Using the Supercomputer
Shen Qi went to the study, spread out the paper roll, and began to deduce and calculate.
The problem of the Navier-Stokes equations is the problem describing the dynamics of incompressible fluids. This problem involves the existence and smoothness of the solutions to the Navier-Stokes equations for incompressible viscous fluids.
The Navier-Stokes equations project team has completed the preceding discussions; now only the final step remains: finding a complete solution to the existence and smoothness of the Navier-Stokes equations for incompressible viscous fluids.
Shen Qi believes that among the four key formulas, formula (1) is the most important key, so Zhou Yu'an listed it as formula (1) and placed it at the beginning.
Zhou Yu'an's achievement in dealing with the Navier-Stokes equations, Millennium Prize Problem that has puzzled people for many years, demonstrates tremendous academic progress and a high level of mathematical ability.
The final blow fell into Shen Qi's hands. He made a new calculation, adding boundary conditions to equation (1), that is, giving periodic boundary conditions and initial conditions for the spatial coordinates.
"Just a little bit more, almost there."
As his thinking became clearer, Shen Qi peeled away the layers of disguise of the problem, revealing the final veil and cracking the code.
After Shen Qi's refinement, equations (1) to (3) determined the initial and boundary conditions. The only condition that could not be determined was the pressure p(t, x).
After several years of preparation, a year of continuous discussion and implementation, and a month of intensive efforts, the NS equation project has progressed to this point, and no pressure can stop Shen Qi's progress.
Shen Qi obtained a set of vector functions f(t, x)=(f1(t, x), f2(t, x), f3(t, x))∈R^3 in Hilbert space. In one go, he calculated the scalar product in Euclidean space, as well as the most important norm.
The final layer of mystery surrounding the Navier-Stokes equations was unveiled tonight in Shen Qi's study.
Shen Qi wrote down an equation on a piece of paper. This was the last equation in the entire project. Compared to the lengthy system of partial Partial Differential Equations before it, the final conclusion was not too complicated:
Σ(∣∣uk(t,)∣∣^2L2(Ω)+∫∣∣graduk(η,)∣∣^2L2(Q)dη)≤Σ∫∣∣fk(η,)∣∣^2L2(Q)dη
"Yes, this is the universal solution to the Navier-Stokes equations! I've finally found you!" Shen Qi clenched his fist and waved it twice. He rushed out of the study, wanting to kiss his daughter and wife.
"Never mind, let's kiss when it's light." Shen Qi glanced at the wall clock; it was 2 a. m., and his wife and daughter were already asleep.
Shen Qi returned to his study and verified it. The general solution satisfies equations (1)-(4), making it undoubtedly the correct answer.
As Shen Qi finished organizing the manuscript, the morning sun rose.
"You stayed up all night?" Ou Ye got up; the child was still asleep.
"The Navier-Stokes equations are done. There won't be many projects that will keep me up all night anymore." Shen Qi sat down at the dining table and had breakfast with Ou Ye.
"It's done! That's great! Let me see it!" Ou Ye was very excited. She breastfed every day and stayed at home during her postpartum confinement. Being a wet nurse for a long time can actually be quite boring.
Although it wasn't directly related to her major, solving the Navier-Stokes equations was like a shot of adrenaline for Ou Ye.
Shen Qi handed the manuscript of the Navier-Stokes equations to his wife. "I've tidied it up, but it's still a bit messy. This is the first draft. Please take a look and make corrections, Dr. Ou."
Ignoring the incredibly complex derivations, calculations, and arguments at the beginning, Ou Ye flipped directly to the conclusion page, staring intently at the general solution of the Navier-Stokes equations without blinking. "That's it?"
"Yes, this is it." Shen Qi nodded, taking a sip of coffee to perk himself up.
It was difficult for Ou Ye to fully understand Shen Qi's manuscript in a short time. Her intuition told her that her husband had done something incredibly impressive again, "Congratulations, husband, but you still need to take care of your health and avoid staying up late."
"My wife is the best." Shen Qi kissed Ou Ye, and suddenly on a whim, he pulled out the last page of the manuscript, the one with the general solution of the Navier-Stokes equation, and wrote on the paper: My beloved daughter Shen Nuo Fei is 64 days old. Her father has broken the Navier-Stokes equation, and her mother has passed the first verification.
"You're keeping this as a souvenir!" Ou Ye chuckled and said, "Nofi's mother, because her profession isn't relevant, just smiled and remained silent; she was just there for show."
Shen Qi handed the manuscript to Ou Ye: "The last page is the most important conclusion page. This page is the first draft of the manuscript that I wrote when I was at my peak. Please keep this page safe and pass it on to Nofi in the future, so that it may be passed down from generation to generation."
Ou Ye took the manuscript. "I'll have Sister Cuiping take it out and frame it later."
Shen Qi hugged his wife and laughed heartily, "That's wonderful!"
Upon arriving at the center, Shen Qi rewrote the last page, scanned the more than 200 pages of manuscript into PDF format, and then convened a meeting with Witten, Zhou Yu'an, and Xiao Junlong.
"I've already calculated the general solution. Now, please verify it yourselves. I'll be waiting for your news." Shen Qi, who had stayed up all night, relied on caffeine to stay awake. He looked energetic, but in reality, he was exhausted.
"The boss really is a big shot!" Xiao Junlong jumped up excitedly, and Witten and Zhou Yu'an were equally excited.
Witten and the other two each found a quiet, private space to verify their respective solutions to the Navier-Stokes equations.
Shen Qi took a nap at the center, and when he woke up, he was fully recharged.
Manual verification cannot be completed in a day or two. While the manual verification was underway, Shen Qi began to contact the Yanjing University Supercomputing Center to reserve the right to use the Shen Teng X8810.
The Deep Computing series of supercomputers are all developed and manufactured by the same Chinese company. Several years ago, Yanda University purchased the then-latest supercomputer, Shen Teng X8810, and named it "Weiming-1". After several years of operation and exploration, "Weiming-1" has become mature and stable, ranking among the top 20 supercomputers in the world.
Manual verification mainly involves logical reasoning and reverse deduction, as well as checking and verifying the massive amounts of data processed and predicted. No human can handle this alone, not even Shen Qi. Supercomputing is necessary.
When the director of the Yanjing University Supercomputing Center heard that Shen Qi to use "Weiming-1" to verify the Navier-Stokes equations, he fully supported him and reserved time for Shen Qi to use it.
Witten, Zhou Yu'an, and Xiao Junlong are very familiar with this NS equation solution, and they were responsible for much of the work.
Even so, it still took Witten, which was the first to complete the manual verification, more than a week.
"Wow, perfect! Amazing! It seems we really solved the Navier-Stokes equations, at least logically." Witten and Shen Qi shook hands and hugged; the two Fields Medal winners had worked together to complete a project.
A few days later, Zhou Yu'an and Xiao Junlong completed the manual verification. No problem, let's go to the supercomputer!
The Yanjing University Supercomputing Center welcomed the NS Equation project team, led by Shen Qi and Witten.
"Weiming-1" has been launched, and the researchers at Yanjing University Supercomputing Center are all ready and waiting for Professor Shen's order to start the computing program.