The top student must be diligent.

Chapter 223: Terence Tao’s Only Problem
"To more efficiently handle the complex behavior of solutions, we need this new function space with the following properties."

“You can control the regularity of the solution, you can control the growth of nonlinear terms, and you can handle more general initial conditions.”

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

"The first is the most important space, the Sobelov space. The 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 there are many other function spaces besides these."

"There is an old Chinese saying, 'Take the essence and discard the dross', which means to extract the useful things and discard the useless things."

"Of course, this is just a metaphor. After all, the parts we removed cannot be called dross. They are just the parts that are not helpful to our research."
"Well, besides Sobelov space, there is also Besov space..."

Through various analytical methods, these function spaces are carefully analyzed and studied.

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

People at the lecture were increasingly amazed at Xiao Yi's strong analytical ability.

If someone else were to do this, they probably would never have thought of integrating so many function spaces.

Of course, it would not be enough if we simply extracted the essence of these different spaces.

So there's still something new in it.

So, 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 don't understand it anymore."

Terence Tao shook his head and said.

Fefferman next to him was stunned: "You don't understand? Are you kidding me!"

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

Tao smiled and said, “I mean, I don’t understand how he came up with that.”

Fefferman suddenly realized, and then he also sighed: "Indeed... I don't understand it anymore."

Tao nodded slightly and said, "Whether it is the explosion trend theory mentioned above, or the steps of using the characteristics of Sobelov's space just now, I can roughly think of his ideas and guess what he was thinking when he was taking those steps."

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

Fefferman nodded slightly, seeming a little 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 imagine that?"

Qiu Chengtong shrugged his shoulders and said, "You think too highly of me. I never thought of that."

"Besides... 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.

Watch Xiao Yi's subsequent deduction process quietly.

Although they had no idea how these ideas came up in Xiao Yi's mind, it did not prevent them from appreciating his on-the-spot deduction process here.
Speaking of which, they haven't seen Xiao Yi hold a lecture for a long time, so naturally they haven't seen Xiao Yi's on-site derivation for a long time.

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

In addition to these math giants, there were also audiences from other fields at the scene.

For example, those who study physics, those who study fluid mechanics, and so on.

To them, what Xiao Yi wrote on the blackboard at this moment was a little bit incomprehensible.

Such a complicated derivation process is simply not something that people like them who are not majoring in mathematics can understand.

For example, Diaby sent their chief engineer to learn. He stared at Xiao Yi's derivation process with dead eyes. Even this top expert with an annual salary of over 10 million couldn't understand the complicated process on the blackboard, and even the mathematical symbols in it. He couldn't help but complain to his colleague next to him: "Wow, these math researchers are really crazy. Can they continue to study this thing?"

His colleague next to him smiled and said, "Look, aren't those math experts in front of you enjoying it?"

He looked in the direction his colleague pointed, and that was where mathematicians like Terence Tao were sitting.

The chief engineer pulled his lips and finally said, "Fortunately, I didn't apply for pure mathematics at Peking University, but for mechanical engineering. I couldn't learn mathematics anyway."

On the other hand, students from various universities in Anhui Province were all drowsy at this time.

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

Especially since this lecture was held in English, after all, there were so many foreigners attending.

And there are many among them who have not even passed the Level 4 English test.

"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 and said, "I can't even understand what he wrote, let alone what he said."

Finally, they all sighed.

"This report is not for us little Karamis."

"It's okay. We are able to meet Xiao Shen and participate in the proof report meeting of the world's top mathematical problems. No matter what, we are witnessing history."

"Indeed, it's free to attend this seminar, so it's not a big deal. Think about it, I'm studying communications."

"Brother who does communications? I saw a guy who made communications antennas on Xianyu these days. Someone asked him if he was that guy. Brother, do you know him?"

The guy who studied communications suddenly looked upset and said, "Damn it, you're just bringing up all sorts of issues, right? The reputation of our communications department has been ruined!"

……

Regardless of where Xiao Yi's fans and audience members have digressed the topic, Xiao Yi on the stage has already completed the entire analysis and deduction process of the Xiao space.

"At this point, a completely new function space was born."

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

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

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

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

"I will not go into details. I believe that the academic community's research on Xiao space will be more extensive than I can imagine. Next, let's move on to the key part of our report, proving the existence and smoothness of the solution to the NS equation!"

When they came to the last step, most of the people present felt relieved, 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 accomplished by just letting nature take its course.

Of course, there are still certain thresholds.

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

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

"The second is the regularity of the weak solution. Using the properties of Xiao space, we can prove that the weak solution has higher regularity in X space."

“Finally, the existence of a strong solution is proved by using an iterative method and the fixed point theorem.”

Xiao Yi began to write on the blackboard that had been erased again. "...Finally, we can use the pseudo-smooth method to construct a sequence of approximate solutions (Un)."

【Un=n∑_(n=1)[ai(t)i(x)]】

"After that, we substitute Un into the NS equations and obtain a finite-dimensional system."

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

"Finally, by energy estimation, we prove the uniform boundedness of the approximate solution sequence (Un), and by using the compact embedding theorem, we can complete the proof of the existence of the solution."

"As for the smoothness of the solution, it has become very simple at this point."

"Here we mainly use high-order energy estimation and regularity improvement."

"First, we use the properties of Xiao space to perform high-order energy estimation."

【d/dt∥u∥^2Hs+2ν∥u∥Hs2≤C(∥u∥^2Hs+∥f∥^2Hs)】

"Then we further improve the regularity of the solution through embedding theorem, and finally prove the smoothness of the solution."

"This is the end of the proof."

After Xiao Yi made a heavy dot on the blackboard, he turned around and looked at the audience.

"So far, we have undoubtedly proved that the flow of liquid is smooth and every liquid molecule is continuous."

"They won't behave in any way that we can't understand, and they won't just sit quietly in this cup and suddenly explode."

Xiao Yi opened the lid of the water cup in his hand, took a sip, and said with a smile: "Similarly, they will not explode after entering my stomach."

"In short, the behavior of fluids can be predicted."

At this point, he also said: "In the same way, I think plasma flow can also be predicted."

Then he bowed slightly and said, "Okay, I have finished my story. If you have any questions, you can ask me now."

Soon, quite a number of people raised their hands.

Xiao Yi picked out a middle-aged man with a foreign face, who looked like a professor.

He stood up, took the microphone handed to him by the staff, and asked, "Hello, Professor Xiao, I am Jimmy Lawrence from the Department of Physics at MIT. What role do Sobolev space and Besov space play in the definition of Xiao space? I noticed that you focused on these two spaces during your presentation."

Xiao Yi nodded: "Good question."

"The reason why we focus on Sobolev space and Besov space is mainly because of their respective advantages. Sobolev space provides good regularity and energy estimation, while Besov space is very effective in dealing with local smoothness and small-scale structures. By combining these two spaces, we can ensure the overall regularity while capturing local complexity and details when dealing with the solutions of the NS equations."

"Of course, this may not be very intuitive, so I will simply demonstrate it to you mathematically."

"Now we simply substitute Sobolev space and Besov space into Xiao space, and we get..."

[X={u∈H^s(Ω)∩B ^s_(p, q)(Ω):∥u∥X<∞}
"Where ∥u∥X is a suitable norm that combines the two spatial norms. Based on this analysis, we can..."

"Finally, it can be seen that with the combination of these two spaces, Xiao space has good regularity and overall energy estimation, as well as local smoothness and the ability to capture small-scale structures."

"Do you have any more questions?"

Xiao Yi turned around and looked at Professor Lawrence from the Massachusetts Institute of Technology.

Professor Lawrence was already a little confused.

He is just a physicist, there is really no need to introduce him from any mathematical perspective!

Now, Xiao Yi has finished explaining, but he hasn’t even finished reading the first line of the formula.

But at this moment, facing the many looks he got, he could only nod and say, "Thank you for your explanation, I understand..."

Then he sat back down.

Xiao Yi nodded slightly, then looked at the others: "Any more questions?"

This time, fewer people raised their hands.
It was not known whether it was because their problems were the same as Lawrence's, or because they were afraid that Xiao Yi would come up and directly perform mathematical derivation for them.

Soon, Xiao Yi answered five questions in a row, and then no one raised their hand anymore.

Just when everyone thought that no one would raise their hand, Terence Tao who was sitting in the front raised his hand.

There are quite a few people who know him, and suddenly many of them became interested and curious about what questions Tao would ask.

The questions asked by such a great person should not be simple ones, right?
On the stage, Xiao Yi raised his eyebrows, then said with a smile: "Professor Tao, please ask your questions."

Tao Zhexuan smiled, took the microphone, and said, "Well, actually I just have one question. When you were deducing the Xiao space, how did the ideas in your mind come about? This is the only question I still don't understand."

He had read the paper, which was written in great detail, and this report had also explained some of the things he had wanted to know before.

For example, the data analysis method of bursting trend theory.

But this was the only question he couldn't figure out.

As he asked this question, those in the audience who understood what he said were all stunned.

They never expected that Terence Tao's final question would be so simple.

What is Xiao Yi’s idea?
What's there to ask?

However, there are still many top scholars who can understand the meaning of Tao's question.

It is very meaningful to study the ideas of other scholars.

Top scholars often think about other scholars' research methods through their papers and reports.

But what surprised them was.

Even a math expert like Terence Tao couldn't come up with Xiao Yi's idea?

Of course, Xiao Yi on the stage was also stunned when he heard this question.

However, he then smiled and said, "My thoughts...are not that complicated."

"Since fluids are so complex, it would be better to use more sophisticated descriptions to explain their complexity."

"That's it."

Tao was shocked, and after a moment, he said with emotion: "As expected of you."

"Thanks for your answer, I understand."

Then he sat back down.

Xiao Yi smiled at him, then looked at the others and asked, "Any more questions?"

Not a single hand was raised.

"Well, that's the end of my presentation."

“Thank you all for participating.”

Bow slightly.

The originally silent conference hall was then drowned out by waves of applause.

(End of this chapter)