Chapter 274 The wind blowing from the world of mathematics to the world of physics
In the study, Xu Chuan carefully inspected the proof process.
After carefully filtering the phased results of the NS equation, it was almost noon.
He originally thought about inputting these manuscripts into the computer by himself, but when he saw the thick stack of manuscripts, he gave up.
Then I thought about it, didn't he still have students? He could just leave such trivial matters to the students he led.
Moreover, organizing 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.
This is a help to 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. There is a paper here that I need your help to input into the computer."
"By the way, remember to bring your computer."
After hanging up the phone, Xu Chuan thought again.
With the NS equation advanced to this point, it can be said that there is only one 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 looking forward to 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 value conditions and boundary conditions, at a sufficiently high Re, the velocity profile will change and be distorted during the flow evolution process.
After strict derivation of the NS equation, the velocity of the fluid is discontinuous on the distorted profile, that is, a singular point appears (this is the beginning of the transition).
And because the flow variables are non-differentiable at singular points, the NS equation has no solution at the singular point, so the smooth solution of the NS equation in the global domain does not exist.
Most of the scholars who believe that there is no smooth and continuous solution to the NS equation basically agree with this concept.
It is a consensus in mathematics that singular points cannot be solved and cannot be breezed.
However, confirmation scholars are different.
They always believe that the solution to the NS equation exists and is continuous and smooth.
In this row, one of the most famous mathematicians has to be mentioned.
That is Kolmogorov of the former Red Soviet Union, known as ‘Ke Laoxie’ in the mathematics community, and a versatile genius in the mathematics community in the 1990s.
If you have studied modern probability theory, you will definitely be familiar with this name.
If Grothendieck laid the foundation for algebraic geometry, then Kolmogorov laid the groundwork for modern probability theory.
But he was not in the mathematics department at the beginning. It is said that when he was about 17 years old, he wrote an article related to Newtonian mechanics, so he went to Cosmo to study.
When he entered school, Ke Laoxie, like Edward Witten, was initially fascinated by history.
Once, he wrote an excellent historical article. After reading it, his teacher told him that in history, you need several or even dozens of correct proofs to prove your point of view.
Ke Laoxie just asked where a proof was needed, and his teacher said it was mathematics, so he began his mathematical life.
In addition to laying the foundation for modern probability theory, the most dazzling work in Kolmogorov's life is the law of thirds of turbulent flow and the idea of scaling.
This achievement has led the development of fluid mechanics in the past hundred years. In the long history of the development of fluid mechanics, he wrote a colorful chapter in the history of the development of modern turbulence with a magical pen.
This is the famous K41 theory.
K41 theory believes that no matter how complex a turbulent system is, its vortex structure is similar, that is, the kinetic energy of the vortex is always applied to the flow field by external forces and injected into the vortex structure of the largest scale (assumed to be L).
Then, the large-scale vortex structure gradually disintegrates and generates small-scale vortices, and at the same time, the kinetic energy is gradually transferred from the large-scale to the small-scale structure, and so on.
However, this process will not continue indefinitely. When the scale of the vortex structure is small enough (assumed to be eta), the fluid viscosity will dominate, and the kinetic energy will be converted into internal energy and dissipated on this scale, and then it will not continue to be transmitted to larger areas. Small-scale vortex structures.
This process is called an energy level string process.
This is the most important and basic knowledge point of contemporary fluid mechanics.
Xu Chuan didn't know about other schools, but when he was at Nanjing University, this knowledge point occupied a full ten minutes in the exam.
It can be said to be the top priority.
As the solutions to the NS equation exist and are continuously smooth, part of the theory is based on the K41 theory.
This time Xu Chuan pushed the NS equation to an unprecedented height and also used this set of theories.
At present, the K41 theory is also suitable for turbulent flow, but I don’t know whether it can still be as popular as it is now when facing the final NS equation solution 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 in the two students.
"Thank you for your hard work. This is to sort out the papers entered into the computer."
Hearing this, Gu Bing looked at the paper on the desk, while Amelia did not move. She looked at Xu Chuan with excitement and asked curiously:
"Professor, you have proved the NS equation?"
As we all know, their mentor has a quirk, that is, when faced with a problem, if he does not solve it, he will almost never leave the house.
But now, it is obvious that there is a result.
Xu Chuan shook his head and said: "No. It is too difficult to prove the NS equation at this stage, and it is basically impossible."
As soon as he finished speaking, Gu Bing's exclamation came from the side: "Professor, have you proved the NS equation?"
Hearing this, Amelia immediately cast a doubtful look at Xu Chuan.
Xu Chuan said that he did not prove the NS equation, so what was the manuscript paper in Gu Bing's hand?
Noticing the doubtful eyes of his students, Xu Chuan shrugged and said: "It's just a phased achievement of the NS equation." With doubts, Amelia grabbed the manuscript paper from Gu Bing's hand and looked at it. on the title.
"Given a finite space, when the initial value is infinitely smooth, a smooth solution to the three-dimensional incompressible Navier-Stokes equation exists! 》
When she saw the title, Amelia's blue pupils suddenly shrank into a small dot, her eyes filled with disbelief.
Is this called "just a phased achievement?"
only?
That's it?
Thinking about the instructor's calm words just now, Amelia wanted to recite the words she learned while studying abroad here.
Pretend! This is definitely pretending!
With the help of two students, Xu Chuan spent two full days entering the proof process into the computer.
After carefully checking twice and confirming that there was no problem with the paper on the computer, Xu Chuan threw it on the Arxiv preprint website.
"Let's go, I've worked hard for you these two days, I'll treat you to a big dinner."
After finishing the paper, Xu Chuan smiled and patted the two students on the shoulders.
It can be said that it is quite tiring to input all the proof papers of more than 200 pages into the computer word by word in two days, and to check every letter and even punctuation mark.
Within two days, these two students had dark circles under their eyes, which is the best proof.
When Xu Chuan took two students to have a big dinner, Arxiv gradually became lively, and undercurrents began to surge in the mathematics community.
On the arxiv preprint website, many scholars who paid attention to tags such as NS equations, seven millennium problems, fluids, turbulence, Xu Chuan, etc. immediately received recommendations from the website.
Among them was Fefferman, who labeled Xu Chuan a 'special concern'.
When he received the push from the Arxiv website, Fefferman was in front of the computer searching for information that would be helpful in studying NS equations.
Last year, he and Xu Chuan made a breakthrough in the NS equation, which gave him a glimmer of hope in overcoming the NS equation.
Even though it was a long shot, Fefferman didn't want to give up.
This is his lifelong dream.
On the screen, Fefferman was searching for papers in the Annals of Mathematics. Suddenly, a small pop-up box popped up in the lower right corner.
Just as he was about to fork it, he noticed with sharp eyes that this pop-up window came from the Arxiv software.
This made him stunned for a moment, and then he clicked to open the pop-up window, ready to see what arixv pushed to him.
"Given a finite space, when the initial value is infinitely smooth, a smooth solution to the three-dimensional incompressible Navier-Stokes equation exists! 》
Fefferman's pupils suddenly shrank when he saw the title displayed after the pop-up window expanded.
Proof of NS equation?
It has advanced to this step!
How is that possible? Who is it?
And why is such an important proof posted on arxiv?
Suddenly, Fefferman thought of something and quickly controlled 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 it right!
Apart from that weirdo Perelman, there is only one person who posts papers proving major conjectures on Arxiv before being reviewed by a journal.
After regaining consciousness, Fefferman quickly downloaded the entire paper and started reading.
"Starting from the compressible Navier-Stokes equation of thermal conductivity, and then using the harmonic equation whose function is the coefficient to deduce the wall flow. This is indeed a good idea, but the S+R decomposition theory has a fatal How will 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 talking about it.
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 good, but it is not that he has not considered it.
However, he finally gave up because there was a fatal flaw in the S+R decomposition theory that he could not solve.
Not only him, but also the mathematicians he had communicated with were not optimistic about this issue.
But this method was used in this paper, and Fefferman is very excited about how the other party will solve this problem that is not optimistic by many top mathematicians.
He was thinking and flipping through the paper in his hand. As he read, the answer he wanted soon appeared in front of him.
".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!"
"Colliding the vortex ring, reconstructing the vortex into a vortex using 3D calculations, and then using a function to solve it, this can completely avoid the fatal flaw in the S+R decomposition theory that cannot approach 0 from 1!"
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"What a great idea! Absolutely brilliant!"
After reading the solution process, Fefferman couldn't help but slap his thigh and admire the ideas and ideas in the paper.
This is a path that no one has ever thought of, perfectly integrating physics and mathematics to solve difficult problems that are important in both 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 own challenge to mathematics. This is a peak that no one has climbed before. Originally he had hope, but now someone has climbed it first.
However, he still has hope!
Although this proof is quite excellent, and in his opinion it can almost be said to have proved a large part of the NS equations, it is not completely completed after all.
There is still one step left to completely prove the NS equation, so let him do the rest of the work!
Thinking about it, Fefferman's eyes showed infinite fighting spirit!
He will definitely put the final roof on this building!
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