Research starts with PhD students

Chapter 179 Turbulence Singularity Argumentation Report, Kongtsevich: This is a miracle!
It is no exaggeration to say that the problem of singularity proof of turbulent transition can get a Fields.

The proof of the smoothness of the solution of the NS equation is one of the seven great mathematical conjectures of the millennium.

The selection criteria for the seven major mathematical conjectures of the millennium are not just the difficulty, but also the impact of the problem. The NS equation problem is related to a large number of applications. Proving the smoothness of the solution of the NS equation will have a very large impact and far-reaching significance in mathematical argumentation and physics.

At the same time, the NS equation is also a typical example of a class of complex partial differential equations.

Regarding the study of the smoothness of solutions to the NS equations, most of the purely mathematical arguments are conducted from the perspective of the weak solution set. They study the set of weak solutions, prove the uniqueness of the weak solutions, and further prove the strong solutions and explain the smoothness of the solutions.

From a mathematical point of view, it is obvious that the "singularity" problem is circumvented.

Most mathematical arguments ignore the "singularity", that is, they assume that there is no singularity, and can prove the smoothness of the solution of the NS equation under the natural boundary.

On the contrary, applied mathematics, which is the research in the field of physical engineering, reaches conclusions that are completely opposite to those of mathematical research.

Turbulence is a typical problem.

The water in the bathtub forms a vortex in the drain, the smoke from the cigarette butt spreads in the air, and the river flows around stones. When an orderly flowing fluid changes into a seemingly unpredictable vortex, it is often associated with turbulence.

Turbulence is one of the most difficult problems to understand in physics, and the NS equations used to describe fluid motion are of great help in solving turbulence problems.

In the research of physics, it is easy to find the problem of turbulence transition.

When laminar flow reaches a certain intensity, it will instantly turn into turbulent flow.

This is why many scholars in applied mathematics believe that there is a singularity in the NS equation. In other words, why does the "sudden" transition occur if there is no singularity?

From the above, we can see that there is a clear divergence between the pure mathematical research and the applied mathematical research of the NS equation.

So when the conference announced that Zhang Shuo had completed a paper on the singularity of the turbulent transition position, the first question many people asked was, "Is this a mathematical argument?"

If it is an argument from applied mathematics, there are already many people in the academic community who have done it, and it makes no sense at all.

Proving the singularity problem of turbulence from a mathematical perspective is an unprecedented study and is also a direct method to solve the singularity problem of the NS equations.

Many people heard the news.

By the next morning, many scholars arrived at Lecture Hall No. 3 in advance to reserve seats, causing the hall to quickly become overcrowded.

Everyone is looking forward to it very much because Zhang Shuo will publish his proof of the turbulence singularity problem here.

Zhang Shuo just won the Fields Medal and set the record for the youngest Fields Medal winner. He did research on numerical simulation of the NS equation and won the award for this.

Therefore, scholars still have high expectations for research. If someone else says that they have completed such research, many scholars may not bother to listen, and their subconscious reaction will be "there must be a problem."

This is mainly because the NS equation is a very broad direction. As long as you are engaged in research in the field of partial differential equations, you have most likely done research in the field of incompressible fluids.

Even if they are not engaged in mathematical research on partial differential equations, many scholars have come into contact with the NS equation problem.

This is such a big topic, and so many scholars have done research on it. If a lesser-known person suddenly says that he has proved the "singularity", who would believe it?
Even for Zhang Shuo, many people are skeptical.

Anton Kapustin is a well-known scholar in the field of partial differential equations. He was also looking forward to Zhang Shuo's report and came early to reserve a seat.

There were several scholars around him, and when they were discussing together, they all asked, "Professor Kapustin, do you think Zhang Shuo can complete the argument today?"

"Hard to say."

Anton Kapustin gave an uncertain answer, "I talked to Zhang Shuo yesterday, and he said he has been studying the NS equations, and this is just part of his research."

"As for my understanding of the NS equation, I believe no one dares to say that they are better than Zhang Shuo..."

“So, all I can say is, I’m looking forward to it.”

This statement was recognized by scholars around him, because Zhang Shuo had completed research on numerical simulation of the NS equation. At least his understanding of the approximate and computational solutions of the NS equation was deeper than others.

Several people around Bent Nielsen also asked about this because he had also discussed the issue of turbulent transition with Zhang Shuo.

Nelson’s attitude was, “If I didn’t believe it, I would have confirmed the existence of a singularity at that location when I reported yesterday.”

That's what Nelson thinks.

He and his team are engaged in research in computational mathematics and do not have much confidence in the direction of pure mathematical arguments.

Since Zhang Shuo said so, he believed it a little.

After all, the basis of their research is still the numerical simulation method developed by Zhang Shuo.

back row.

Qi Zhixiang, Wang Hui, Sun Xingli and others sat together. When others asked about Zhang Shuo, Sun Xingli said, "He should still be outside the venue."

He added, "I suggest he come later, otherwise the other reports can't be carried out."

The others nodded.

Zhang Shuo was not the first to give a report, but many people at the scene came for his report.

If he were in the lecture hall, he would definitely attract most of the attention and would affect other people who were giving reports first.

"Where is Luo Yongjun?" Qi Zhixiang looked around and asked again.

Sun Xingli pointed in a direction.

Luo Yongjun was in the row in front of them, with a lot of people around him. If you listen carefully, you can vaguely hear a loud voice shouting, "No problem!"

"Don't you even look at whose students you are?"

"Zhang Shuo is my student, a PhD candidate I supervised. We worked together on the boundary proof of the Monge-Ampere equation when he was in his first year of PhD..."

“It’s been a year since I graduated!”

"The more you study the field of partial differential equations, the deeper you will get..."

"Ula Ula~~~"

Qi Zhixiang listened from a distance and commented, "Lao Luo is..."

"Although, anyway... he seems more popular than us!"

"With his personality, he might be popular among foreigners..."

The others nodded in agreement.

They all think that Luo Yongjun is too high-profile and likes to brag everywhere, but some foreigners like this. Just look at the people around him.

Amid the discussion among the crowd, the morning report meeting had already begun.

Zhang Shuo was not in the lecture hall, he didn't even enter the venue, but it still caused a great impact.

Most scholars were waiting for Zhang Shuo's report, which made the scholars who gave their reports first very embarrassed. There were hardly any people listening to them while they were standing on the stage giving their reports.

They could only insist on continuing, and after finishing their speech, they left the stage hastily.

The scholars waited for two reports, and it was almost ten o'clock before people in the back row noticed that Zhang Shuo walked into the conference hall.

"I'm coming!"

"Zhang Shuo is here!"

Someone in the back row shouted, and the scholars in the front turned around.

Zhang Shuo specifically asked, "Is the second report finished?"

"It just ended. You came at the right time."

"That's good."

Zhang Shuo breathed a sigh of relief. He walked into the lecture hall at the right time because he was worried that his arrival would affect other people's reports.

Now you can go directly to the stage.

Under the gaze of everyone in the lecture hall, Zhang Shuo walked step by step to the stage. He felt that he had long been accustomed to the feeling of being the center of attention, but when he was truly being watched by the entire audience, he still felt a little nervous.

In fact, it was mainly because the news got out.

Zhang Shuo's plan was just to give a report, talk about the direction of singularity demonstration, and prove the singularity problem of turbulent position, that's all. The current situation is somewhat different from what he expected. He took a deep breath while standing on the podium, and then calmed down.

He took out the information slowly, then operated the computer, and then stood there steadily, looking down at the time from time to time.

Finally, the time has come.

Zhang Shuo nodded to the whole audience. As he did so, the noise suddenly became quieter, and the chaos originally caused by the discussion was quickly suppressed.

Under everyone's gaze, the title of the report appeared on the screen:

"NS Equations: Singularity Argument for Turbulent Transitions".

Zhang Shuo immediately said, "There's no need to introduce the turbulence problem. Let's get straight to the point."

"The proof starts with the mapping of the solution set of the two-dimensional NS equation."

These words made the lecture hall completely quiet.

Many people were stunned after hearing this.

The most difficult part of proving the singularity problem of the NS equation from a mathematical point of view is finding a starting point. Due to the complexity of the equation, most proofs start with a weak solution.

Starting from the weak solution, it is difficult to connect to the 'singularity' problem.

Zhang Shuo’s report brought a surprise right at the beginning - two-dimensional mapping.

This is a starting point that many scholars can think of, but no research has ever been able to achieve.

When Zhang Shuo talked about the mapping of two-dimensional solution sets, the screen turned to the next page and entered the main topic, "The famous female mathematician Radskaya of the former Soviet Union proved the regularity of the two-dimensional NS equation."

"The two-dimensional NS equation is the projection of the three-dimensional NS equation under specific values."

"My method is to use the projection of the three-dimensional NS equation to prove it, setting a plane H that intersects with the plane where the two-dimensional NS equation lies..."

"A solution has a unique projection on two intersecting surfaces, which proves the existence and uniqueness of the solution..."

"Similarly, if the projection of a solution set on two intersecting surfaces has the characteristic of smoothness, it also represents the smoothness of the solution set itself."

"At the turbulence transition point, we can take values..."

"Set bounds on the equation..."

Zhang Shuo had already organized his thoughts in his mind. After giving a brief description of the proof method, he quickly got into the main topic of the argument.

In the lecture hall, everyone listened very attentively.

Everyone watched the proof process on the screen and listened carefully to every detail, but many people still couldn't keep up with the pace.

Zhang Shuo's proof direction is very clear, and the proof process is not long. Put together, there are only a few pages of information. He only studies the solution set problem near the turbulence transition position, that is, studying the "special numerical substitution" and special boundary situations.

This is a study of the "special case" of the NS equation, which is much simpler than the argument for a wide range of values.

However, the logical reasoning contained therein is very difficult to deduce line by line, and it is not so easy to understand.

The proof process includes function theory, geometry, and a small part of algebraic geometry methods. It involves many fields, which increases the difficulty of understanding.

Bent Nielsen, who does research in applied mathematics, was one of those who quickly fell behind.

He couldn't understand the report after only ten minutes. He turned to Bob James next to him and asked, "Do you understand?"

Bob James pursed his lips and continued to stare at the stage. After a long while, he finally spoke, "If I could understand everything, I would definitely say: 'Please shut up'!"

He laughed as he spoke.

"I can't believe it."

Bob James shook his head and continued, "Functions, geometry, even algebraic geometry? I only know these methods, but I don't understand them."

"This research covers a wide range..."

He sighed.

There are many branches of mathematics, and most scholars will choose a single direction for research.

Some scholars are able to dabble in multiple fields, but generally when they get older and find that they can no longer make progress in a single field, they will start to study other fields. On the one hand, they want to find a new direction, and on the other hand, they also hope to expand their fields to benefit their original research direction.

Zhang Shuo is only 26 years old. He is too young. How could his research involve so many fields?

Even if you are not a professional researcher, just knowing those methods and applying them is amazing.

It is not just a saying that changing industries is like crossing a mountain.

In different fields of mathematics, top scholars may not be as good as doctoral students.

Of course there are scholars who can keep up with the pace.

For example, Anton Kapustin.

Anton Kapustin is a well-known scholar in the field of partial differential equations. He listened to the lecture very carefully and took notes with a pen.

Just like what Bob James described, those who truly understand are afraid of being disturbed in the slightest.

Kapustin listened very attentively.

When Zhang Shuo finished laying the groundwork for the argument and entered into the comparison of two two-dimensional solution sets, the proof process became clear.

This also makes the thinking of scholars who understand it clearer.

Anton Kapustin let out a sigh of relief and continued to take notes with a smile on his face.

You don't need to listen to the rest of the part.

By the time the report got here, he was sure that it was correct.

Those who understood felt the same way.

Maxim Kontsevich, Fields Medalist and one of the judges in Hall 3, was sitting in the first row, and he and Kapustin showed the same smile.

Someone nearby noticed Koncevic's smile and immediately asked, "How is it?"

"No problem."

Kong Cevic said confidently, "You don't need to listen any further. The proof is over."

"Is that right?"

"of course."

Kong Caiwei laughed and said, "I remember that I was a reviewer for Zhang Shuo's paper submitted to Advances in Mathematics. Now I am a reviewer again for the Mathematicians' Conference."

"From the general algorithm for second-order partial differential equations to the singularity proof of the NS equations, it took only a little over a year..."

"This is a miracle!"

Kong Cevic said with a sigh.

The explanation had reached its final part, and the summary appeared on the screen.

Zhang Shuo finished the final explanation and made a summary, "Based on the above conclusions, we can determine that the solutions of the turbulence transition position increase exponentially and are very dense, but there are no jumping singularities. The solution set still exists, is unique, and is smooth!"

When the words fell, the lecture hall was silent for a moment.

Kongtsevich stood up immediately, stretched out his hands and clapped them together vigorously.

a bit.

Two clicks...

Several scholars who understood the process, including Anton Kapustin, stood up and applauded.

The applause quickly broke out and soon filled the entire lecture hall.

(End of this chapter)