Mathematicians found the problem in the famous equations for describing liquids

Two mathematicians have proved that under certain extreme conditions, the Navier-Stokes equations give nonsense

The Navier-Stokes equations with the help of several laconic terms describe one of the most common phenomena of the physical world: the flow of fluids. Today, these equations, which appeared back in the 1820s, are used to describe everything, from ocean currents and turbulence that follows an airplane to blood flow in the heart.

Although physicists consider these equations as reliable as a hammer, mathematicians treat them with suspicion. For mathematics, the fact that these equations seem to work does not mean much. They need proof that the equations are infallible: that for any fluid and for a long-term prediction that is extended as far as possible into the future, the mathematics of equations will not fail. This guarantee was not easy to find. The first person or team that can prove that the Navier-Stokes equations will always work — or provide an example proving that they do not work — will be able to receive an award for solving one of the Millennium Problems , announced by the Clay Institute of Mathematics , and a million dollars addition [as of 2017, only one of the seven tasks of the millennium (Poincare conjecture) was solved by Gregory Perelman / approx. trans.].

Mathematicians have developed many ways to solve this problem. The new work, published in September, poses serious questions about whether one of the most popular approaches to the problem, developed over many years, can succeed. The work that Tristan Backmaster and Vlad Vikol of Princeton University wrote is the first result that shows how, under certain conditions, the Navier-Stokes equations give a controversial description of the physical world.

“We are trying to understand the specific problems inherent in these equations, and why people will probably have to rethink them,” says Backmaster.

The work of Backmaster and Vikola shows that if you take very rough assumptions when solving the Navier-Stokes equations, they begin to produce nonsense: they say that the same liquid with the same initial conditions can come in two or more different states. It can flow in one way, or completely different. If so, then these equations cannot reliably describe the physical world for which they were developed.

Blasting equations

To understand how equations can break, imagine an ocean current. Within it, there may be local currents, with the result that some of its parts can move in one direction and at one speed, while others - in another direction with another speed. Local currents interact with each other in the constant mutual action of friction and pressure of water, determining its flow.

Mathematicians model this interaction with a map that tells you about the directions and flow rates at any point in the fluid. This map, called the vector field, is a snapshot of the fluid’s internal dynamics. The Navier-Stokes equations take this snapshot and reproduce it like a video, telling exactly how the vector field will look like at each subsequent point in time.


The wind map (windy.com) works in a way similar to a vector field. At each point the wind has a certain direction and strength.

These equations work. They describe fluid flow as reliably as Newton's equations predict the future positions of the planets; physicists constantly use them, and they constantly coincide with the results of experiments. However, mathematicians need more than episodic confirmation - they need proof that the equations are not violated, that no matter what vector field you start from, and how far into the future you will reproduce it, the equations will always give you new, unique vector field.

This is the topic of the Millennium Problem, asking whether the Navier-Stokes equations have solutions (a solution, in fact, has a vector field) for all starting points at all times. These solutions should provide accurate direction and force of flow at each point of the fluid. Solutions that provide information with such infinitely small resolution are called “smooth”. In a smooth solution, each point of the field has a vector associated with it, which allows you to “smoothly” travel around the field without getting stuck at points where the vector is absent - at a point where further movement from which you will not understand.

Smooth solutions are a complete representation of the physical world, but from a mathematical point of view, they may not always exist. Mathematicians working on equations like this worry about this situation: you run the Navier-Stokes equations and watch for changes in the vector field. After some finite time, the equations tell you that a certain particle of fluid moves at infinite speed. Then you will have problems. The equations include the measurement of changes in properties such as pressure, friction, fluid velocity — in a slang language, they take the derivatives of these quantities — but the derivative of an infinite quantity is not simpler to take than to divide by zero. So if the equations give an infinite value, you can say that they refused you, or “exploded”. They can no longer describe the subsequent states of your fluid.

Such an “explosion” is evidence that the equations lack a description of some properties of the physical world, which they should describe. “It’s possible that the equations do not cover all the effects of a real fluid, because in a real fluid we don’t expect” infinite speed of motion of particles, as Bakmaster says.

The solution to the Millennium Problem is either to show that the Navier-Stokes equations never explode, or to find the conditions under which this happens. One of the strategies used by mathematicians is to soften the requirements for how exactly these equations should describe the required solutions.

Flow violation

The Navier-Stokes equations should describe the flow of any fluid, with any initial conditions, and spread the description infinitely far into the future. Trying to prove this ability of theirs, mathematicians sometimes “weaken”, that is, they use approximate descriptions of vector fields that describe a fluid. But with this there are difficulties.

Ideally, mathematicians want to prove that applying the Navier – Stokes equations to any continuous, “smooth” fluid will produce one unique result.



However, it is easier to work with "weak", not such detailed vector fields. And here mathematicians found that some weak descriptions produce non-unique results — they allow the same fluid to flow in the same initial conditions in two ways.

From weak to smooth

When mathematicians study equations like these, they sometimes begin to broaden the definition of what is considered a solution. Smooth solutions require maximum information - in the case of Navier-Stokes, they need a vector to exist at every point of the vector field associated with the fluid. But what if you weaken the requirements and say that you only need to count vectors for some points of the field, or do you only need to get approximate values ​​of the vectors? Such decisions are called "weak." They allow mathematicians to feel the behavior of the equation without the tedious work of finding absolutely all solutions (which in practice may be impossible).


Tristan Backmaster, a mathematician from Princeton University

“From a certain point of view, weak solutions are even easier to describe than real ones, because you need to know much less,” said Camillo De Lellis, in collaboration with Laszlo Schekelihidi, who wrote several important works that laid the foundation for the work of Backmaster and Vikola.

Weak decisions come in different gradations. If we imagine a smooth solution in the form of a mathematical image of a fluid with infinite resolution, then weak solutions will be something like 32-bit, 16-bit or 8-bit versions of this image.

In 1934, the French mathematician Jean Leray defined an important class of weak solutions. Instead of working with exact vectors, “Leray solutions” take the mean value of the vectors in a small neighborhood of the vector field. Leray proved that you can always solve the Navier-Stokes equations, allowing your solutions to take the form of this form. In other words, Leray’s decisions do not explode.

Leray's achievement has defined a new approach to the Navier-Stokes problem: start with Leray's solutions, the existence of which is already known, and see if they can be turned into smooth solutions whose existence you want to prove. This process resembles the one where you start with a rough picture, and see if it is impossible to gradually tweak the resolution in order to achieve the perfect representation of reality.

“One of the possible strategies is to show that these weak solutions to Lera are smooth, and if you can show that they are smooth, you will solve the Millennium Problem,” said Backmaster.


Vlad Vcol is a half of the team that revealed problems in the approach to the verification of the Navier-Stokes equations.

There is another trick. Solutions of the Navier-Stokes equations correspond to real physical events, and physical events occur in one possible way. Given this, I would like your equations to have only one set of unique solutions. If the equations give you many possible solutions, they do not cope with their task.

Therefore, mathematicians will be able to use Leray's solutions to solve the Millennium Problem only if Leray's solutions are unique. Leray's non-unique solutions will mean that, according to the Navier-Stokes rules, the same fluid with the same initial conditions can come to two different physical states, which has no physical meaning, and implies that the equations do not really describe what should.

The new result of Backmaster and Vikola is the first hint that for certain definitions of weak solutions this is exactly what can happen.

Many worlds

In their new work, Backmaster and Vikol consider even weaker solutions than Leray’s solutions — solutions that use the same averaging principle as Lera’s, but weakened another additional requirement (known as energy inequality). They use the “convex integration” method, originating from the work on the geometry of mathematician John Nash, and later attracted to the study of liquids by De Lellis and Schekelihidi.

Using this approach, Backmaster and Vikol prove that these very weak solutions of the Navier-Stokes equations are non-unique. For example, they demonstrate that if you start with a completely calm liquid, for example, with a glass of water next to the bed, there are two possible developments. The first is obvious: the water begins with a calm state and always remains calm. The second one is fabulous, but mathematically possible: the water starts from a calm state, explodes in the middle of the night, and then returns to a calm state.

“This proves the lack of uniqueness, since at least two objects can be constructed from the initial data,” says Vikol.

Backmaster and Vikol proved the existence of a set of non-unique weak solutions (not just the two described above) of the Navier-Stokes equations. The importance of this proof remains to be understood. At some point, weak solutions may become so weak that they will no longer be associated with smoother solutions that need to be imitated. If so, then the result obtained by Backmaster and Vikol will lead to little.

“Such a result is definitely a warning, but one can argue that this warning concerns the weakest idea of ​​weak solutions. There are many layers of stronger solutions that can be pinned on much better behavior in the case of the Navier – Stokes equations, ”says De Lellis.

Buckmaster and Vikol also think in terms of layers, and he set his sights on Leray's decisions - to prove that they allow for multiple physics in which the same liquid from the same state can come to different forms in the future.

“Tristan and I believe that Leray’s solutions are not unique. We have not yet proved this, but our work lays a springboard for attacking this task, ”said Vikol.