Great Country Academician

After dismissing the four students, Xu Chuan once again stood in front of the blackboard where Professor Fefferman wrote mathematics.

The N-S equation, full name - Navier-Stokes equation, is a motion equation describing the conservation of momentum of a viscous incompressible fluid.

Broadly speaking, it is not an equation, but a system of equations composed of several equations.

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

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

Or St. Venant and Stokes independently proposed a form in which the viscosity coefficient is a constant in 1845, which is called the Navier-Stokes equation.

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

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

So far, the advancement of the mathematical circles is nothing more than the existence of the overall smooth solution of the N·S equation under the assumption that a certain norm of the given initial value is appropriately small, or the fluid motion area is appropriately small "That's all.

For the overall NS equation, it can be said that there is almost no progress at all.

After all, when the Reynolds number Re≥1, the viscous force is much smaller than the inertial force outside the boundary layer of the flow object, 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, the solution at this level does not meet Xu Chuan's requirements for the NS equation.

For the N·S equation, he did not ask to completely solve the problem, to prove the smoothness of the solution, nor dream of calculating the final solution.

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

This is a fundamental requirement for controlling the flow of ultrahigh-temperature plasma in controllable fusion reactor chambers.

If this cannot be done, let alone the subsequent turbulence model and control system.

Fefferman asked the professor to list these calculations on the blackboard in front of him, which can bring hope to advance to this step.

If this isospectral problem can be solved, he and Fefferman can push the NS equation down a small step.

At least, in the surface space, given an initial condition and boundary conditions, it can be determined that the solution exists and is smooth.

Don't underestimate that it's just a small step, but it took one hundred and fifty years for the mathematics world to do something that hasn't been done.

So Xu Chuan urgently hopes to solve this problem.

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

For the equispectral non-isometric isomorphism conjecture, he has no idea for the time being. Whether it is Laplacian operator, ellipse operator, or 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 on breaking through the problem of losing the equal spectrum, and began to sort out the communication with Fefferman during this period.

Maybe what Fefferman said is right, maybe the inspiration came out of me who was sorting out the data?

But unfortunately, the inspiration for this prophecy did not emerge until he finished sorting out his thoughts and ideas.

Fortunately, he is not a quick-tempered person. His long-term scientific research experience has taught Xu Chuan that the more he faces such a world-class problem, the more he must calm down and calm down.

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

The best way is to clarify your thinking and start from the basics.

Problem solving is about finding the key, and one way to solve math problems is to break them down into smaller, more manageable parts.

This approach is called "divide and conquer".

By breaking a problem into smaller pieces, it becomes easier to understand and solve.

Additionally, breaking a problem into smaller pieces can help identify patterns and relationships that might not be immediately apparent when looking at the problem as a whole.

Of course, this approach doesn't work for all mathematical conjectures.

Because some mathematical conjectures cannot be split.

But for the equispectral non-isometric isomorphism conjecture, it is not a problem that cannot be split. Its foundation is built on the mathematical problems of modern differential geometry, which combines spectral theory and isospectral problems, curvature and topological incompatibility. Mathematical knowledge of variables and other directions.

On this basis, Xu Chuan split it into the original mathematical structure, and then started from the most familiar spectral theory and equispectral mathematics in this life, to perfect and solve these problems little by little.

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 for analyzing physical processes is to stratify complex problems and resolve them into multiple interrelated "sub-processes" for research.

This method is not only useful in the student era of junior high school, high school, but also can be adapted to various physics fields even after entering graduate students and doctoral students.

The splitting method of mathematics is similar to the analytical method of physics.

So Xu Chuan is quite handy in using it, at least it takes a lot of time to learn a new mathematical research method.

For the next week or so, Xu Chuan devoted himself to trying to use this method to solve the isospectral non-isometric isomorphism conjecture, and he gave Princeton's weekly lectures to the older Roger Dean .

Roger Dean, who is 31 years old this year, has almost finished his Ph.D degree at Politecnico di Milano in Italy, and even has prepared his graduation thesis. He came to Princeton for advanced studies, and it is not a problem to replace him to give lectures to those undergraduates. .

Of course, Xu Chuan did not prostitute other people’s labor force for free. Although according to the unspoken rules of academia, it’s okay for him to prostitute for free, he still applied for the student’s position as an intern assistant in Princeton.

With this position, Roger Dean can enjoy some subsidies from Princeton. Although it is 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 Xu Chuan gave this student. After all, he is not the kind of unscrupulous tutor who oppresses students in various ways, and he can't do things like prostitution of students' labor.

Of course, not everyone is like this. For some doctoral supervisors, it is a matter of course to arrange the students they bring to attend classes instead of themselves.

I'm afraid they never thought about the reward.

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

In the office, Professor Fefferman, who hadn't been here for more than ten days, came here again.

"Professor Fefferman."

Xu Chuan said hello and asked Amelia to make two cups of coffee.

"Thank you." After taking the coffee from Amelia, Fefferman blew on the froth on it, took a small sip, and looked at Xu Chuan: "Xu, about the problem of equal scores last time , I might have an idea."

"you say."

Xu Chuan nodded, indicating that he was listening.

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

Fefferman pondered for a while, 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, Lapp All spectra of Las operators consist of eigenvalues ​​with finite multiplicity, whereas for complete noncompact manifolds the situation is much more complicated."

"Suppose 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."

"Approximate by smooth functions, so that Pm also includes non-smooth functions. It is called u ∈ Dm, if there is a regular Borel measure and a monotonically decreasing smooth function sequence {uj} Pm such that Hm(uj )→, and denoted as Hm (u)="

"."

"If we start from this aspect, it may be hoped that we can go deep into the isomorphic non-isomorphism conjecture."

"I don't know what you think?"

After expressing his thoughts, Fefferman looked at Xu Chuan expectantly.

Xu Chuan didn't answer immediately, but tapped his fingers regularly on the desk. From Fefferman's words, he saw another way to the equal notation problem.

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

But this path came out of Fefferman's mouth, and he was keenly aware that it seemed to be equally feasible.

After pondering 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, this direction is something I have never thought about. "

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

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

Xu Chuan smiled and said, "Don't worry, I also have some ideas on the equispectral and non-isometric isomorphism conjecture, do you want to hear it?"

There was a trace of surprise in Fefferman's eyes, but it was quickly covered by curiosity, and he quickly replied: "Of course."

Xu Chuan got up, walked to the edge of the office, dragged the blackboard he had used before from the corner, picked up a piece of chalk, sorted out his 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 nonempty open set in Rn, or in general an n-dimensional region with finite Lebesgue measure, △ is a Laplace operator, and both T1 and T2 are nonempty .we define"

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

"."

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

Mathematicians who have reached their level do not need the reporter to introduce their ideas in detail, as it can be seen from the written formulas.

With Xu Chuan's writing, Fefferman's eyes gradually brightened, from curiosity at the beginning, to surprise, and then to astonishment.

Just as Xu Chuan saw a road leading to the isomorphic non-isometric isomorphism conjecture from his narration, he also saw a completely different road from Xu Chuan's writing.

This line of thinking is also likely to solve the difficulties that hinder their progress.

No!

In terms of possibility, the idea on the blackboard is more likely to solve the isospectral problem.

After all, he only proposed a seemingly feasible path, but Xu Chuan has already pioneered another path.

It's like one person points to a vacant lot and says I'm going to build a house here, and another person has already leveled the vacant lot with an excavator.

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

After restating the thoughts and sorted out thoughts in his mind these days on the blackboard in front of him, Xu Chuan turned to look at Fefferman.

"This is my idea, by constructing a pairwise disjoint set of bounded open domains, and then using the Laplacian operator to complete the isospectral non-isomorphic isomorphism for the two mixed boundary value conditions of R2 and R3 The structure of the region."

"Perhaps it is also a path that can lead to solving the isospectral problem."

"I don't know what you think?"

Fefferman's ideas and his own ideas are two completely different paths, but Xu Chuan doesn't think Fefferman is wrong.

Of course, he didn't think his own ideas were wrong.

Different routes lead to the same goal. For this top-level mathematical problem, it involves a lot of things, and there is no unique way to solve the problem.

It is not like 1+1=2 is always constant, no matter starting from the Dirichlet function and nonlinear partial differential equations, or constructing a bounded open domain set, using the Laplacian operator to complete the non-isometric isomorphic area constructs, both are solutions to the problem.

Although the two methods differ greatly.

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

Number theory, algebra, geometry, topology, mathematical analysis, function theory, ordinary differential equations, partial differential equations, these mathematical classifications have long been me in you, and you in me.

Today's mathematics, starting from a seemingly unrelated field, but solving major problems in another field is no longer uncommon.

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

Just like Pope Grothendieck laid the foundation of modern algebraic geometry, countless mathematicians have tried to complete the great unification of algebra and geometry.