From University Lecturer to Chief Academician

Buckmaster, professor at MIT, winner of the 'Ramanujan Prize', and academician of the American National Academy of Sciences.

He is a well-known expert in the field of application of partial differential equations, and is also recognized as an authority in the field of research and application of NS equations. He has been committed to the research of theoretical applications of NS equations.

As early as five years ago, Buckmaster began to try to question and challenge whether the main method of NS equation research could be successful, and published the research results of himself and his colleagues.

The results at that time were not perfect, just demonstrating the inconsistency of NS equations describing the physical world under certain assumptions.

The current research result is to prove that the output of the NS equation is unreasonable, that is, the deviation value is too large and unstable, under the condition of "allowing the solution set of the NS equation to be rough".

To illustrate with an example, for example, if a certain parameter is adjusted to 5, the output value is 10; if the parameter is adjusted to 6, the output value becomes 60; The value does not change with the slow change of the parameter, but fluctuates greatly.

This means that the deviation value is too large and unstable.

In the case of 'allowing the solution set of the NS equation to be rough', the value output by the equation is not stable. To a certain extent, it can be inferred that the equation itself is also unstable, that is, to a certain extent, it denies the solution set of the NS equation. smoothness.

Buckmaster himself was also interviewed, and he explained, "Smooth solution sets are complete to describe the physical world, but mathematically speaking, they don't always exist."

"Many times, we can only study equations with rough solution sets, that is, weak solutions."

"It's like sketching a face. Each line is not necessarily drawn in a fixed position, but the overall trend is fixed."

"If the line of the face is drawn on the nose, we think it's not a successful sketch, but a low-level mistake."

"If this kind of error occurs on the weak solution set, then it can be considered that the smooth solution set is also incomplete (smooth) to a certain extent."

Whether the logic of Buckmaster's explanation in the interview is reasonable or not depends on personal judgment, but the proof he made is logically rigorous.

Wang Hao downloaded the original version of the paper and read it carefully for more than two hours, but he couldn't find any problems.

As for the details of the derivation, it takes two rounds of review to be published in the top academic journals of mathematics, and it is almost impossible to make similar low-level mistakes.

"impossible!"

Wang Hao frowned and thought, "There can be no mistakes in the process, and there is no logical problem..."

"Is it proven correct?"

"This is impossible!"

If Buckmaster's argument is correct, his research is wrong.

How is this possible?

The human brain may make mistakes, but the system's judgment of knowledge inspiration can't catch up with Buckmaster's logical rigor?

Or, Buckmaster overstepped the system?

"impossible!"

Wang Hao was determined to get on with this paper, and he reviewed it from beginning to end, but still couldn't find any problems, so he simply created a task——

【Task 4】

[Research Project Name: Find out the research problem of Buckmaster (difficulty: C). 】

[Inspiration value: 0. 】

"!!"

"Difficulty C? You are worthy of being recognized as a top expert in the NS equation!"

Wang Hao was shocked when he saw the difficulty of the task. He was just looking for a problem in a research paper, and the difficulty actually caught up with a research paper. No wonder he couldn't find anything after examining it for three hours.

Let Buckmaster find this question by himself, probably he can't find it himself!

...

Buckmaster's research impact is indeed high.

Although it has not reached the level of shock in the international mathematics community, scholars related to the study of partial differential equations and NS equations will read his papers, and even some scholars who use NS equations will also read his papers.

Including some scholars of aerodynamics and fluid mechanics, as well as experts in the field of application.

etc.

Buckmaster's research negates the NS equation to some extent.

In fact, there are many studies every year to deny the NS equation, but this time it is Buckmaster, a recognized top expert in the field of NS equation research.

In addition, Buckmaster's paper was published in "Basic Mathematics and Applied Mathematics", and the authoritative journal is naturally convincing.

Then again, his thesis proved logically rigorous.

When no one finds the problem, they will be very surprised. Some people even propose to find realistic examples of NS equations that are not smooth based on Buckmaster's research.

Of course, most people are still calm.

In many cases, there is still a difference between mathematical logic and physical reality, because in terms of application, as long as the tool used is effective, it does not need to prove that it will always be effective.

Now it is only a theoretical research in the mathematics field, and the paper does not deny the NS equation 100%, but only demonstrates that the NS equation may be invalid through the research on the rough solution set.

For Wang Hao, this is not the case.

Buckmaster's research directly conflicted with his own, and he had to find out where the other was wrong, otherwise it would be tantamount to denying his own research.

Wang Hao went to class.

Taking classes can greatly increase the inspiration value.

Research with C-level difficulty can often accumulate 100 inspiration points in one class. His course is "Modern Partial Differential Equations", which is strongly related to the research of NS equations.

This is the last class at the end of the semester.

Wang Hao explained the content very carefully, and finally sorted out the whole course, so that students can have a better understanding of the course as a whole.

This can help them have a deep understanding of the content, not just know the application of some basic mathematical methods.

One class, two class hours.

[Inspiration value: 37. 】

"Very few!"

This class brings surprisingly little inspiration.

Wang Hao was also very surprised. He thought that one class would be enough to complete the research, but found that the increased inspiration value was only one-third.

This means that the key was not found.

After returning to the Mason Number Laboratory, he was bored in the office and looked at Buckmaster's research again. Later, when Zheng Yaojun came over, he simply studied with Zheng Yaojun.

Zheng Yaojun has also been engaged in research in the field of partial differential equations for a long time, and has a certain personal understanding of NS equations.

He also knew about Buckmaster's research.

The two of them reviewed and discussed the paper from beginning to end, hoping to find errors in the process or logic, but there was no progress.

"The process is presumably correct, and if there is an error, it may be logical."

"The final conclusion has also been drawn, but there are still some places that need to be carefully thought about."

Zheng Yaojun frowned and said.

At this time, Helen knocked on the door and walked into the office. She also came to discuss Buckmaster's research issues. Because she couldn't find any questions, she wanted to ask Wang Hao's opinion.

"We are also studying this issue, and I think the conclusion must be problematic." Wang Hao said while pursing his lips.

Helen said, "I combed through the process carefully and found no problems, but this conclusion..."

"It's hard to accept."

The reaction of an ordinary mathematician is like that of Zhou Qingyuan. He cannot accept the conclusion that the NS equation is not smooth, even if it is only an analysis of the rough solution set, he still cannot accept it.

It's like seeing a perfect artwork with huge flaws, which makes people feel very depressed.

Zheng Yaojun suddenly became interested. He knew that Helen was Wang Hao's student, so he said it from a position he was not sure about, "The process may not be all correct. Look at this position."

He pointed to a position and said, "There may be a problem with the logic here. The deviation value analysis he mentioned may not be perfect."

Helen looked at Zheng Yaojun and said, "There are no uncertainties in mathematics, only right and wrong."

"...?"

It was just a word of "education", which made Zheng Yaojun unable to react for a while.

Helen continued, "I also thought about the position you pointed out. The deviation value analysis they did is very perfect. Does it really prove that there is a big difference?"

"But, how do you define it?" Zheng Yaojun immediately asked back when he found out that he was being educated by the little girl.

Helen said, "Just look at the degree of separation of the curve. This data is enough to explain any problem. Study the deflection of the value of the curve. Judging from the direction, the degree of deviation exceeds the defined value."

"Ah~~"

Zheng Yaojun thought about it, and it was true, but he felt humiliated when he was pointed out by a female student. He immediately found another place, "What about here? He used an algebraic analysis method, but he was not sure to include all the thresholds. "

"Of course not all thresholds need to be included."

Helen said, "You only need to divide a part, and a part cannot represent everything, but the content is just an explanation, not an argument."

Zheng Yaojun immediately said, "As you said just now, there are only right and wrong in mathematics. Even if it is just an explanation, this explanation is not perfect."

"I don't think you understand the problem..."

"Ula Ula~~"

Helen and Zheng Yaojun argued about the content.

One sentence from you, one sentence from me, no one can convince anyone.

Looking at this scene, Wang Hao touched his forehead helplessly. Helen was a bit inquisitive, and she was very unwilling to admit defeat.

Zheng Yaojun seems to be a little bit too.

What is a big professor and a little girl arguing about?

When the debate got to the end, Zheng Yaojun obviously stopped talking about martial arts, and talked about some "completely super-class" content, some of which even involved his own research.

Then, he won.

Because Helen couldn't understand something behind. After all, she was a girl in her teens. Even if she was a genius and had a high IQ, she couldn't catch up with Zheng Yaojun in the field of knowledge involved.

In the end, Helen's cheeks were flushed with anxiety, and it was Wang Hao who comforted her, "Helen, don't argue with this guy, in two years, he won't be your opponent anymore!"

Helen seemed to have listened, pointing at Zheng Yaojun, gritted her teeth and said, "Just wait for me!"

"!!"

Helen is gone.

Zheng Yaojun was obviously a little proud, like a general who won a battle.

Wang Hao broke the cold water for him, "Old Zheng, Helen is only sixteen years old..."

Zheng Yaojun's smiling face immediately disappeared, and he realized that it was his students who should be compared with Helen, not himself.

But his student, Hu Lidan?

and Helen...

"What a genius!" Zheng Yaojun finally sighed, "Why do you have such a talented student? You are only 16 years old, and after two years, you will be better than me."

Wang Hao shrugged, "Helen is indeed a genius, but I think another student, Qiu Hui'an, is the best."

"Why?"

"He's working on the Legendre conjecture."

One sentence will make it clear.

Zheng Yaojun pursed his lips forcefully, "Even if he can't prove it, he will definitely be very good in the future."

"yes."

"I envy you... There are so many gifted students, next time you find such a gifted student, can you recommend them to me?" Zheng Yaojun said, "Although I am not a genius, I want to have a gifted student."

A short, fat and small-eyed figure suddenly appeared in Wang Hao's mind.

no!

The young man is very talented, it's a pity to follow Zheng Yaojun.

Zheng Yaojun didn't know what Wang Hao was thinking, but continued, "Wang Hao, do you think a genius like Helen is a normal person?"

"Um……"

This feels like a philosophical question.

Wang Hao thought about it carefully, is a genius a normal person?

A genius is the same as a normal person, with two arms and two legs, and the appearance is the same. The difference is that the brain is very developed?

But at the same time, some people are born with great strength and outstanding physical development, but the development of modern society has led to more emphasis on mental genius.

So a genius is also within the range of "normal person" judgment bias...

yes!

Wang Hao's thinking eyes lit up, he slapped the table excitedly, and suddenly shouted, "Boom!"

"I see!"

Zheng Yaojun was so frightened that he trembled all over.

I heard Wang Hao say, "Even a genius like Helen, compared with you, is still within the normal range!"

Zheng Yaojun was stunned for a long time with his mouth slightly open, then he recovered and pointed at himself, "You mean..."

"I am a fool?"

...

After Wang Hao found the inspiration, he already discovered the problem. Buckmaster's thesis is indeed correct, but being correct doesn't mean anything.

They take the conclusion too seriously.

Perhaps even Buckmaster himself is the same. He found that in the case of "allowing the solution set of the NS equation to be rough", the numerical value output by the equation is not stable. It is taken for granted that this denies the smoothness of the solution set of the NS equation to a certain extent.

This logic itself is problematic. To a certain extent, it does not mean 'certainty'.

As Helen said, mathematics is only correct and incorrect, there is no vague definition.

"To a certain extent", is it proved or not proved?

After Wang Hao discovered the problem, he contacted his own research, and immediately thought of the key point, and knew how to refute the research. He could prove that the output of the 'rough solution set' equation is boundedly convergent. In other words, for the 'rough solution set' ' research, the output of the equation determines that there is an unstable situation, and it is also within a certain range, rather than completely unstable.

The sketch example is really nice.

For the normal values ​​of the NS equation, it is impossible for a stroke to be drawn on the nose.

Therefore, Buckmaster's research can't explain any problems. It has nothing to do with whether the solution set of NS equation is smooth, and can't prove anything.

Wang Hao did not take notes to refute Buckmaster's research.

Because he has enough inspiration and the research is in the same direction, he can even prove on the spot that "the problem of bounded convergence of equation output when rough solution sets are allowed".

He was doing an inspired record of his own research.

【Task 1】

【Research Project Name: Navier-Stokes Equation Research (Difficulty: S+). 】

[Inspiration value: 60. 】

Looking at the inspiration value of the system task, Wang Hao couldn't help showing a smile on his face, and even said he was a little excited.

That last bit of inspiration didn't come easily.

Zheng Yaojun looked at Wang Hao's continuous records and asked curiously, "Do you know the problem with that paper? Are you going to deny his paper?"

"of course not."

Wang Hao shook his head and said, "What's the point of denying other people's papers? They can't be published as results."

"Then you are..."

"My own research." Wang Hao said, "I already know how to prove the smoothness of the NS equation solution set within a fixed range of value conditions."

Zheng Yaojun was stunned for a moment, and pondered carefully, "Buckmaster proved that under the range of values, the NS equation is not smooth to a certain extent."

"Now it is to prove the smoothness of the NS equation solution set under the range of values."

"These two..."

He suddenly widened his eyes and realized, "It's completely the opposite! You said you didn't deny his research!"