Calculating and modeling the motion of fluids has been done by mathematicians for hundreds of years. Researchers have been aided by the equations that describe how ripples crease the surface of a pond. When written in the right language, these equations are easy to understand. Making sense of basic questions about their solutions can be hard.
The oldest and most prominent of these equations was formulated more than 250 years ago. According to a mathematician at Duke University, almost all of the fluid equations are derived from the equations in the book. You could say they're the first ones.
Whether the equations are always an accurate model of ideal fluid flow is one of the unknowns. One of the biggest problems in fluid dynamics is figuring out if the equations ever fail, outputting nonsensical values that make it impossible to predict a fluid's future states.
There are initial conditions that cause the equations to break down, according to mathematicians. They haven't been able to show it.
A pair of mathematicians showed in a preprint that a particular version of the equations sometimes fails. While the proof doesn't completely solve the problem for the general version of the equations, it offers hope that a solution is within reach. The mathematician who was not involved in the work said it was an amazing result. In the literature, there are no results of its kind.
There's only one thing left to do.
Significant use of computers is made by the 177-page proof. It's hard for other mathematicians to verify it. Many experts think the new work will be correct, but they are still in the process of doing so. They have to think about what a proof is and what it will mean if computers are the only way to solve important questions in the future.
If you know the location of the particles in the fluid, the equations should be able to predict how the fluid will change over time. They want to know if that is true. In some cases, the equations will produce precise values for the state of the fluid at any given moment, only for one of those values to suddenly increase in value, and cause the fluid to go to zero. The equations are said to give rise to a "singularity" at that time.
The equations won't be able to calculate the fluid's flow once they hit that point. Charlie Fefferman said that what people were able to do fell far short ofproving blowup.
It gets more difficult if you want to model a fluid that has a certain amount of elasticity. The Clay Mathematics Institute is offering a million-dollar prize if anyone can prove that the same failures happen in the same equations.
Thomas Hou, a mathematician at the California Institute of Technology, and Guo Luo, a mathematician at the Hang Seng University of Hong Kong, proposed a scenario in which the Euler equations would lead to a singular existence. The top half of a cylinder swirled clockwise while the bottom half swirled counterclockwise. The simulations started to show more complicated currents. It led to strange behavior along the boundary of the cylinder. The fluid was growing so fast that it appeared to be about to blow up.
Their work was not a true proof. It's not possible for a computer to calculate infinite values. The solution might be very accurate, but it is still an approximation since it can only get very close to a singularity. The value of the vorticity might only be increasing because of an artifact of the simulation. It is possible that the solutions will grow before subsiding.
Simulations used to show that a value in the equations blew up, only for more advanced methods to show otherwise. The road is littered with the wrecks of previous simulations. Hou got his start in this area by proving the existence of hypothetical singularities.
When he and Luo published their solution, most mathematicians thought it was a real thing. Sverak is a mathematician at the University of Minnesota. They tried to establish that this is a real scenario. The work done by Sverak and others strengthened that conviction.
A proof was hard to come by. Fefferman said that he had seen the beast. You attempt to take it. It was necessary to show that Hou and Luo's simulation of the solution is very close to the solution of the equations.
Nine years after that first glimpse, Hou and his former graduate student have finally been able to prove the existence of that area.
Hou and Chen were able to take advantage of the fact that the approximate solution from the previous year seemed to have a special structure. The solution displayed a self-similar pattern as the equations evolved, only re-scaled in a specific way.
The mathematicians didn't need to look at the thing themselves. They could look at it from an earlier point in time. They could model what would happen later on by zooming in on that part of the solution.
It took a few years for them to come up with a similar scenario. A group of mathematicians, including Buckmaster, used different methods to find a similar approximate solution. They are using that solution to create a proof of singularity formation.
Hou and Chen needed to show that an exact solution existed nearby. There is no way to escape a small neighborhood around the approximate solution if you were to slightly perturb it and then evolve it. Hou described it as a black hole. You will be sucked in if you begin with a profile close by.
A general strategy was only the first step. Fefferman said that ussy details mattered. After several years of working out the details, Hou and Chen discovered that they had to rely on computers again.
Their first challenge was figuring out what they had to prove. The output wouldn't be able to stray far if any set of values were plugged into the equations. Is it possible for an input to be close to the solution? There are many ways to define the notion of distance in this context. They had to choose the correct one.
De la Llave is a mathematician at the Georgia Institute of Technology. A deep understanding of the problem is required for it to be chosen.
Hou and Chen had to prove the statement because it boiled down to a complicated inequality involving terms from both the re-scaled equations and the approximate solution. The values of all those terms had to be balanced out so that the other values were not large.
The whole thing can break down if you make something too big or small. It is very delicate work.
The fight is a fierce one.
The inequality was broken into two major parts by Hou and Chen. They were able to take care of the first part by hand, using techniques that date back to the 18th century. It was striking thatHou and Chen used it for this.
The first part of the disparity was left. To tackle it, computer assistance is required. The amount of work you'd have to do with pencil and paper would be staggering. To get various terms to balance out, the mathematicians had to perform a series of Optimization problems that are very time consuming for humans. Since the approximate solution was calculated using a computer, it's easier to perform additional computations on top of that.
Gmez-Serrano said that if you try to manually do some of the estimates you will probably lose. The numbers are very small and the margin is very small.
Computers can't manipulate an infinite number of digits. Hou and Chen had to make sure they didn't interfere with the rest of the balancing act.
They were able to find bounds for all the terms and complete the proof.
The existence of a cylindrical boundary and the Navier-Stokes equations can make the equations more complex. This work gives me hope. I think there's a way to resolve the full Millennium problem.
Buckmaster and Gmez-Serrano are working on a computer-assisted proof of their own that they hope will be more general and able to tackle many other problems.
The use of computers is growing in the field of fluid dynamics.
Susan Friedlander is a mathematician at the University of Southern California.
Computer-assisted proof is new in fluid mechanics. The first computer-assisted proof that tackled toy problems in the area was by Hou and Chen.
Peter was of the opinion that such proof are not so controversial as a matter of taste. A proof is needed to convince other mathematicians that a line of reasoning is valid. It should improve their understanding of why a particular statement is true, rather than just provide validation that it's correct. Do we know the answer to the question or do we learn something new? The person said, "ELEVENDI said." If you think mathematics is an art, then this is not so nice.
A computer is capable of helping. It's great. I get insight from it. It doesn't give me a complete understanding. Understanding can come from us.
He wants to work out an alternative proof of blowup by hand. He said he was happy that Hou and Chen's work existed. I think it's more of a motivation to do it in a less computer dependent way.
Computers are seen as a vital new tool by other mathematicians. The work is not just paper and pencil anymore. It's possible to use something more powerful.
According to him and others, there is a good chance that the only way to solve big problems in fluid dynamics is with computer help. Trying to do this without using computer-assisted proof is like tying one or two hands behind your back, according to Fefferman.
If that is the case and you don't have any choice, then people like myself should be quiet. The skills needed to write computer-assisted proof would need to be learned by more mathematicians. Buckmaster said that a lot of people were waiting for someone to solve the problem before they invested their own time into the approach.
It isn't that you need to pick a side when it comes to debates about the extent to which mathematicians should use computers. The proof would not work without the analysis and the computer assistance.
De la Llave said there was a new game in town.
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