| Today's scientists and engineers are pretty good at predicting the trajectory of hurricanes or calculating how much water will be delivered to your faucet. But the equations for how fluids move continue to flummox mathematicians.
All of fluid motion's unimaginable complexity was captured about 200 years ago in the seemingly simple Navier-Stokes equations. For any initial state of a fluid, these equations can tell you what will happen to it. At least in theory. When meteorologists in the 1960s tried using them to predict the weather, they found that even the tiniest bit of uncertainty about the initial state quickly ballooned, ruining their forecasts. This chaotic behavior, now famously known as the butterfly effect, is the reason why forecasts more than a week out are so unreliable.
For mathematicians, the situation is far worse. They don't even know whether the Navier-Stokes equations always have solutions. Can they effectively describe any fluid's flow? Or is there a certain set of initial conditions that could make the equations "blow up" — that is, explode to infinity? If that happened — if the equations "became singular," as mathematicians like to say — they would no longer be able to model the behavior of the fluid at hand. It would mean that mathematicians are missing something in their theory of how fluids flow.
The Clay Mathematics Institute has offered $1 million to the first person to prove whether or not the Navier-Stokes equations can have singular solutions. (I gave a roundup of the seven "Millenium Prize Problems" in an earlier edition of this newsletter.) A big part of the difficulty is turbulence, where vortices both large and small become hopelessly intertwined, interacting to produce unpredictable but powerful behavior.
Mathematicians have found creative ways to navigate the tempest of mathematical difficulty at the heart of fluid dynamics. While they remain far from achieving a complete understanding of fluids (if such a thing is possible at all), they continue to make intriguing and important progress toward that distant goal.
What's New and Noteworthy
In the early 20th century, mathematicians were stymied by the complexity of fluids. So they decided to pull away from the fluids themselves and instead consider what would happen to something dropped into a fluid's tumult: weather balloons released into a hurricane, rubber ducks dropped in a rushing river, black paint drizzled into a swirling sea of white. They were shocked to find that many of turbulence's difficulties washed away, allowing them to make concrete, reproducible predictions about how the fluid would influence these objects' trajectories.
One of these predictions, called Batchelor's law, deals with the sizes of the swirls that appear in the paint-mixing example. It predicts how the number of small-scale features will compare to the number of larger-scale features — a ratio that continues to hold as you zoom further outward and consider increasingly large swirls. Mathematicians proved the law for a vast number of turbulent systems in 2019, as Kevin Hartnett reported for Quanta.
More recently, mathematicians studying a simplified model of a fluid proved another hallmark of turbulence. In 1926, a scientist named Lewis Fry Richardson observed that the aforementioned balloons and rubber ducks get scattered by turbulence with stunning efficiency. Physicists later confirmed that this phenomenon, now known as superdiffusion, does indeed occur. But mathematicians couldn't rigorously prove it until last year, as I described last month.
Mathematicians have also approached the Navier-Stokes equations more directly, often studying their foundations under some simplifying assumptions. One classic simplification involves removing friction from Navier-Stokes; these simplified equations are called the Euler equations. In 2022, two mathematicians proved — relying heavily on computers — that in a specific setting, the Euler equations can blow up. This marked a small but significant step toward the full Millennium Problem.
Even the most straightforward aspects of fluids are hard to prove mathematically. Depending on what you want to know about a fluid, you might model it in different ways. At the microscopic level, for instance, a fluid is made up of individual molecules that move around like billiard balls. But at a macroscopic level, the fluid acts as a single entity. (The Navier-Stokes equations describe this macroscopic behavior.) In 1900, David Hilbert challenged mathematicians to prove that these different models are compatible. For 125 years, no one could. But now, three mathematicians claim to have done just that. If the proof holds up, it will mark a major advance in the history of mathematical physics. Leila Sloman's detailed account of the groundbreaking proof ran last week in Quanta.
Despite the notorious difficulty of fluid mechanics — some consider it "the place where careers go to die" — the field is intensely active. Our greatest minds are continuing to probe the mysteries of fluids' roiling depths. | 
