Chapter 320: Existence! And smooth!
Institute for Advanced Study, Princeton, USA.
Professor Fefferman was communicating with him in the office about the mathematics of partial differential equations.
In recent months, in order to study the last step of the NS equation, he has been working hard for the second time in his old age. Not only has he been studying the NS equation every day, but he has also turned down most of his work.
It can be said that we are determined to win the last step of the NS equation.
The two were communicating when suddenly, Deligne's cell phone on the table vibrated. He subconsciously picked it up and looked at it, and his brown-green pupils shrank slightly.
Then, without hesitation, he unlocked his phone 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 his friend's character very well. Unless he encountered something important, he would not be able to leave the person he was communicating with and look at something else.
Deligne did not reply immediately. He read through the message in his hand before slowly raising his head and looking at Fefferman, with a trace of hesitation and pity in his eyes.
"Perhaps, you have no chance."
"Why no chance?" Fefferman looked confused. He had no idea what Deligne was talking about.
"NS equation."
Fefferman: "????"
Deligne hesitated for a moment before forwarding the message on his phone to him.
"I sent you the message, you better read it."
With a question mark on his face, Fefferman took his hand out of his pocket and unlocked the screen.
The first thing that caught his eye was the message sent to him by Deligne.
"Professor Xu Chuan hit the last step of the NS equation in the NTU classroom, and may have solved this millennium problem!"
The title of the message made Fefferman's heart suddenly stop. With a look of disbelief in his eyes, he quickly opened 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 have no chance."
Deligne shrugged and said nothing.
Based on his understanding of his student, if he formally starts to study a certain problem, he will probably not stop until he succeeds.
Judging from the calculations on the pictures attached to the message, I'm afraid he already has some ideas on how to solve the NS equation.
Perhaps, after some time, they will be able to see the NS equation completely solved.
This is of great significance to the world of mathematics, physics, and industry.
To be honest, he was looking forward to it!
It's just a pity for his good friend.
Ever since he started cooperating with Xu Chuan to study NS equations, he has always been a step behind, from two phased results to the final step today.
If the opponent were someone else, his friend might still be able to fight.
But when I met his student
Thinking about it, Deligne couldn't help but shook his head.
Perhaps Fefferman would still have a chance to compete if he were thirty or forty years younger, but now, I'm afraid he has no chance.
On the other side, Huaguo, Jinling.
Xu Chuan ignored the news on the Internet. Even if some media reporters wanted to interview him, they were stopped by Zheng Hai.
After returning from the classroom, he locked himself in the study room and began to study the last step of the NS equation.
To be honest, he never thought that the study of NS equations would come so soon.
Because before that, he had almost reached the end of using Kolmogorov's K4 theory to prove the staged results of the NS equation.
When the viscosity coefficient ν approaches zero, does the solution to the initial boundary value problem of the Navier-Stokes equation tend to the corresponding solution of the ideal fluid inside the fluid motion region, the characterization of the fluid boundary layer problem, and the three-dimensional infinite In space, the fluid flow speed is getting faster and faster, and the speed tends to infinity. The final problem is that it exceeds the common sense in reality.
This step is both the last and the hardest part.
Before the correct answer is found, whether a smooth solution to the three-dimensional incompressible Navier-Stokes equation exists is still a mystery. No one knows whether the divergence of turbulent flow will eventually calm down. Otherwise, when Fefferman invited him, he wouldn't have refused outright.
But Xu Chuan didn't expect that after only five or six months, new inspiration and new paths would come so quickly.
A basic mathematics class gave him a new way of thinking.
If each fluid-emitting microfluidic unit is regarded as a mathematical value, then using microfluid mathematics he can construct a set that contains these numbers.
In the Poincaré conjecture or Poincaré's 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 bounded three-dimensional space; and simple connectivity means that every closed curve in this space can continuously shrink to a point.
In other words, in a closed three-dimensional space, if every closed curve can be contracted to a point, this space must be a three-dimensional sphere.
Using micro-element fluids, he built a mathematical tool that included all the fluid diffusion in the NS equations. He then used Ricci manifolds to expand the fluid topology, construct geometric structures, and transform them from irregular manifolds into Regular manifold.
This path spans the most basic micro-element fluids, complex diffusion fluids, and ultimate turbulent fluids, and finally successfully builds a brand-new mathematical tool.
A brand new road and a brand new tool were his answer to the last step of the NS equation.
This is completely different from using mathematics and practical physics to climb the NS equation.
This time, he took the path of pure mathematics.
After a long and winding climb, we returned to the starting point.
However, there is no fixed solution when faced with the seven millennium problems that challenge the pinnacle of the human mind, such as the NS equation.
Although mathematics was often used as a tool to solve physical problems in the past, no one has ever stipulated that physics cannot be used as a tool to solve mathematical problems.
For this kind of problem that stands at the pinnacle of mankind, as long as we can take a step forward, even if it is one centimeter or one millimeter, no matter what method is used, it is worth it.
In the study, Xu Chuan looked at the manuscript paper on the desk.
The tools to cross the abyss are already there, all that’s left is to complete the summit.
If we compare the NS equation to a towering snow peak, he had climbed halfway up the mountain before. But it was blocked by an abyss crack.
The tools he originally used to climb the snowy peak were not enough to support him in crossing this bottomless abyss, but now, after circling halfway up the mountain, he miraculously found it in the mountain col. A forest.
Cut down trees, build bridges, and cross the abyss bit by bit.
The mathematical tools derived from micro-element fluids were the bridge for him to conquer the last step of the NS equation.
With the help of this tool, he can finally move forward towards the summit.
After sorting out the manuscript papers on the desk, he took out a new stack of A4 paper from the drawer and laid it flat in front of him.
He picked up the pen and wrote the last title on the manuscript paper.
[Proof of the existence and smoothness of solutions to the three-dimensional incompressible Navier-Stokes equation! 】
It’s time to head towards the final summit!
I don’t know how much time has passed, but time seems to have been suspended in this small study.
For Xu Chuan, the pen in his hand has never stopped since he wrote that title.
Finally, when the last line quietly appeared on the white manuscript paper, a satisfied smile appeared on his lips.
It's time to draw final conclusions.
With a smile, Xu Chuan gently moved his palm to lower the pen tip in his hand.
[.When the viscosity coefficient ν approaches zero, the solution to the initial boundary value problem of the Navier-Stokes equation tends to the corresponding ideal fluid state inside the fluid motion region. That is, there is an initial boundary value solution to the Euler equation! 】
[To sum up all the above inferences, we can easily know that in the three-dimensional incompressible Navier-Stokes equation, the solution exists! And smooth! 】
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(End of chapter)