Water rushing through pipes, currents in the ocean, weather patterns, airplanes taking off, blood running in veins, chemicals churning in a mixer, even milk being stirred into coffee: all are governed by a single phenomenon – turbulence. Scientists and mathematicians have been studying turbulence since the legendary 18th-century mathematician Leonhard Euler formulated the first mathematical equations for describing liquid flow in non-linear patterns. This formula was refined in the next century by Claude-Louis Navier and George Gabriel Stokes, who noted that viscosity, even when minimal, influences fluid motion and incorporated terms to reflect its effect. The Navier-Stokes equations, as well as a number of equations derived from them, are still used today in scientific fields ranging from chemical and aerospace engineering to climate modeling.
Nonetheless, these useful equations still hold some puzzles for mathematicians, and the prestigious Clay Institute in Cambridge, Mass., has included the Navier-Stokes equations in a short list of the seven most challenging mathematical problems for the new millenium. Whoever solves any one of them is promised a $1 million prize. The burning question for the Navier-Stokes equations is whether the solutions to them “break” or “blow up” within a finite time period or whether they remain “smooth” ad infinitum. By “breaking,” mathematicians mean that as the solutions progress in time, some of the quantities might become infinite and the mathematical equation would no longer be a valid model for the physical process. In fact, some mathematicians believe that turbulence is linked to an infinite number of solution breaks, which can occur at any moment. However, proof for Navier-Stokes solutions breaking or remaining smooth has never been established – hence the challenge.
Prof. Edriss Titi of the Computer Science and Applied Mathematics Department is fascinated by the ways in which mathematics contributes to solving real-world problems: “Physicists ask: ‘What are the mechanisms that underlie this phenomenon?’ Engineers ask: ‘How can I control this phenomenon?’ But it is mathematicians who study the observations of the other two, develop a proper mathematical framework (and once in a while a completely new theory) and apply rigorous methods, to simplify and answer the others’ questions.”
Titi has shown that several equations, variations on Navier-Stokes that apply to specific patterns of turbulent flow, are able to produce smooth solutions at all times, with no breaks. The first, which he proved while at Cornell University, was for helical flows. As the name implies, in helical flows symmetrical lines of flow swirl around a central vortex. But this kind of flow, says Titi, is not truly three-dimensional. One can treat the flow lines as though they lie on two-dimensional surfaces, and this reduces the complexity of the problem. Titi and his colleagues proved that helical flows are invariant – once they start they’ll go on forever without breaking, according to the formula.
The second set of equations is truly three-dimensional, making it more complex, and Titi’s proof for smoothness at all times in these equations settles a major open problem in the mathematical theory of geophysical fluid dynamics and adds significantly to the under-standing of the field.
The equations were originally formulated in the 1920s, for weather prediction, by Lewis Fry Richardson – one of the first to try modeling the complex movement of air around the globe. In doing so, he had to deal with movement along an enormously wide, thin curving layer of fluid. (If the earth were a very large apple, the atmosphere would be the thin peel.) Taking advantage of the shallow atmosphere, Richardson distorted the Navier-Stokes equations, replacing the equation for vertical motion with the so-called “hydrostatic balance” equation, which is based on a simplified balance between the rate of change in pressure with respect to depth and the buoyancy forces.
“Richardson’s ‘Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics’ ruined the elegant symmetry of Navier-Stokes,” says Titi. Mathematically, these equations appear to be more difficult to approach than the original Navier-Stokes, and it was only last year that mathematicians succeeded in proving they possess smooth solutions for even a short period. So it was quite a surprise when Titi and his student were able to prove that these equations are eternally smooth.
Titi: “The charm of challenging mathematical problems such as Navier-Stokes is that they can be simply formulated and explained, yet they keep us busy trying to solve them for decades, centuries – sometimes millennia.”
From Akko to Rehovot
Prof. Edriss S. Titi was born and raised in an Arab family in the Old City of Akko. His parents barely finished grade school, but they insisted on providing their four children with the best education available, at Akko’s private Franciscan Terra Sancta school. There, with the help of teachers he’s in touch with to this day, Titi excelled in mathematics and physics. In 1974, he began studies at the Technion, in Haifa.
After receiving an M.Sc. in theoretical mathematics from the Technion, Titi switched to applied mathematics, completing his Ph.D. at Indiana University and postdoctoral research at the University of Chicago. He was a lecturer for two years at Cornell University, then moved to the University of California at Irvine, where he became a full professor in 1989. Titi first came to the Weizmann Institute as a visiting scientist in 1999; he joined the Mathematics and Computer Science Faculty in 2003.
The Coin Toss Paradox
Alice and Bob are two fictional characters who love to play games with a mathematical bent. Here, for example, is one game they play: They toss coins and figure their odds of winning. What are their chances, for example, if Bob tosses a coin and Alice has to guess which side is up, and then Alice flips a coin and Bob has to guess, and they play as a team, so that both must be correct to win?
In the coin-toss game, says Dinur, one might think that after both had flipped and guessed they would have a 25% chance of winning the game, since each had a 50% chance of guessing correctly. But there is a clever strategy that Bob and Alice can use: If each guesses that the other has flipped exactly the same as their own coin flip, they raise their odds to 50%.
Now imagine Bob and Alice continue playing the game – they must guess correctly every time to win. Are there more clever tricks they can use to increase their chances of winning? The answer, sadly, is no. As they continue to flip coins, their odds decrease round by round: Mathematical calculations reveal that in all cases, as the game is repeated, the chance of winning quickly approaches zero.
Changing the rules
Alice and Bob’s information was now in the shape of quantum particles; to understand the game, one must understand a few basic principles of quantum mechanics. First, there is superposition: A quantum particle can be in more than one state at a time. But when it is observed or measured, it “collapses” into a single state. So the possible information held by each superimposed particle can be much greater than “yes” or “no.”
Alice and Bob’s particles were also entangled: When two particles are entangled in a quantum setup, they can be placed at a distance from each other, but their states remain in perfect sync, so that any change in the state of one results in an instantaneous change in the state of the other.
Both of these ideas were proposed in the early 20th century, and Einstein was famously opposed to the concept of entanglement, calling it “spooky action at a distance.” A paper he coauthored in 1935 presented the “EPR paradox,” which suggested that, because information cannot travel between the particles faster than the speed of light, there must either be some hidden variables controlling the process, or else the outcome is already “known” before the measurement is performed. Today, entanglement has been proven experimentally, and Einstein’s hidden variables did not pan out. But the paradox remains: How do two particles “share information,” coordinating their states with no time lag?
Dinur and her colleagues showed, mathematically, that a game based on entangled bits of information will eventually follow the pattern of the other games: As the rounds are repeated, the chances of winning will drop significantly. Entanglement may give them some advantage in the beginning, as Bob now has a piece of information about Alice’s knowledge. But, like the coin toss, Alice’s information will still be random: Measuring will collapse her quantum system to a particular state, but she will not be able to control or predict what that state will be. Thus while the odds for each individual round will be better, the pattern will remain the same, moving toward zero as rounds are added to the game. In other words, entanglement would only be a partial cheat, at best, and the EPR paradox is not quite the paradox it seemed.
If quantum communication becomes a reality, Alice and Bob will be able to use it to ensure encryption – for example, to detect any interference in messages sent from one to the other. Although the technology is still far in the future, it will rely on today’s mathematics to set the rules and the limits on its operation.