New Developments in the Navier-Stokes Equations

Shen Nuo Fei will be one month old tomorrow, and today happens to be the day before the start of the new school year for universities, primary and secondary schools across the country.

At the request of relevant departments, primary and secondary school students across the country will watch the "Back to School Classroom" program at 8 p. m. tonight.

This year's "Back to School Classroom" is different from previous years. It starts broadcasting promptly at 8 p. m. without any commercial breaks.

The warm-up act wasn't a heavily made-up entertainment talent show singer, but rather the Chinese men's basketball silver medalist team from the World University Games.

The young men on the basketball team come from universities across the country. In the recently concluded men's basketball semifinals of the World University Games, they staged a stunning comeback from a 16-point deficit, relying on their tenacious fighting spirit and impeccable teamwork to eliminate the strong Russian team and make history by advancing to the finals, winning a precious silver medal.

The Chinese university men's basketball team displayed tremendous spirit and demonstrated the passionate and masculine power of Chinese youth, moving the entire nation.

The Chinese university men's basketball team, returning with honors, introduced themselves at the beginning of the "Back to School Classroom" program. They are all university students, some majoring in materials science, some in mechanical engineering, and some in law, among whom are many academic high-achievers.

The program then played a video clip, a compilation of highlights from renowned Chinese scholars.

Physicists are constantly breaking records in quantum entanglement, astronomers are building national instruments to explore the universe, writers are expressing their inner thoughts through ink and brush, and mathematicians are solving century-old problems with great enthusiasm.

Shen Qi was holding his daughter in front of the TV watching "Back to School Classroom," and the mathematician on the TV screen was him.

Shen Qi was very satisfied with this year's program, saying, "It's right, this is real back-to-school education." He added, "Patriotic and scientific education should start from a young age."

After the program aired, no parents across the country complained; they all gave it a thumbs up.

The next day, school started.

Shen Qi held a one-month celebration for his daughter today, inviting a small number of relatives and friends to the banquet.

Zhou Yu'an brought over a bunch of gifts, including milk powder, baby clothes, dolls, and all sorts of other items.

"My dear niece, Nuofei, I wish you a healthy and happy life," Zhou Yu'an said to Shen Nuo Fei.

Shen Nuo Fei smiled but didn't say anything. She was developing relatively quickly; at one month old, she would smile at things she found interesting, but she couldn't laugh out loud. It would probably take another month or two for her to laugh out loud.

Xiao Junlong gave her a piece of jade, saying it had a special history and was suitable for a girl to wear, "Little Nuofei, beautiful jade for a beautiful woman."

A one-month-old baby can't laugh out loud, but can cry. Shen Nuo Fei burst into tears; Xiao Junlong's appearance frightened her.

Ou Ye's best friend, Xiao Lian, gave her a baby hat.

Shen Nuo Fei smiled again. She would smile whenever she saw men and women with normal appearances, which was quite endearing.

Witten arrived in China a few days ago; he had been resting in the United States for more than two months.

When in Rome, do as the Romans do. Witten gave them a red envelope containing 1,000 RMB. What a kind person!

After the full moon celebration, Shen Qi, Witten, Zhou Yu'an, and Xiao Junlong came to the center to discuss important matters.

Shen Qi said, "Zhou Yu'an has done a fantastic job recently. He proved the Cornerstone Principle and obtained an important set of data on the Navier-Stokes equations using the Infinite Flow Algorithm for Partial Differential Equations. Zhou Yu'an, why don't you explain it?"

Zhou Yu'an started the presentation by showing the PowerPoint slides. "Based on Professor Witten's algorithm framework and Professor Shen's supplementary improvements, I have arrived at the following conclusions."

Four sets of formulas are displayed on the screen:

(1)uj/t+Σukuj/xk=△uj-p/xi+fj, (t, x)∈Ω, i=1,2,3; divu=Σuk/xk=0,(t, x)∈Ω

(2) u(t, x)|t=0=0

(3)u(t, x)∣xk=-π=u(t, x)=π, u/xk∣xk=-π=u/xk∣xk=π, k=1,2,3,0≤t≤a

(4)∫Qp(t, x)dx=p0, p0=const>0

"The problem of the Navier-Stokes equations is transformed into the problem of finding the velocity vector and scalar function at the point x∈Ω and time t∈(0, a)." Zhou Yu'an switched the PPT to the next page.

This page contains a corollary. "Corollary 1: For f∈L2(Ω), if we find a solution to equations (1)-(4), that is, we have solved the generalized solution to the Navier-Stokes equations, and this solution has universal significance."

"Awesome, Lao Zhou!" Xiao Junlong stood up abruptly. Zhou Yu'an had been in seclusion for the past few months and had accomplished something extraordinary.

Zhou Yu'an has completed all the previous derivation and proof work. Now the only problem, and the most crucial one, is to find the solution to equations (1)-(4) and prove this corollary.

"Looks like there's nothing for me to do," Witten said with a smile, gesturing to acknowledge Zhou Yu'an's efforts.

Shen Qi said, "So now we only need to do one thing, as Zhou Yu'an just explained very clearly: prove corollary 1, that is, find the solution to equations (1)-(4). So Zhou Yu'an, you'll be the one to deliver the final blow."

"I can't handle it." Zhou Yu'an shook his head and looked at Xiao Junlong: "Old Xiao, why don't you go?"

Xiao Junlong shook his head. "Old Zhou, if you can't handle it, I definitely can't either. The later parts are no longer purely Mathematical Physics, Partial Differential Equations, or Fluid Mechanics; they'll require the use of topology and functional theory. Actually, Old Zhou, we both have strong strengths, but we also share a weakness: our expertise is relatively narrow. For comprehensive handling, we'll still need Professor Shen and Professor Witten to step in."

All four individuals have contributed to the advancement of the NS equation project to this stage.

The current situation is that Generals Zhou and Xiao have cleared out all the smaller monsters, and only one big boss remains that requires the two of them to deal with.

The two big shots exchanged a glance and agreed, "Alright, let's get started."

The NS equation project has entered a critical phase, and Shen Qi and Witten have divided the work.

Witten has a clear mind and a good overall perspective, but he is getting older and struggles with high-intensity, detailed calculations.

During the critical phase, Shen Qi personally handled the tedious and complex derivations and calculations, while Witten was responsible for verification, identifying problems, and finding loopholes.

After expanding equations (1)-(4), a large number of partial differential and Ordinary Differential Equations are derived, as well as more than a dozen definitions and lemmas that need to be proven.

To simplify the calculations, Shen Qi invented a new tool and discussed it with Witten.

"From Lemma 2 to Lemma 9, the normal approach is to consider the cases of u2, u3, and infinity. This would require us to use Lemma 3 infinitely. The most troublesome case is the non-negative monotonically increasing function, which means our workload will increase by more than a hundred times."

"So I invented this new tool. It's a processing method that works, and it's simple and efficient." Shen Qi showed Witten his new method. Let PN be an N-dimensional orthogonal projective operator in H. Then, for any x∈H, as N→∞, PNx→x.

Witten has verified that the new tool satisfies the commutativity of P and A, and can be used in the computational proofs of Lemmas 2 to 9 to simplify the computational process and improve efficiency.

Even with simplified procedures, the work continued throughout September, and Shen Qi still failed to calculate the generalized solution to the Navier-Stokes equations.

Regardless of whether he could solve the Navier-Stokes equations, Shen Qi would leave work and go home promptly at 5 p. m. every day.

Today, when Shen Qi was playing with his two-month-old daughter, she laughed.

"Nofi, I know you laugh at me for being useless, for not even being able to solve the Navier-Stokes equation. How about this, I won't solve this equation for now. You can solve it when you grow up, okay?" Shen Qi stroked his daughter's rosy cheeks.

The daughter continued to laugh, drooling with laughter.

Shen Qi: "Daughter, can you give me a hint?"

Shen Nuo Fei: "Hehehe."

Shen Qi was pleasantly surprised. "Nofi laughed out loud! She's already laughing out loud after just two months!"

Seeing Shen Qi like this, Shen Nuo Fei couldn't stop laughing. This was a particularly cheerful baby girl. "Lelal...

"Lelouch lap lap lap lap, periodic boundary?" Shen Qi suddenly realized. His daughter was talking about periodic boundary, so she must be right!

Shen Qi handed his daughter to Ou Ye: "Honey, take the baby, I'll be right back."