Chapter 413 Lu's Manifold?

Staring blankly at Lu Zhou for about half a minute, Molina suddenly stretched out her hand.

Looking at the hand touching his forehead, Lu Zhou subconsciously avoided it.

"What do you want to do?"

Withdrew her hand as if nothing had happened, Molina said solemnly: "It's nothing, I just want to see if you have a fever?"

Lu Zhou: "..."

Looking at Lu Zhou seriously, Molina continued: "Seriously, although I haven't studied partial differential equations, why do you make the problem so complicated?"

Lu Zhou patted the grass on his pants and stood up.

"I'd like to make it easier too, but it can't be helped, it's just that complicated."

Molina also stood up and walked in front of Lu Zhou: "If a calculation result has violated basic common sense, then there is a high probability that something went wrong."

Lu Zhou did not deny her statement.

"Maybe you are right, because I think so too. However, rather than whether the solution of the three-dimensional NS equation has global regularity at a particular point, what I want to know is why?"

Pausing for a moment, Lu Zhou, who was staring at the lake, continued.

"Why our equation exploded."

...

"Explosion" can also be called divergence in the field of computational fluid dynamics. Some authors in many foreign literatures like to use the word "Blow-up" to describe this troublesome phenomenon.

In mathematics, it generally refers to many problems. For example, the denominator of the solution process may be 0, or the solution matrix may not converge...

For the NS equation, the so-called explosion problem, or the divergence problem, refers to a certain time point and a certain space point, the fluid flow rate is getting faster and faster, and then the speed tends to infinity, which is beyond the common sense in reality .

Lions et al. proved half a century ago that this point does not exist in the two-dimensional case, that is, the uniqueness, regularity and stability of the overall weak solution of the NS equation in the two-dimensional case. But what is the situation of the NS equation in the three-dimensional situation? There is still no unified conclusion in the academic circle.

The general view in the mathematical community is optimistic about the existence and smoothness of the solution of the NS equation in the three-dimensional situation. People who are engaged in computational fluid dynamics certainly agree with this because of the ass problem—otherwise, they will use it according to the experimental data. Aren't the established phenomenological models tantamount to using lies to explain lies?

Returning home covered in sweat, Lu Zhou threw his clothes into the washing machine, turned around and went to the bathroom to take a shower.

The feeling of hot water flowing from his head calmed down the impetuousness in his heart a lot.

There may be problems with the idea of ​​indirect proof through abstract bilinear operators. Instead of repeatedly entangled in uncertain issues, it is better to make two-handed preparations, such as trying an additional way of thinking in another way.

This kind of game that challenges the pinnacle of human mind does not have any formula for solving problems.

Before the Calabi conjecture was solved, the differential geometry community never thought that partial differential equations and Riemannian geometry could play like this. After the Calabi conjecture was solved, the geometric analysis based on the PDE method came into being.

Maybe, while solving the NS equation, it is not necessarily that he can discover something greater from it?

After returning to the study, he turned on the computer and began to search for literature on the NS equation.

After all, it is a century-old problem offered by the Clay Institute. The NS equation plays a pivotal role in the field of partial differential equations. Therefore, scholars in the field of partial differential equations have also made many beautiful research results around this equation.

Whenever research hits a bottleneck, Lu Zhou will try to find the missing piece of the puzzle by retrieving papers from the database.

Just as Perelman immediately applied the method to solving the Poincaré conjecture after seeing Hamilton's paper on understanding Ricci flow singularities, so he was looking for something similar.

However……

Finding this piece of the puzzle is obviously not that simple.

The sunset outside the window has been covered with stars all over the sky, and the hour hand of the clock on the wall has passed 12 o'clock and started to shift towards 1 o'clock.

Heaving a sigh of relief, Lu Zhou leaned back on the chair and pinched his sour brows.

The erratic thoughts in his mind were like almost solidified ink for a while, and turned into a plume of smoke for a while, causing headaches.

However, in this vagueness, a hint of enlightenment suddenly appeared in Lu Zhou's heart.

"If you don't have tools, why not build one yourself..."

If each molecule is abstracted into a point, and the collection of these points is abstracted into a space with local Euclidean space properties, he can completely construct an approximate three-dimensional manifold based on this, and use the method of topology go in...

This seems to make the "simple" problem more "complex".

But it seems...

is it going to work?

Eyes are getting brighter.

Grasping the faint inspiration, Lu Zhou quickly grabbed the ballpoint pen and wrote a line on the paper.

【Lu manifold】

Then, the pen in his hand couldn't stop...

...

Time always flies when you are fully immersed in a job.

In a blink of an eye, it was April.

For more than a month, Lu Zhou, who locked himself in the house, also spent a short and monotonous spring break.

During this period of time, except for Vera who came to his house once and sent him a teaching report during this period, Lu Zhou almost cut off communication with the outside world.

In fact, although he asked Vera to send those things, he put them in the corner of the study after they were delivered, and almost never looked through them.

In Princeton, Professor Lu's unique way of delving into problems can almost be regarded as an anecdote. Even undergraduates who have just entered the school have heard of it from old students.

Perhaps because he knew that his research had entered a critical period, Professor Fefferman thoughtfully did not disturb him, but temporarily stopped regular exchange meetings and started independent research at the same time.

And now, those efforts are finally bearing fruit.

Stopping the pen in his hand, looking at the stack of draft paper in front of him, a smile appeared on the corner of Lu Zhou's mouth.

The tense string in his mind finally relaxed, and Lu Zhou began to think of some unimportant things.

For example, is the name Lu Manifold a bit unpleasant?

How about changing it to LZ stream or LuZhu stream?

After thinking about it, Lu Zhou felt that it would be better not to embarrass future candidates.

The former seems to be prone to strange ambiguities, and the latter does not read smoothly.

"The Chinese name is Lu's Manifold, and the English translation is L Manifold, or L Flow for short!"

Satisfied with the name, Lu Zhou casually changed the title on the manuscript paper, then stacked the papers and placed them on the corner of the table, preparing to organize the contents on the computer one by one.

Just as he turned on the computer and was about to start the work, a string of bubbles suddenly popped up from the work bar in the lower right corner of the screen.

Xiao Ai: [Master, there is a new email! ()】

Seeing this message, Lu Zhou casually clicked on the link thrown by Xiao Ai, and logged into his mailbox.

The email is from the Annals of Mathematics.

As for the content, it is naturally the paper on the proof of the Kakutani conjecture.

After reading the email from beginning to end, a smile appeared on Lu Zhou's face again.

Although it was expected, after seeing this email, he was really happy for his students.

According to the decision of the editorial board of "Annual of Mathematics", their papers will be published in the latest issue of the journal and will be examined by the entire mathematical community...