Great Country Academician

In the study, Xu Chuan carefully checked the proof process.

After carefully filtering the staged results of the NS equation, it was almost noon.

Originally, he thought about inputting these manuscripts into the computer by himself, but when he saw the thick pile of manuscripts, he was discouraged.

After thinking about it, doesn't he still have students, and it would be better to leave such trivial matters to the students he leads.

Moreover, sorting out the manuscript and inputting it into the computer can also allow them to deeply understand the core of the paper and learn more knowledge points.

Here's to helping them!

Thinking of this, Xu Chuan showed a smile on his face, took out his mobile phone and called the two students.

"Hey, Gu Bing, call Amelia to come to my villa, here is a paper that needs your help to input into the computer."

"By the way, remember to bring your computers."

After hanging up the phone, Xu Chuan began to think again.

The NS equation has advanced to this point. It can be said that there is only the last step left to the conjecture proposed by the Clay Mathematics Institute. He is also thinking about how to take this step.

But for the NS equation, today's mathematical physics community does not have a unified and complete proof idea.

It’s not that everyone is expecting the ‘existence and smoothness of the Navier-Stokes equation’, there are also a large number of mathematicians or physicists who are falsifying it.

That is, they believe that there is no smooth and continuous solution to the NS equation.

This comes from the properties of the fluid.

In transitional flow and turbulent flow, given smooth initial and boundary conditions, at sufficiently high Re, the velocity profile will change and distort during the flow evolution.

After the strict derivation of the NS equation, the velocity of the fluid is discontinuous on the distorted section, that is, a singularity appears (this is the beginning of the transition).

And because the flow variable is not differentiable at the singularity, the NS equation has no solution at the singularity, so the smooth solution of the NS equation on the global domain does not exist.

The school of scholars who believe that there is no smooth continuous solution to the NS equation basically agrees with this idea.

It is a consensus in mathematics that the singularity is unsolvable and impossible.

Verification scholars, however, are different.

They always believed that the solution of the NS equation existed and was continuous and smooth.

And in this row, one of the most famous mathematicians has to be mentioned.

That is the former Hongsu's Kolmogorov, known as "Ke Laoxie" in the mathematics circle, who was an all-rounder in the mathematics circle in the 1990s.

If you have studied modern probability theory, you must be familiar with this name.

If Grothendieck laid the foundation for algebraic geometry, then Kolmogolov laid the foundation for modern probability theory.

But he was not from the Department of Mathematics at the beginning. It is said that he wrote an article related to Newtonian mechanics when he was about 17 years old, so he went to Cosmo to study.

When he entered school, Ke Laoxie, like Edward Witten, was quite fascinated by history at first.

Once, he wrote an excellent article on history. His teacher read it and told him that in history, several or even dozens of correct proofs are needed to prove his point of view.

And Ke Laoxie asked where a proof was needed, and his teacher said it was mathematics, so he started his life in mathematics.

In addition to laying the groundwork for modern probability theory, Kolmogorov’s most dazzling works in his life are the law of thirds in turbulence and scaling ideas.

This achievement has led the development of fluid mechanics for nearly a hundred years. In the long river of development of fluid mechanics, he has written a chapter in the history of modern turbulent development with a magical pen.

This is the famous K41 theory.

The K41 theory holds that no matter how complex a turbulent flow system is, its vortex structure is similar, that is, the kinetic energy of the vortex is always applied to the flow field by an external force and injected into the largest scale (assumed to be L) vortex structure.

Then, the large-scale vortex structure disintegrates one by one and produces small-scale vortices, and at the same time, the kinetic energy is transferred from the large-scale to the small-scale structure step by step, and so on.

But this process will not go on indefinitely. When the scale of the vortex structure is small enough (assumed to be η), the fluid viscosity will take the dominant position, and the kinetic energy will be converted into internal energy and dissipated on this scale, and then will not continue to be transmitted to further Small-scale vortex structures.

This process is called the energy level string process.

This is the most important and basic knowledge point of contemporary fluid mechanics.

Xu Chuan from other schools didn't know about it, but when he was in Nanjing University, this knowledge took up a full ten minutes in the exam.

It can be described as the most important thing.

However, the solution of the NS equation exists and is continuous and smooth, and part of the theory is based on the K41 theory.

This time Xu Chuan pushed the NS equation to an unprecedented height, also using this set of theories.

At present, the K41 theory is also suitable for turbulent flow, but it is not known whether it can kill all directions as it does now when facing the final solution of the NS equation in the future.

After receiving the call, Gu Bing and Amelia rushed over quickly.

"Professor, we are here, please open the door."

In the study, Xu Chuan received a call from Gu Bing, got up and went out to bring the two students in.

"Thank you for making the trip, this is to sort out the papers entered into the computer."

Hearing this, Gu Bing looked at the thesis on the desk, but Amelia did not move. She looked at Xu Chuan excitedly, and asked curiously:

"Professor, you have proved the NS equation?"

Their mentor has been known to have a quirk that, when faced with a problem, hardly leaves the house without addressing him.

And now, apparently, there are results.

Xu Chuan shook his head and said, "Not at all. The NS equation is too difficult to prove at this stage, basically impossible."

As soon as the words fell, Gu Bing's exclamation came from the side: "Professor, have you proved the NS equation?"

Hearing this, Amelia immediately cast a puzzled look at Xu Chuan.

Xu Chuan said that he did not prove the NS equation, so what is the manuscript paper in Gu Bing's hand?

Noticing the puzzled eyes of his students, Xu Chuan shrugged and said, "It's just a phased achievement of the NS equation."

With doubts, Amelia snatched the manuscript paper from Gu Bing's hand in doubt, and her eyes fell on the title.

《Given a finite space, when the initial value is infinitely smooth, the three-dimensional incompressible Navier-Stokes equation smooth solution exists! "

Seeing the title, Amelia's blue pupils suddenly shrank into a small dot, and her eyes were full of disbelief.

This is called "It's just a phased result?"

only?

That's all?

Thinking of the teacher's calm and calm words just now, Amelia really wanted to say something she learned when she was studying here.

Pretend to be b! This is definitely pretending to be b!

With the help of two students, it took Xu Chuan two full days to input the proof process into the computer.

After double-checking carefully and confirming that there was no problem with the paper in the computer, Xu Chuan threw it on the Arxiv preprint website.

"Let's go, I have worked hard for you these two days, I invite you to have a big meal."

After processing the thesis, Xu Chuan patted the shoulders of the two students with a smile and said.

In two days, it is quite tiring to input more than 200 pages of proof papers into the computer with good words, and to check every letter and even punctuation mark.

In two days, the dark circles under the eyes of these two students is the best proof.

When Xu Chuan took two students to have a big meal, Arxiv gradually became lively, and the mathematics world also began to have an undercurrent.

On the arxiv preprint website, many scholars who paid attention to the NS equation, the seven millennium problems, fluid, turbulent flow, Xu Chuan, etc. received recommendations from the website immediately.

Among them, Fefferman, who defined Xu Chuan as a "special attention" label.

When he received the push from the Arxiv website, Fefferman was searching and flipping through the information that would be helpful to the study of NS equations in front of the computer.

Last year, he and Xu Chuan made a breakthrough in the NS equation, which gave him a glimmer of hope to overcome the NS equation.

As slim as it is, Fefferman doesn't want to give up.

This is his lifelong dream.

On the screen, Fefferman was searching for papers in the "Annual Journal of Mathematics". Suddenly, a small bullet box popped up in the lower right corner.

Just as he was about to fork it off, he noticed with keen eyes that this pop-up window came from the Arxiv software.

This made him involuntarily stunned for a moment, and then clicked to open the pop-up window, ready to see what arixv pushed him.

《Given a finite space, when the initial value is infinitely smooth, the three-dimensional incompressible Navier-Stokes equation smooth solution exists! "

Seeing the title displayed after the pop-up window expanded, Fefferman's pupils suddenly shrank.

Proof of the NS equation?

Still advanced to this step!

How can it be? who is it?

And why is such an important proof thrown on arxiv?

Suddenly, Fefferman thought of something, and quickly manipulated the mouse to click on the pop-up window to enter Arxiv. Immediately, the name of the publisher of the paper came into his eyes.

Xu Chuan

The familiar name made him stare at the screen for a long time.

Sure enough, he guessed right!

There is only one person who throws a major conjecture proof paper on Arxiv before journal review, except for the weirdo Perleman.

After recovering, Fefferman quickly downloaded the entire paper and began to read it.

".Start from the compressible navier-stokes equation of thermal conductivity, and then use the harmonic equation as the coefficient to calculate the wall flow. This is indeed a good idea, but the S+R decomposition theory has a fatal How would you solve the problem?"

Fefferman's eyes fell on the paper, and the proof process page by page came into his mind, and he kept chanting.

As a top mathematician in fluid mathematics, he can easily understand the proof process and ideas in this paper.

The proof method in the paper is indeed very clever, but it is not that he has not thought about it.

However, in the end he gave up, because there was a fatal flaw in the S+R decomposition theory that he could not solve.

Not only him, but also mathematicians who have communicated with him are not optimistic about this problem.

But this method is used in this paper, and Fefferman is looking forward to how the other party will solve this problem that many top mathematicians are not optimistic about.

He was thinking, and he kept flipping through the papers in his hands, and as he read, soon, the answer he wanted appeared in front of his eyes.

".By approximating the Fourier function in Fourier space, then approximating it through the function, and then converting it into a wave function describing the momentum p, and then using it"

In the office, Fefferman stared at the manuscript paper in his hand and kept mumbling to himself, his eyes gradually revealed a bright light.

"Wonderful!"

"Collide the vortex ring, use 3D calculation to reconstruct the eddy current into a vortex pattern, and then use the function to solve it, so that the fatal flaw that cannot be approached from 1 to 0 in the S+R decomposition theory can be completely avoided!"

"What a great idea! Simply brilliant!"

After reading the solution process, Fefferman couldn't help but pat his thigh, admiring the thoughts and ideas in the paper.

This is a path that no one has ever thought of. It perfectly integrates physics and mathematics to solve difficult problems that are both important in mathematics and physics.

Looking at the paper in his hand, Fefferman's eyes were full of satisfaction and admiration.

But then, he remembered something again, and there was a trace of sadness in his eyes.

The NS equation is also his challenge to himself in mathematics. This is a peak that no one has ever climbed. Originally, he also had hope, but now someone has reached it first.

However, he still has hope!

Although this proof is quite excellent, in his opinion, it can almost be said that it has proved a large part of the NS equation, but it is not completely completed after all.

The NS equation is still one step away from being able to prove it completely, so, leave the rest of the work to him!

Thinking about it, Fefferman's eyes showed infinite fighting spirit!

He will certainly put the final roof on this mansion!

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