Great Country Academician

United States, Princeton Institute for Advanced Study.

Professor Fefferman de Ligne communicated with him in the office about the mathematics of partial differential equations.

In the past few months, he has been working hard for the last step of his research on the NS equation. He has been studying the NS equation every day, not to mention, and has turned away most of the work in his hands.

It can be said that the last step of the NS equation is determined.

The two were talking, when suddenly, Deligne's cell phone on the table vibrated, he picked it up subconsciously and looked at it, his brown-green pupils shrank slightly.

Immediately afterwards, he unlocked the phone without hesitation and clicked on the message.

On the opposite side, Fefferman stopped talking, looked at his friend curiously, and asked, "What's wrong? What happened?"

He knew the character of this friend very well, unless something important happened to him, he would not be able to cast aside the self who was communicating to look at other things.

Deligne didn't reply right away. After going through the news in his hand, he slowly raised his head and looked at Fefferman with a trace of hesitation and pity in his eyes.

"Perhaps, you have no chance."

"Why is there no chance?" Fefferman looked confused, he didn't understand what Deligne was talking about at all.

"NS equation."

Fefferman: "????"

Deligne hesitated, but forwarded the message on the phone to him.

"I sent you the message, you should take a look."

Fefferman took out his hand from his pocket with a question mark on his face, and unlocked the screen.

The first thing that caught his eye was the message Deligne sent him.

"Professor Xu Chuan hit the last step of the NS equation in the classroom of Nantah University, and may have solved this millennium problem!"

The title of the message made Fefferman's heartbeat stop suddenly, with disbelief in his eyes, he quickly clicked on the message and entered the details.

After a long time passed, Fefferman raised his head and looked at his friend with a complicated expression.

"Maybe, I really don't have a chance."

Deligne shrugged and said nothing.

Based on his understanding of his student, if he formally started to study a certain problem, he might not give up without success.

Judging from the formulas on the pictures attached to the news, I am afraid that he already has a certain idea of ​​how to solve the NS equation.

Perhaps, after a while, they will be able to see the NS equation completely solved.

This is of great significance to the fields of mathematics, physics, and industry.

Honestly, he's looking forward to it!

However, it is a pity that his friend is gone.

Since the beginning of cooperation with Xu Chuan on NS equations, he has always been a step behind, from the two phased results to the last step now.

If the opponent was someone else, his friend might still be able to fight.

but met his student

Thinking about it, Deligne couldn't help but shook his head.

Perhaps Fefferman still has a chance to fight when he is thirty or forty years younger, but now, I am afraid that he has no chance.

On the other side, Huaguo, Jinling.

Xu Chuan didn't pay attention to the news on the Internet, even if some media reporters wanted to interview him, they were stopped by Zheng Hai.

Since he came back from the classroom, he locked himself in the study and began to study the last step of the NS equation with all his strength.

To be honest, he never thought that the study of NS equations would come so soon.

Because before that, he had almost come to the end of using Kolmogorov's K4 theory to prove the staged results of the NS equation.

When the viscosity coefficient ν tends to zero, whether the solution of the initial boundary value problem of the Navier-Stokes equation tends to the solution of the corresponding ideal fluid in the fluid motion area, how to describe the fluid boundary layer problem, and in the three-dimensional infinite In space, the fluid velocity is getting faster and faster, and then the velocity tends to infinity, which is beyond the common sense in reality and is the final problem.

This step is both the last and the hardest part.

Before the correct answer is found, whether the smooth solution of the three-dimensional incompressible Navier-Stokes equation exists is still a mystery, and no one knows whether the divergence of turbulent flow will eventually return to calm.

Otherwise, when Fefferman invited him, he would not have refused directly.

It's just that Xu Chuan didn't expect that after only five or six months, new inspirations and paths would come so quickly.

A basic mathematics class gave him a new way of thinking.

If it is said that each fluid emission microfluidic unit is regarded as a mathematical value, then he can construct a set to accommodate these numbers by using the microfluidic mathematics.

In the Poincaré conjecture or Poincaré theorem, any simply connected, closed three-dimensional manifold must be homeomorphic to a three-dimensional sphere.

Simply put, a closed three-dimensional manifold is a three-dimensional space with boundaries; and simple connectivity means that every closed curve in this space can be continuously shrunk to a point.

Or in a closed three-dimensional space, if every closed curve can shrink into a point, this space must be a three-dimensional sphere.

Using the micro-fluid, he built a mathematical tool that included all the fluid diffusion in the NS equation in the set, and then used the Ricci manifold to expand the fluid topology and construct the geometric structure, changing it from an irregular manifold to a regular manifold.

This path has spanned the most basic micro-fluids, complex diffusion fluids, and ultimate turbulent fluids, and finally successfully constructed a brand-new mathematical tool.

A brand-new path and a brand-new tool are the answer sheet he handed over in the last step of the NS equation.

This is completely different from the previous use of mathematics and practical physics to climb the NS equation.

This time, he took the path of pure mathematics.

After climbing for a long time, I returned to the original point.

However, there is no fixed solution to the seven millennium problems that challenge the pinnacle of the human mind, such as the NS equation.

Although in the past, mathematics was usually used as a tool to solve physical problems, but no one has ever stipulated that physics cannot be used as a tool to solve mathematical problems.

For this kind of problem standing at the pinnacle of human beings, as long as you can take a step forward, even if it is a centimeter by a millimeter, no matter what method you use, it is worthwhile.

In the study, Xu Chuan looked at the manuscript paper on the desk.

The tools to cross the abyss are already available, and the rest is to complete the summit.

If we compare the NS equation to a towering snow peak, he has already climbed halfway up the mountain before that. But it was blocked by an abyss crack.

And the tools he used to climb the snow peak were not enough to support him to cross this bottomless abyss, but now, after he circled halfway up the mountain, he miraculously found it in the col. a forest.

Logging, building bridges, stepping across the abyss little by little.

The mathematical tools derived from micro-fluids are the bridge for him to conquer the last step of the NS equation.

With the help of this tool, he can finally move on towards the summit.

After tidying up the manuscript paper on the desk, he pulled out a new stack of A4 paper from the drawer and laid it flat in front of him.

He picked up his pen and wrote the last title on the manuscript paper.

[A proof of the existence and smoothness of the solution of the three-dimensional incompressible Navier-Stokes equation! 】

Time to move towards the final summit!

I don't know how long it has passed, time seems to be suspended in this small study.

For Xu Chuan, the pen in his hand has never stopped since he wrote the title.

Finally, when the last line quietly appeared on the white manuscript paper, a satisfied smile appeared on his lips.

It's time for the final conclusion.

With a smile on his face, Xu Chuan moved his palm lightly, letting the tip of the pen in his hand drop down a space.

【.When the viscosity coefficient ν tends to zero, the solution of the initial boundary value problem of the Navier-Stokes equation tends to the corresponding ideal fluid state in the interior of the fluid motion region. That is, there is an initial boundary value solution to the Euler equation! 】

[To sum up all the inferences above, we can easily know that in the three-dimensional incompressible Navier-Stokes equation, the solution exists! And smooth! 】

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