Chapter 203 Two Different Paths

After sending away the four students, Xu Chuan stood in front of Professor Fefferman's blackboard again to write mathematics.

Navier-Stokes equation, full name - Navier-Stokes equation, is an equation of motion that describes the conservation of momentum of viscous incompressible fluid.

In a broad sense, it is not a single equation, but a system of equations consisting of several equations.

For example, Navier first proposed the equation of motion for viscous fluid in 1827;

For example, Poisson proposed the equation of motion for compressible fluid in 1831;

Or Saint-Venant and Stokes independently proposed the form of viscosity coefficient as a constant in 1845, all of which are called Navier-Stokes equations.

These equations reflect the basic mechanical laws of viscous fluid flow and are of great significance in fluid mechanics.

But its solution is very difficult and complicated. Before the solution ideas or technologies are further developed and broken through, its exact solution can only be obtained in some very simple special case flow problems.

So far, the progress made by the mathematical community is only the step of "under the assumption that a certain norm of a given initial value is appropriately small, or the fluid motion area is appropriately small, the overall smooth solution of the N·S equation exists".

This can be said to be no progress at all for the overall NS equation.

After all, when the Reynolds number Re≥1, the viscous force outside the boundary layer of the flow object is much smaller than the inertial force, and the viscous term in the equation can be almost ignored.

After ignoring the viscous term, the N-S equation can be simplified to the Euler equation in ideal flow.

If it is simply to solve the Euler equation, it is not difficult.

But obviously, this level of solution does not meet Xu Chuan's requirements for the NS equation.

For the N·S equation, he does not require to completely solve this He did not dream of calculating the final solution, nor did he want to prove the smoothness of the solution.

But at least, he wanted to be able to determine the flow of the fluid given certain initial conditions and boundary conditions.

This is the basic requirement for controlling the flow of ultra-high temperature plasma in the chamber of a controlled nuclear fusion reactor.

If this cannot be done, the subsequent turbulence model and control system will be even more out of the question.

And the formulas that Feferman asked the professor to list on the blackboard in front of him can bring hope for advancing to this step.

If this equispectral problem can be solved, he and Feferman can advance the NS equations a small step further.

At least, it can be done in the curved space, given an initial condition and boundary condition, to determine the existence of the solution and that it is smooth.

Don't underestimate it as just a small step, but the mathematical community has spent 150 years to... There is no time to do it.

So Xu Chuan is eager to solve this problem.

Standing in front of the blackboard, Xu Chuan pondered for a long time, and finally shook his head.

For the equal-spectral non-isometric isomorphism conjecture, he has no idea for the time being. Whether it is the Laplace operator or the elliptic operator, or the bounded connected region, he can't see any hope.

At least, these directions did not bring him any bright ideas or ideas.

Shaking his head, Xu Chuan returned to his desk, temporarily gave up the breakthrough of the equal-spectral problem, and began to sort out the communication with Feferman during this period.

Maybe Feferman is right, maybe the inspiration came to him while he was sorting out the materials?

But unfortunately, the inspiration of this prophecy did not come until he sorted out his thoughts and ideas. There are some.

Fortunately, he is not an impatient person. His long-term scientific research experience has taught Xu Chuan that the more he faces such world-class problems, the more he needs to keep calm and steady.

When a person is in a hurry and panic, the choices and decisions he makes are not 100% wrong, but the probability of making the wrong choice is undoubtedly quite large.

The best way is to sort out your thoughts and start from the basics.

To solve a problem, you need to find the key, and one way to solve mathematical problems is to break them down into smaller, more manageable parts.

This method is called "divide and conquer".

By dividing a problem into smaller parts, it can be made easier to understand and solve.

In addition, dividing a problem into smaller parts can help identify patterns and relationships that may not be immediately apparent when looking at the problem as a whole.

Of course, this method does not apply to all mathematical conjectures.

Because some mathematical conjectures cannot be split.

But for the isospectral non-isometry conjecture, it is not a problem that cannot be split. Its foundation is built on the mathematical problems in modern differential geometry, integrating mathematical knowledge in the directions of spectral theory and isospectral problems, curvature and topological invariants.

On this basis, Xu Chuan split it into the original mathematical framework, and then started from the spectral theory and isospectral mathematics that he was most familiar with in his life, to improve and solve these problems bit by bit.

This method is also very common in the field of physics. Generally speaking, complex physical processes are composed of several simple "sub-processes".

Therefore, the most basic method to analyze physical processes is to hierarchicalize complex problems and resolve them into multiple interrelated "sub-processes" for research.

This method is not only useful in the student age of junior high school, high school and university, but also can be adapted to various physical fields even if you enter graduate school and doctoral students.

The mathematical splitting method and the physical analysis method have the same purpose.

So Xu Chuan was quite comfortable with it, at least it took a lot of time to learn a new mathematical research method.

For more than a week, Xu Chuan tried to use this method to solve the isospectral non-isometric isomorphism conjecture, and he handed over the weekly lectures at Princeton to the older Roger Dean.

Roger Dean, who is already 31 years old this year, is almost a doctoral student at the Polytechnic University of Milan in Italy, and even has his graduation thesis ready. He came to Princeton for further studies, and there was no problem in replacing him to teach those undergraduates.

Of course, Xu Chuan did not take advantage of others' labor for free. Although according to the unspoken rules of the academic world, it didn't matter if he took advantage of others for free, he still applied for an internship assistant position for this student at Princeton.

With this position, Roger Dean can enjoy some subsidies from Princeton. Although not much, it is enough to support his daily life.

And with this experience, it will be much easier for Roger Dean to apply for an assistant professor at Princeton in the future.

This can be regarded as some compensation from Xu Chuan to this student. After all, he is not the kind of unscrupulous tutor who exploits students in various ways, nor does he do things like taking advantage of students' labor for free.

Of course, not everyone will do this. For some doctoral tutors, it is natural to arrange their own students to go to class instead of themselves.

I am afraid they have never thought about compensation.

There are even a very small number of tutors who want to occupy every achievement of their students' independent research and development.

In the office, Professor Feferman, who has not been here for more than ten days, came here again.

"Professor Feferman."

Xu Chuan greeted him and asked Amelia to make two cups of coffee.

"Thank you." After taking the coffee from Amelia, Feferman blew the foam on it, took a small sip, and looked at Xu Chuan: "Xu, I may have some ideas about the equispectral problem last time."

"You say."

Xu Chuan nodded, indicating that he was listening.

In fact, it is not only Professor Feferman who has ideas and inspirations. These days, he has been splitting and studying the equispectral non-isometric isomorphism conjecture, and he has some ideas in his mind.

Feferman pondered for a moment, organized his thoughts, and then said: "Studying the spectrum of a manifold is a basic problem in Riemannian geometry. For compact Riemannian manifolds, all spectra are point spectra, that is, all spectra of the Laplace operator are composed of eigenvalues ​​with finite multiplicity, while for complete non-compact manifolds, the situation is much more complicated."

"Assume that Ω is an open region of Cn, u is a smooth function defined on Ω, the Hessian matrix of u is (u/zjzk), its eigenvalues ​​are λ1, λ2λn, and the complex Hessian operator is defined as."

"Through smooth function approximation, Pm also includes non-smooth functions. Call u∈ Dm, if there exists a regular Borel measure and a monotonically decreasing smooth function sequence {uj} Pm such that Hm(uj )→, and it is recorded as Hm(u)=”

“.”

“If we start from this aspect, we may be able to delve into the isospectral non-isometric isomorphism conjecture.”

“What do you think?”

After speaking out his thoughts, Feferman looked at Xu Chuan expectantly.

Xu Chuan did not answer immediately, but tapped his fingers regularly on the desk. He saw another path to the isospectral problem from Feferman’s words.

Green’s function of a second-order completely nonlinear partial differential equation, this is a path he had not thought of before.

But when this path came out of Feferman’s mouth, he keenly realized that it seemed equally feasible.

After thinking for a while, Xu Chuan stopped tapping his fingers on the mahogany desk and said, "Starting from the direction of nonlinear partial differential equations, using Dirichlet functions to study isospectral problems is a direction I have never thought of."

"But from an intuitive point of view, this may be a feasible path, and it is totally worth a try."

Hearing this, Feferman raised a smile at the corner of his mouth: "Then let's go."

Xu Chuan smiled and said, "No hurry, I also have some ideas about the isospectral non-isometric isomorphism conjecture. Do you want to listen?"

Feferman's eyes flashed with a trace of surprise, but it was soon covered by curiosity. He quickly replied: "Of course."

Xu Chuan stood up and walked to the edge of the office. I dragged the blackboard I had used from the corner, picked up a piece of chalk, sorted out my thoughts, and wrote on it:

“(p){-△u=λu,x∈Ω;u=0,x∈Γ1;δu/δn=0,x∈Γ2”

“Here Γ is the boundary of Ω, and Γ=Γ1UΓ2, Ω is a bounded non-empty open set in Rn, or a general n-dimensional region with limited Lebesgue measure, △ is a Laplace operator, and T1 and T2 are both non-empty. We define”

“The spectrum б(P) is discrete, and can be arranged into 0≤λ1≤λ2≤…≤λk≤… according to the finite multiplicity of its eigenvalues, and when k→00, let k→0, define N(O,-λ,λ)= # {k∈N]ょ.

“.”

In the office, Xu Chuan was writing his thoughts and ideas on the blackboard with chalk in his hand, while Professor Fefferman stood behind him and watched.

Mathematicians at their level did not need the presenter to introduce their ideas in too much detail, as they could clearly see them from the written formulas.

As Xu Chuan wrote, Fefferman's eyes gradually brightened, from curiosity at first, to surprise, and then to astonishment.

Just as Xu Chuan saw a path to the problem of equal-spectral non-isometric isomorphism conjecture from his narration, he also saw a completely different path from Xu Chuan's writing.

This idea is also likely to solve the difficulties that hinder their progress.

No!

If we only talk about the possibility, the idea on the blackboard is more likely to solve the equal-spectral problem.

After all, he only proposed a seemingly feasible path, while Xu Chuan has already opened up another path.

It's like one person pointing to a vacant land and saying I want to build a house here, while the other person has already leveled the vacant land with an excavator.

Both parties are building houses on vacant land, but the latter gives people much more credibility than the former.

After recounting the ideas and ideas in his mind these days on the blackboard in front of him, Xu Chuan turned and looked at Fefferman.

"This is my idea. By constructing a set of bounded open domains that do not intersect each other, and then using the Laplace operator to complete the construction of the isospectral non-isometric regions of the two mixed boundary conditions R2 and R3."

"Perhaps it is also a way to solve the isospectral problem."

"What do you think?"

The idea proposed by Feferman and the idea he thought of are two completely different paths, but Xu Chuan does not think Feferman is wrong.

Of course, he does not think his own idea is wrong.

Different paths lead to the same destination. For such a top mathematical problem, it involves a lot of things, and there is no single way to solve the problem.

It is not like 1+1=2 is always constant. Whether it is starting from the Dirichlet function and nonlinear partial differential equations, or constructing a bounded open domain set and using the Laplace operator to complete the construction of non-isometric regions, both are ways to solve the problem.

Although the difference between the two methods is very different.

But the boundaries of mathematics have long been blurred since its development.

Number theory, algebra, geometry, topology, mathematical analysis, function theory, ordinary differential equations, partial differential equations, these mathematical categories have long been intertwined.

In today's mathematics, it is no longer uncommon to start from a seemingly unrelated field and solve major problems in another field.

There are even many mathematicians who are specifically trying to connect two different fields.

Just like after Pope Grothendieck laid the foundation of modern algebraic geometry, countless mathematicians have been trying to complete the grand unification of algebra and geometry.