After writing down the title and introduction, Xu Chuan began to step into the main text.

Citing the paper on the compressible navier-stokes equation of thermal conductivity by professors Pan Ronghua and Zhang Weizhe, the initial value condition is relaxed."

"Then (v,υ,θ)(×)∈H*H*H becomes (v,θ)∈H(0,1),υo∈H(0,1)"

"There is some normal number C and there is no η>0 such that for any (x,t) ∈(0,1)(0,∞)."

"C≤υ(x,t)≤C,C≤θ(x,t≤C), and ||(υ-∫υdx，υ，θ-∫υdx) (·，t)||H(0，1)≤Ceηt”

In the study, Xu Chuan began to explore the NS equation.

It's a puzzle that spans three centuries, and it's incredibly difficult to solve.

Since Saint-Venant and Stokes independently proposed a formal equation with a constant viscosity coefficient in 1845 and named it the Navier-Stokes equation, there have been many mathematicians and physicists who have studied it for two centuries.

However, there are only a handful of major breakthroughs in this area.

At present, the greatest progress in the mathematical community on the NS equation is the phased achievement that he and Fefferman advanced together when he was in Princeton.

It is possible to determine the existence of a solution in the surface space, given an initial condition and a boundary condition.

Now, Xu Chuan wants to take it a step further, so that he can give a finite boundary and a Dirichlet boundary, and in three-dimensional space, the Navier-Stokes equation has a real solution and the solution is smooth.

If this can be done, it will almost be possible to build a mathematical model of the plasma turbulence in the chamber of a controlled nuclear fusion reactor and use a supercomputer to control it.

For Xu Chuan, he is not looking forward to solving the NS equation or anything at the moment, and it is not a good and reliable idea.

It has been nearly 200 years since the NS equation was proposed, and it still stands like a peak with no end in sight.

Countless climbers do not even approach the base of the mountain, and people cannot see its summit, but can only look through the fog from afar.

Xu Chuan did not dare to say that he would be able to complete the solution of the NS equation in his lifetime.

Not only because it is difficult, but also because it is a huge systematic project.

The Clay Institute's definition of 'the existence problem of smooth solutions of N-S equations in three-dimensional space' is just a prelude to the NS equation.

In the villa, Xu Chuan has not been out for more than a week.

His progress on the NS equation was relatively smooth at the beginning, partial differential equations were one of his research areas in his previous life, coupled with the fact that he had taken mathematics as his major in this life, in this area, he had successfully gone beyond the previous life.

But that didn't make it easy for him to go down the NS equation, and two days ago, he was stuck in a bottleneck and is still looking for a way to solve it.

In the study, Xu Chuan frowned and stared at the equation on the manuscript paper.

“U``=-(1/v)(1-cosA)U。”

This is a very simple formula, which is a harmonic equation with a function as a coefficient, which is derived from the deformation of the deformation tensor S+R decomposition theory of Chen Zhida for the wall flow with zero pressure gradient, and the deformation of the velocity profile U(y) theoretical equation.

From this equation, it can be concluded that with the increase of wall distance, the scale of turbulence evolves from a small scale of ultra-high wavenumber to a super-large scale that tends to zero wavenumber.

In general, it can be applied to almost all turbulences instead of Euler's equations, resulting in a universally valid system of equations.

In addition, for this equation, it has been confirmed that Plante's logarithmic law velocity is the theoretical solution of the equation.

Therefore, it can be considered that for the ideal wall flow, the theoretical solution is consistent with the experimental solution.

To put it simply, in an ideal world, the turbulent operation state calculated by the mathematical formula is exactly the same as the actual operation.

If this can be done, it can be used to build mathematical models to predict and control turbulence.

However, it has a fatal problem!

That is, the turbulent region is the region where cosA evolves from a region that cannot be approximated to 1 to a region close to 0, and it is difficult to obtain a universally valid analytical solution.

This is the deadliest point for the bizarrely shaped chamber of a controlled nuclear fusion reactor.

Xu Chuan wanted to find a way to make up or replace it, but so far he has not been able to do so.

More crucially, mathematically, the strict formula for acceleration is proved using the Lie derivative.

Therefore, although the acceleration of the microelement derived by S+R is essentially the same as the Lie derivative, it is very different in the mechanical (physical) interpretation.

At present, the scientific community generally accepts Euler's equation based on the Lie derivative, or NS equation.

Therefore, there is little supporting literature in the theoretical community for the wall flow equation given here, as well as for the general equation of turbulence.

In other words, Xu Chuan couldn't do it if he wanted to consult and learn from previous literature and papers.

This is an almost blank field.

In the study, after crumpling the manuscript paper in his hand into a ball and throwing it into the trash can on the side, Xu Chuan stared at the brand-new A4 paper and breathed a long sigh of relief.

He has been stuck on this issue for almost ten days since the derivation reached a bottleneck, but nothing has been found.

Of course, it can't be said completely, at least in the past ten days or so, he has eliminated a variety of methods that cannot be used.

Shaking his head, he was about to continue writing, but after thinking about it, he threw the pen in his hand aside.

Looking up at the ceiling for a while, Xu Chuan pushed away the chair and stood up.

Perhaps, he needs a little help.

He thought of a previous life solving the problem of the existence and mass interval assumptions of the Yang-Mills gauge field.

At that time, as this time, it was limited by a bottleneck for a long time.

The NS equation, like the Yang-Mills gauge field existence and mass interval assumptions, is not just a mathematical problem, but also a physical problem.

Perhaps, he can think of a way from a physical point of view.

Mathematical thinking aside, physically speaking, the fastest way to study a problem is to practice.

Turbulence is everywhere, in the wake of high-speed airplanes and in bathtubs filled with water.

Its essence lies in the injection of energy from the largest scale to the smallest scale through the formation, interaction and demise of vortices.

Put simply, the orderly flow of fluid forms a vortex that interacts and splits into smaller vortices, which continue to interact with each other, and so on......

But this chaos has puzzled scientists for centuries.

There is currently no mechanistic framework that can resolve how the interaction between vortices drives such an energy cascade.

And for physicists, there is a solution to a difficult problem that physicists often use!

That is to put these things together and "smash" them completely!

For example, in order to understand the basic components of the universe, theoretical physicists have built large strong particle colliders to accelerate microscopic particles and cause them to collide to obtain data.

This time, in order to reveal the basic mechanism of turbulence and find a solution to the NS equation, Xu Chuan decided to let the vortex collide with the vortex and see its structure and motion at the microscopic level.

At Nanjing University, Xu Chuan went straight to the School of Physics, found Yu Yongwang, the dean of the School of Physics, and made a request to borrow the equipment of the Institute.

Regarding Xu Chuan's request, Dean Yu agreed directly without thinking about it.

In the physics experiment building, Xu Chuan called two of his students and asked them to help them. Under Yu Yongwang's arrangement, NTU also called two doctoral students to help.

In fact, it is not difficult to create turbulent collisions.

A variety of marine organisms can make vortex rings underwater with air and fast-moving water.

This is because when the circular bubble moves forward, it will be subjected to the squeezing force of the front water and the side backward water surface friction, which will cause the originally round bubble to be flattened, and the edge will be disturbed by the backward force, the air at the edge will be disturbed to rotate, thus forming a vortex at the edge, and gradually the middle is separated, forming a vortex ring.

The difficulty of the experiment is to use an ultra-high-resolution camera to record the collision of the two turbulent flows, and then use a 3D visualization program to reconstruct the collision process to determine the basic mechanism of turbulent evolution.

"Professor, I've adjusted it here, the A1 vortex ring uses green material, and the A2 vortex ring uses red material."

In the laboratory, Gu Bing reported loudly to complete the work in his hands.

Xu Chuan nodded and said, "Okay." ”

On the other hand, with the help of students majoring in photogrammetry and remote sensing, Amelia also successfully completed the erection and commissioning of the ultra-high-resolution camera.

"Professor, the super-resolution camera is ready and ready to record."

Under the command of Xu Chuan and the help of Nanda, the equipment used for the vortex ring collision experiment was quickly assembled.

The experiment was opened, and under precise control, the vortex ring makers located on both sides of the tank simultaneously launched a bubble forward, and under the high-speed movement, the bubble evolved into a vortex ring, and then collided together in the center.

The red and yellow vortex rings formed a visible melange ripple and ring at the moment of collision, but in just a second, these ripples and rings dissipated into a piece of dye.

But for Xu Chuan, this is enough.

In this laboratory, Xu Chuan specially found a powerful scanning laser film, synchronized on a high-speed camera, and the combination of the two allows it to capture hundreds of thousands of images per second.

The ultra-high-resolution high-speed camera accurately recorded the entire experimental process and transmitted it to the computer.

All that remains is to reconstruct the collision process using a 3D visualization program.

"Professor, is this experiment done?"

In the laboratory, Amelia looked curiously at her classmates who were disassembling the equipment, and asked Xu Chuan.

Xu Chuan nodded and said, "Well, it's done." ”

"Can I ask what this is researching? Eddy? Or turbulence? ”

Being called over in a hurry, Amelia and Gu Bing were a little curious about what their mentor had been doing since he disappeared for most of the month.

Xu Chuan smiled and replied, "Study the NS equation." ”

Amelia opened her mouth, looked at Xu Chuan with some surprise, and then looked at the equipment being disassembled: "Just use this?" ”

Xu Chuan said with a smile: "Of course, the NS equation is the study of fluid mechanics, and eddy currents are also part of fluid mechanics. ”

In fact, physicists have been using the Vortex Collider to study turbulence since the '90s, but previous experiments have failed to slow down and mechanically reconstruct the moment of chaos when a collision occurs.

The reason why Xu Chuan did this was also because of the experience brought by rebirth.

In later generations of aerodynamics, it was very common to study the reconstruction of chaotic systems in the system system, so he added it easily.

"Professor, can I join your research?" Amelia asked expectantly.

She studied mathematical physics in college, and she is also quite interested in NS equations, and joining Xu Chuan's research will definitely learn a lot even if it doesn't help much.

On the side, Gu Bing also cast an expectant gaze.

Noticing the eagerness of the two students, Xu Chuan smiled and said, "You better complete the task I gave you before." ”

It's not that he doesn't want two students to work on his own project, but they probably don't have enough energy and time.

Last year, he didn't take many students, but this year is different, and at the beginning of the year, he personally deployed a Hodge-like math problem and handed it to them.

This problem is estimated to consume all their daily time.

If it can be solved, they will not be far from graduation.

After a few days of tossing and turning, the 3D visualization of the vortex collision was finally completed.

NTU immediately sent the reconstructed data.

After receiving the data, Xu Chuan made tea and turned on the computer.

Since he was inspired by the tea mist in Qiu Chengtong before, he has now also started to make tea and drink tea, hoping to continue to get inspiration and ideas from it.

Although this was not useful, Xu Chuan unexpectedly found that drinking tea allowed him to maintain a certain level of concentration in his daily research, so he began to get used to making a cup of tea before engaging in research.

Holding his teacup, he took a sip and opened the reconstructed vortex ring collision experiment.

This is a completely different picture from the eye, and after the reconstruction, the color of the vortex ring is completely disappeared or unified.

But Xu Chuan keenly noticed that when the vortex rings collided with each other, they would be stretched outward, and their edges would form an antisymmetric wave.

The wave peaks of these waves develop finger-like filaments that grow along the core perpendicular to the collision where the collision occurred.

These fingers then rotate in the opposite direction to their neighbors, forming a new array of micro-vortices that interact with each other for a few milliseconds.

If it weren't for the extreme slow motion, it would be difficult to spot these.

But it also gave Xu Chuan a vague inspiration.

With a slight click of the mouse, he pulled the screen to the very beginning and replayed it.

When the new vortex array and ripples were formed, Xu Chuan's eyes became brighter and brighter, but there was still a trace of doubt in his bright eyes.

He always felt that these things gave him an inexplicable sense of familiarity in mathematics, but for a while he couldn't remember where he had seen them.

The mouse pulled back to progress again, and he watched the video in front of him over and over again.

Suddenly, in his mind, a piece of manuscript paper appeared in his mind, which made his eyes brighten suddenly!

He remembered where he had seen this familiar thing, and he knew how to proceed with the NS equation!

PS: Yesterday's monthly pass plus more, when it comes to it, let's get a little more monthly pass, dear, there are still 300 monthly passes to 2,000, \( ̄) ̄*\)), let the eight tails draw next month!