.

Yet, after almost a quarter of a century of effort since the Millennium Prize Problems were posed, only one has been solved: the Poincaré conjecture by Grigori Perelman in 2010 (though he refused the cash prize). For the others, including the Navier–Stokes existence and smoothness problem, purported proofs pop up on a regular basis from amateurs and experts alike, but so far each and every one has been shown to possess a fatal error. As a consequence, progress has been slow.

For his lecture at the 11th Heidelberg Laureate Forum on Tuesday, 24 September, entitled “Three-dimensional Fluid Motion and Long Compositions of Three-dimensional Volume Preserving Mappings,” Dennis Sullivan (Abel Prize – 2022) wanted to discuss this problem, which has, like a magnet, repeatedly attracted his attention over the course of more than 30 years.

“When I was an undergraduate in the State of Texas where I grew up, I worked summers in the oil industry,” he recalled during his lecture. “And they used this model to increase the production of oil, and it worked perfectly.” But much later, Sullivan discovered how shaky the foundations were for these equations, that although the Navier–Stokes existence and smoothness problem had been solved in two dimensions based on the Riemann mapping theorem, the problem in three dimensions remained unresolved: “When I heard this question in the early 90s, I was quite surprised … by this lack of knowledge.”

Space and Number

Sullivan’s background is far removed from the Navier–Stokes problem. He received the 2022 Abel Prize “for his ground-breaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects.” At a very fundamental level, his work has always reduced problems to two basic building blocks: space and number. In his own words, from a short 2022 Abel Prize interview: “I always look for those elements in any math discussion, what’s the spatial aspect, or what’s the quantitative aspect of numbers?”

This approach began to pay dividends early in his career when working on surgery theory. Surgery theory applies geometric topology techniques to produce one finite-dimensional manifold from another in a ‘controlled’ way. A manifold is a shape that is the same everywhere; no end points, edge points, crossing points or branching points.

For shapes made from one-dimensional strings, for example, the letter ‘o’ is a manifold, but ‘a’ and ‘z’ are not. For shapes made from two-dimensional sheets, a sphere and a torus are manifolds, but a square is not. Surgery theory comes into play at a higher and more abstract level, for manifolds of dimension five and over. Sullivan’s input helped provide a full picture of what manifolds there are in five and more dimensions, and how they behave.

He later made key contributions to a wide variety of topics, not least with his fellow mathematician and wife Moira Chas, also attending and contributing to the 11th Heidelberg Laureate Forum, who together developed the field of string topology in the late 1990s. String topology can be defined as a certain set of operations on the homology of a manifold’s free loop space, i.e. the space of all maps from a circle into the manifold. Not only is this field interesting from a mathematical perspective, it has also been applied to advance topological quantum field theories in physics.

It All Starts with Euler

Given his most important contributions did not relate to fluid flow, much less the Navier–Stokes equation specifically, Sullivan wanted to approach the problem sensibly, first asking: what makes solving it so difficult? Why is it so much harder to understand equations that can model the flow of water in a garden hose than, say, Einstein’s field equations?

To understand why the Navier–Stokes existence and smoothness problem is so difficult to solve, Sullivan turned to the related Euler equation. Over 250 years ago, Swiss polymath Leonhard Euler formulated equations describing the flow of an ideal, incompressible fluid. “When there’s no friction or diffusion term, it’s called the Euler equation, and this is a special case of the whole problem,” Sullivan said. “The Euler equation simply says that vorticity (a mathematical object whose nature needs to be discussed) is transported by the fluid motion.”

In effect, the Euler equation represents a type of flow involving vorticity where a vector field rotates while being transported along the flow line in physical space. “I liked this idea of transport of structure,” said Sullivan. Perhaps the Navier–Stokes equations could be posed along similar lines, reformulating the problem to make it easier to solve, he thought.

In the conventional formulation, the Navier–Stokes equations describe how an initial velocity field representing the fluid, which specifies the speed and direction of flow for each point in 3D space, evolves over time. This description leaves the possibility open that after some time, the velocity fields could abruptly and unphysically change from one point to another, generating sharp spikes skyrocketing to infinite speed, for example. This situation is known as ‘blow-up’, where the equations completely break down.

A New Approach 

Sullivan instead replaces velocity as an innate property of the fluid with vorticity. He argued that vorticity twists the fluid at every point, giving it rigidity in an analogous way to how angular momentum provides the stability that keeps a bicycle from falling over. This rigidity, or resistance to deformation, allows the fluid to be thought of as an elastic medium, with motion deforming this elasticity. In the physical three-dimensional case, vorticity can be thought of as a vector field, pointing in a different direction to the velocity field, which points in the direction of motion.

“The idea is to think of the fluid as an elastic medium, with the vorticity giving it its structure, and then in the theory of elasticity, study the Jacobian of the motion,” he explained. “This gives you a new tool for deriving any qualities related to this discussion, and that’s what I’m working on now.”

Sullivan’s approach provides hope that a proof can be derived revealing that the solutions to the Navier–Stokes equations always remain smooth and well-behaved, and therefore always accurately represent real-world fluid flow. But success is far from guaranteed, and many others including the likes of 2006 Fields Medallist Terence Tao are devising ingenious methods to prove the opposite: that the Navier–Stokes equations do not fully capture real-world fluid flow.

Whatever the outcome, attacking the problem from very different directions using innovative methods will no doubt lead to interesting mathematics, and perhaps even a deeper understanding of the very basic but important physical phenomenon of how a fluid flows. 

You can view Sullivan’s entire lecture from the 11th Heidelberg Laureate Forum in the video below.

The post A New Take on the Navier–Stokes Equations originally appeared on the HLFF SciLogs blog.