Section 256

"As Professor Terence Tao mentioned in his blog, after we have completed the interpretation of the singularity problem, it seems that we can try to prove the NS equation from more general initial conditions."

"As a mathematician, I am naturally more inclined to this method of proof."

"The more general it is, the more abstract it is. The more abstract it is, the more general it is."

"Then how to choose the appropriate function space at this time has become a key issue."

Xiao Yi smiled slightly: "Obviously, the various function spaces at present cannot help us adapt to the explosion trend theory well."

“We needed a more appropriate space.”

"The question is, what kind of function space is the most suitable function space for NS equations?"

Xiao Yi turned around, erased all the previous data analysis processes on the blackboard, and then started analyzing the function space again.

This time the analysis is more complex and abstract than the previous analysis.

At this time, it can no longer simply be called a show of skill, but a brand new analysis system.

Looking at those wonderful analysis steps, Tao Zhexuan spread his hands.

"The only one who can defeat Xiao Yi is probably him, right?"

Fefferman said: "He is the king of analysis!"

Chapter 223 Terence Tao’s only problem

"In order to deal with the complex behavior of understanding more effectively, we need this new function space with the following properties."

“The ability to control the regularity of solutions, the ability to control the growth of nonlinear terms, and the ability to handle more general initial conditions.”

"First, let's list several common function spaces."

"The first is the most important space, Sobelov space. Sobelov space is often used in the study of fluid mechanics, so using the properties of this function space can provide us with a lot of help."

On the podium, Xiao Yi wrote Sobelov space on the blackboard.

"And beyond that, there are many other function spaces."

"There is an old saying in China, which is to extract the essence and remove the dross, which means to extract the useful things and then remove the useless things."

"Of course, this is just a metaphor. After all, the parts we removed cannot be called dross, they are just parts that are not helpful to our research."

"Then, in addition to Sobelov space, there is also Besov space..."

Through various analysis methods, these function spaces are analyzed and studied in detail.

Gradually, the prototype of Xiao's space gradually emerged.

People at the report meeting were also more and more amazed by Xiao Yi's powerful analytical ability.

Anyone else would never have imagined that they could combine so many function spaces.

Of course, if we just extract the essence of these different spaces, it will definitely not be enough.

So, there's still something new in it.

Therefore, after Xiao Yi completed the analysis of various function spaces, he began to add new things to the prototype of Xiao's space.

"Well, I can't understand it at this point."

Terence Tao shook his head and said.

Fefferman next to him couldn't help but was stunned: "You can't understand? You're kidding!"

After Xiao Yi's paper was published, Tao Zhexuan gave his evaluation immediately, so how could he not understand it?

Terence Tao smiled and said, "I mean, I don't understand how he thought of doing this."

Fefferman suddenly understood, and then sighed with emotion: "Indeed... I don't understand it anymore."

Tao Zhexuan nodded slightly and said: "Whether it is the previous blasting trend theory, or the steps just now using the characteristics of Sobelov's space, I can roughly think of his thoughts, and guess what he was thinking when he was doing those steps. What do you think?"

"But now...I just really can't think of it."

Fefferman nodded slightly and seemed slightly silent.

Tao Zhexuan turned his head and glanced at Qiu Chengtong next to him, and asked: "Professor Qiu, you and Xiao Yi were both born in China, can you think of it?"

Qiu Chengtong shrugged his shoulders and said, "I wouldn't have thought that you would look down on me."

"What's more...this kind of thinking has absolutely nothing to do with where you were born."

Tao Zhexuan laughed and said, "Maybe?"

Afterwards, they said no more.

Watching Xiao Yi's next derivation process quietly.

Although they had no idea how Xiao Yi came up with these ideas in his mind, it did not prevent them from appreciating his on-site derivation process here.

Speaking of which, they haven't seen Xiao Yi hold a report meeting for a long time, and naturally they haven't seen Xiao Yi's live deduction for a long time.

No matter how many times they watched it, they couldn't help but sigh that every derivation by Xiao Yi was very exciting.

In addition to math experts like them, there were also audiences from other fields.

For example, those who study physics, or those who study fluid mechanics, etc.

For them, what Xiao Yi wrote on the blackboard at this moment was a bit mysterious.

Such a complicated derivation process is simply not something that non-mathematics majors like them can understand.

For example, Diaby sent their chief engineer to study. At this time, he stared at Xiao Yi's derivation process with dead fish eyes, the complicated processes on the blackboard, and even the mathematical symbols of the game. He, a top expert with an annual salary of over 10 million, couldn't understand it. He couldn't help but complain to the colleague next to him: "Wori, these people who study mathematics are really crazy. Can they continue to study this thing?"

The colleague next to him smiled and said, "Don't you look at it? Aren't those math experts in front of you looking at it with gusto?"

He looked in the direction pointed by his colleagues, which was where mathematicians such as Terence Tao were sitting.

The chief engineer twitched his lips and finally said: "Fortunately, I didn't apply for pure mathematics at Beijing University, but applied for mechanics. I couldn't learn this mathematics anyway."

On the other side, students from major universities in Hui Province were all drowsy at this time.

Although, they can all be considered Xiao Yi's fans, it is absolutely impossible for them to listen to the entire report, which is more difficult than a math class.

Especially this report meeting is still in English. After all, there are so many foreigners.

And there are quite a few among them who have not even passed CET-4.

"Do you understand what Xiao Shen is talking about?"

Someone asked the classmate next to him in a low voice.

The classmate rolled his eyes: "I can't even understand what he wrote, so there's no need to talk about what he said."

Finally, they all sighed.

"This seminar is not for us little Karami."

"It's okay, we can meet Xiao Shen, and we can also participate in the proof report of this world's top mathematical problem. In any case, we can be regarded as witnessing history."

"Indeed, it doesn't cost much to attend this lecture anyway, so it's not a big problem. Come to think of it, I study communications."

"Brother who is in communications? I saw on the Internet in the past two days that there is someone who makes communications antennas in Xianyu, and then I was asked if he was doing that. Brother, do you know?"

The guy who was studying communications suddenly had a dark look on his face and said angrily: "Damn it, don't pick up the pot, right? The reputation of our communications has been ruined!"

…

No matter where Xiao Yi's fans and audience are taking the topic, Xiao Yi on the stage has completed the entire analysis and derivation process of Xiao's space.

"At this point, a brand new function space is born."

"We can easily see that Xiao space has many excellent characteristics in terms of fluid mechanics."

"For example, the three conditions I gave earlier."

"In addition, it can also well restore the basic properties of various fluid models."

"I believe that in recent academic circles, many papers have appeared to verify this."

"I won't go into more details. I believe that the academic research on Xiao's space will be broader than what I can imagine. Next, let's enter the key part of our report to prove that NS Existence and Smoothness of Equation Solutions!”

When it came to the last step, most of the people present felt relaxed, because with the completion of Xiao's space, all the key tools to prove the NS equation had been gathered.

Everything is ready, the east wind has come, and the rest can be done just by letting nature take its course.

Of course, there are still certain thresholds.

"First of all, we need to know the key steps in proving the existence of the solution..."

"First, the existence of weak solutions. In this step, we only need to use energy inequality and compact embedding theory to prove the existence of weak solutions."

"Secondly, it is the regularity of the weak solution. Using the properties of Xiao space, it is proved that the weak solution has higher regularity in the X space."

"Finally, the existence of strong solutions is proved through the iteration method and the fixed point theorem."

Xiao Yi began to write on the blackboard that had been erased again.

"...Finally, we can construct a sequence of approximate solutions (Un) using the pseudo-smooth method."

【Un=n∑_(n=1)[ai(t)i(x)]】

"After that, Un is substituted into the NS equation to obtain a finite-dimensional system."

[(Un/t,j)+((Un)Un,j)=(pn,j)+ν(ΔUn,j)+(f,j)]