Obviously from the results point of view, compared to proving a Millennium Prize Problems, just raising Greenwald's limit is not that important.
After all, one is one of the most important core problems in the field of mathematics, and it is also the last problem in classical physics. It has countless applications in reality. It can play a very important role in any problem related to fluids.
The other is just a slightly important issue in the field of nuclear fusion. For nuclear fusion, what is more important are the major material issues. Only by solving these material issues can controllable nuclear fusion be realized. As for the Greenwald limit problem, it will have to wait until controllable nuclear fusion is officially realized before it can be meaningful.
Therefore, in order to solve a Greenwald limit problem and solve the smoothness problem of NS equation, it can really be called making a meal of dumplings for a plate of vinegar.
"Well..."
Xiao Yi thought for a moment, and then thought, "Let's see if we can solve it through other methods first."
Then he temporarily put this issue aside and began to study how to combine NS equation with the multi-fluid model from other angles.
In this way, time passed quietly for a long time.
It wasn't until it was almost night that he raised his head again and sighed helplessly.
"It looks like it won't work anymore."
For nearly half a day, he considered at least 4 angle methods, but either it didn't work, or the final model did not achieve the effect he wanted, or it also showed singularity and instability.
Until finally, he focused on that issue again.
NS equation solution existence and smoothness.
"It seems that we really need to solve this obstacle."
Director Zhao, Director Zhao, you really gave me a difficult problem.
Xiao Yi sighed a little in his heart.
I guess Zhao Zhanyou didn't expect that a problem he asked Xiao Yi to help solve would eventually lead to this top problem in mathematics.
But no matter what, since he had promised it in the first place, he had to give it a try.
Anyway, it's just Millennium Prize Problems.
It's not like he hasn't solved it before.
"It just so happens that I haven't studied mathematics for a long time."
Although most of the projects he had solved before, he used mathematical methods to solve various problems, but it has indeed been a while since he studied purely theoretical mathematical problems like this.
"Come on!"
This is one of the Seven Millennium Problems, the last problem in classical physics!
Xiao Yi The research on the existence and smoothness of the solution of NS equation has officially begun.
NS equation, which refers to the Navier-Stokes equation, is named after the French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Stokes. It is a set of partial differential equations that describe the motion of fluids such as liquids and air.
The history of research on fluids has begun more than 20 centuries ago. Everyone has heard the story of Archimedes using buoyancy to measure gold. The resulting Archimedes' principle can be regarded as the earliest scientific theorem of fluid mechanics.
Later, Da Vinci also observed fluids and made the earliest description of vortices and turbulence. Later, Newton, Euler, and Bernoulli made more and more contributions to fluid mechanics.
It was not until the 19th century that Navier and Stokes summarized various previous studies on fluid mechanics and finally came up with the most important equation NS equation.
It can describe the conservation of momentum and mass during the movement of various types of fluids, and use it to analyze the behavior of fluids during the flow process.
It can be said that this equation is constantly playing a role in all walks of life.
From aircraft flying in the sky, to pipeline design under cities, or submarines hundreds of meters under the sea, NS equation is needed to analyze the fluid problems caused by them.
As for why it is said to be the last problem in classical physics, it is because other related problems in classical physics have been well solved. However, using NS equation to study fluid mechanics, since every fluid molecule in the fluid is completely free and can be affected by various other molecules around it, there are still places that people cannot analyze.
Especially at high Reynolds number, the fluid will become very complex and form turbulent flow. This is also one of the most complex phenomena in fluid mechanics, because its flow is highly random and irregular, just like a chaotic system. Its precise description and understanding is still a huge challenge.
Unlike other classical mechanics problems, they are not as complex as those in fluid mechanics, and can basically be solved quite completely, such as in construction engineering and so on.
Of course, precisely because the problem of NS equation is difficult enough and has a very important position in the industrial field, there are quite a few people studying this problem, which can be said to be one of the best in the mathematical physics industry.
"However, the complexity of this problem is indeed quite high..."
As Xiao Yi began research on NS equation, various difficulties followed.
The existence and smoothness of NS equation solution are two issues.
Existence requires proof: for given initial conditions and boundary conditions, can a solution that satisfies the Navier-Stokes equations always be found?
Smoothness requires proof: is the found solution smooth over the entire time scale? That is, whether all derivatives of this solution exist and are continuous.
Of course, there is a more perverted problem above this, which is to solve NS equation.
As an equation, it can naturally be solved. However, NS equation is too complex, so the possibility of obtaining its analytical solution is too great. The mathematical community generally believes that this is basically impossible, so the academic community basically only requires that the existence and smoothness of its solution can be proven.
"For NS equation in the two-dimensional case, its global existence and smoothness have been proven. However, in the three-dimensional case, it has only been proved that there is a local smooth solution, that is, the solution exists and is smooth within a certain limited time interval. However, for the global case, it has still not been done."
"When proving the overall situation, there is always the problem of singularity."
After consulting the relevant information about NS equation, Xiao Yi analyzed the various problems encountered by the current academic community in the process of studying this issue.
Singularity is also a problem he encountered when combining NS equation with the multi-fluid model. Its main manifestation is that the fluid velocity becomes infinite or its derivative becomes infinite in a limited time, as if it suddenly exploded. Therefore, this problem is also called the explosion solution problem.
This issue can be said to be the key problem in the process of proving NS equation.
Xiao Yi began to deduce the process of explosive decomposition on the scratch paper.
[{[u/t+(u)u=p+νΔu], [u=0]...]
[lim(t→T)∥u(, t)∥Hk=∞]
At some point, the speed becomes infinite.
In a physical sense, it means that at a certain moment, the speed or energy density of the fluid becomes infinite.
It was as if the originally calm water in the pond suddenly exploded. He even used his imagination to turn the grenade into a water grenade. Before throwing the water grenade, he shook it vigorously to make the fluid inside it turbulent enough, and then threw it out. After finally throwing it next to the enemy, it exploded directly. The power caused may even be stronger than the grenade.
It can even be used to explain the principle of ability of those who have mastered water-related superpowers, that is, they can control the fluid force in the water and accurately control it to reach the "explosion cracking" condition at a certain moment.
But obviously, this is a completely impossible thing.
This issue has been discussed in academic circles for many years, but there has been no good way to explain the conditions for the formation of explosive cracks.
"For now, some related mechanisms have been mainly discovered, well... one is called energy cascade and turbulence, which means that in turbulence, energy is transferred from large scales to small scales, which may cause the velocity or velocity gradient on the small scale to become very large, eventually leading to an explosion."
This is easy to understand. Because of the conservation of energy, the fluid energy of one cubic meter was condensed into one cubic millimeter. This naturally led to a sharp increase in the fluid energy within this cubic millimeter, and then an explosion occurred.
"Furthermore, the influence of initial conditions, such as certain initial flow fields that are highly concentrated and have strong rotational characteristics, may produce explosive solutions."
"Therefore, finding a good initial condition is the key to proving NS equation."
"Initial conditions... initial conditions..."
Xiao Yi entered into deep thinking.
Until finally, he stood up suddenly.
"Need data!"
Since you want to find appropriate initial conditions, a simple method is to set various initial conditions, and then use numerical simulation to determine the subsequent impact on the fluid under different initial conditions, such as when it will undergo explosive solution, and the changes in its data during the period before the explosive solution occurs.
And this requires the help of computers.
Heading to the computer room, he set up the model and started running it on the laboratory server.
But this time, a problem occurred.
The computing power of the laboratory server is not enough!
Looking at the slowly advancing progress bar, his expression darkened.
When Wang Hao next to him saw this scene, he couldn't help but was stunned and said, "Our server is worth 10 million. Professor, what do you mean by this?"
Xiao Yi twitched his lips and said, "Some direct numerical simulation models of NS equation."
"Oh, that's not surprising."
Wang Hao nodded.
The direct numerical simulation of NS equation can be regarded as one of the models with the highest computing power requirements, and Xiao Yi does not only need to calculate one model, but hundreds of models, covering various situations of fluid operation, such as pipe flow, boundary layer flow, open channel flow, etc., and the computing power requirements are even higher.
"Remember, when we expand the laboratory next time, we will also expand the computer room. By the way, how much does it cost to build a supercomputing center? What is the scale of 100 million?"
Xiao Yi waved his hand and was ready to build a supercomputing center for their laboratory.
I have no choice, I have too much money and I am willful!
Wang Hao next to him swallowed.
Good guy, 100 million, it seems that their laboratory is really rich.
"Yes, Professor!"
It is inconvenient to do it in the laboratory. After all, the laboratory server is not only used by him, but also by other people sometimes. He is not that dictatorial.
The amount of data this time requires supercomputing.
Coincidentally, Fei City has a supercomputing center that has been built in recent years, and its name is also very nice, called Chaohu Mingyue.
It looks like you can go take a look.
Chaohu Mingyue Supercomputing Center is located in Fei City High-tech Zone. Its name naturally comes from a famous lake in Hui Province - Chaohu.
"Professor Xiao, I have admired you for a long time!"
Chaohu Mingyue Director Tu Ran of the Supercomputing Center stood at the door. When he saw a group of people walking over, he smiled and greeted the leader.
That person was naturally Xiao Yi. After determining that he needed to use supercomputing, he directly contacted the Chaohu Mingyue Supercomputing Center and prepared to come and observe.
"Hello, Director Tu."
Xiao Yi stepped forward and shook hands with Director Tu, and then looked at a glass cube in front of him.
Chaohu Mingyue supercomputer, placed in this cube, looks pretty good.
After exchanging a few words, Xiao Yi pointed at the glass cube in front of him and asked, "Is this Chaohu Mingyue?"
Tu Ran laughed, then nodded and said, "Yes, our Chaohu Mingyue is right here, and we can see it from the outside."
"Let me introduce to Professor Xiao the 'Super Brain' of our Fei City."
Xiao Yi nodded, "I'm all ears."
Then, Tu Ran began to introduce, "This glass cube is 16.8 meters long and 13.5 meters high..."
While listening to Tu Ran's introduction, Xiao Yi also walked in.
Tu Ran has many of them, but Xiao Yi only remembers the most critical data among them.
Double precision calculation peak 12PFlops.
"How high is this peak computing power ranking in the world?"
Xiao Yi asked.
Tu Ran was stunned for a moment, then replied, "In terms of ranking, our country, China, is not participating in this ranking comparison now, but judging from the data released now, it can probably be ranked around 100 in the world."
Xiao Yi nodded.
This kind of computing power should be enough to calculate his model.
Then he said, "Well, Director Tu, as I said before I came here, I plan to borrow Chaohu Mingyue to calculate a model. Can I go up and try it now?"
"Haha, of course no problem."
Tu Ran smiled and nodded, then led the way. "Come with me, the computer room is over here."
Arriving at the computer room, Xiao Yi also took out the laptop he was carrying.
Connect the laptop to the supercomputer host, then upload the model, and soon the calculation will begin.
But this time, the progress was faster.
Tu Ran was a little surprised, "Professor Xiao, what is your calculation model? The occupancy is so high."
Connecting to one server is not enough, it requires connecting to four servers.
"Fluid mechanics stuff."
"I see." Tu Ran expressed understanding.
At this time, Xiao Yi waved to Wang Hao behind him. Wang Hao understood, and then put the box he was carrying on the table next to him. When he opened the box, he could see that it was filled with hard drives.
Then Xiao Yi said, "I would also like to ask Director Tu to help copy some of the calculated data to these hard drives."
To him, the calculation results of these models are not important, because these results are destined to "explode" suddenly within a certain period of time.
The most important thing is the changes in the process from the beginning of calculation to the occurrence of explosion.
The data generated in this process is the most important.
Seeing this, Tu Ran didn't know whether to laugh or cry, and said, "Okay, I finally understand why Professor Xiao, you want to run here specifically for a model. It turns out it's for data."
Xiao Yi also smiled knowingly.
If he just calculates the model results, of course he can directly connect the computer to the supercomputer through the network in the laboratory, and then he can calculate it.
But the problem this brings is that the speed at which the data calculated by the supercomputer is transmitted to his side is limited by bandwidth and will be very slow.
Therefore, the fastest way is to physically transport the hard drive.
It only takes half an hour for him to drive back to the laboratory from here. He can also visit the supercomputing center here. If he waits for the data to be uploaded here and then downloads it over there, the transmission speed will probably be about the same, or even lower.
This is what is called physical transportation speed being faster than network transmission speed.
For example, in the original Black Hole photo, the amount of data finally obtained by several observatory observation centers was almost 5PB. In the end, several vehicles directly drove the hard drives to the research center.
Just like that, while waiting for the calculation to be completed, Tu Ran continued to take Xiao Yi to visit their supercomputing center.
Xiao Yi also asked about his opinions on building a supercomputer for Science Island Laboratory.
In this way, time passed quickly.
Until the evening.
The data calculation is completed, and during the calculation process, the data are also stored in the hard drives brought by Xiao Yi.
There is a total of more than 300 terabytes of data.
With so much data, the server in their laboratory was really trying to pull a big cart.
In this way, after bringing the data and expressing their gratitude to Tu Ran, they left the supercomputing center directly.
Next, comes the main event.
I hope these data will not disappoint him.