After sending away the four students, Xu Chuan once again stood in front of the blackboard where Professor Fefferman was writing about mathematics.
The n-s equation, full name - Navier-Stokes equation, is an equation of motion that describes the conservation of momentum in viscous incompressible fluids.
Broadly speaking, it is not an equation, but a system of equations composed of several equations.
For example, Navier first proposed the equations of motion for viscous fluids in 1827;
For example, Poisson proposed the equation of motion of compressible fluids in 1831;
Or Saint-Venant and Stokes independently proposed in 1845 that the viscosity coefficient is a constant, which is called the Okes equation.
These equations reflect the basic mechanical laws of viscous fluid flow and are of great significance in fluid mechanics.
】
However, its solution is very difficult and complex. Before further development and breakthroughs in solution ideas or technologies, its precise solution can only be obtained for some very simple special case flow problems.
Up to now, the mathematical community's promotion of it is only the existence of an overall smooth solution to 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.
"This is just one step.
This can almost be said to have no advancement at all for the overall ns equation.
After all, when the Reynolds number re ≥ 1, outside the boundary layer of the flowing object, the viscous force is much smaller than the inertial force, and the viscosity term in the equation can almost be ignored.
After neglecting the viscosity term, the n-s equation can be simplified to the Euler equation in ideal flow.
If you simply 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 does not require a complete solution to the problem to verify the smoothness of the solution, nor does he dream of calculating the final solution.
But at least, he wanted to be able to determine the flow of fluid given certain initial conditions and boundary conditions.
This is the basic requirement for controlling the flow of ultra-high temperature plasma in the controllable nuclear fusion reactor chamber.
If this is not possible, the subsequent turbulence model and control system will be even less useful.
Fefferman asked the professor to list these calculations on the blackboard in front of him, which can bring hope to this step.
If this equal spectral problem can be solved, he and Fefferman can take the ns equation a small step forward.
At least, it can be done in the surface space, given an initial condition and boundary condition, to determine the existence and smoothness of the solution.
Don't underestimate it, it's just a small step, but it has never been done in mathematics for 150 years.
So Xu Chuan urgently hopes to solve this problem.
.......
Standing in front of the blackboard, Xu Chuan thought for a long time, and finally shook his head.
Regarding the equispectral non-isometry isomorphism conjecture, he has no idea at the moment. Whether it is the Laplacian operator, the elliptic operator, or the bounded connected area, he does not see any hope.
At least, these directions did not bring him any dazzling ideas or ideas.
Shaking his head, Xu Chuan returned to his desk, temporarily gave up on the breakthrough of the equal spectrum problem, and began to sort out the exchanges with Fefferman during this period.
Maybe Fefferman is right, maybe the inspiration comes to him while he is sorting out the information?
But unfortunately, the inspiration for this prophecy did not come out until he finished sorting out his thoughts and ideas.
Fortunately, he is not impatient. His long-term scientific research experience has taught Xu Chuan that the more he faces such a world-class problem, the more he must stay calm and steady.
The choices and decisions a person makes when he is in a hurry or panic are not necessarily 100% wrong, but the probability of making the 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 into smaller, more manageable parts.
This approach is called "divide and conquer".
By breaking a problem into smaller parts, you make it easier to understand and solve.
Additionally, breaking the 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 broken down.
But for the equispectral non-isometric isomorphism conjecture, it is not a problem that cannot be broken down. It is based on the mathematical problems of modern differential geometry, integrating spectral theory and equispectral problems. Curvature and topology are not the same.
Mathematical knowledge of variables and other directions.
On this basis, Xu Chuan split it into the original mathematical structure, and then started from the spectral theory and isospectral mathematics that he is 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 of analyzing physical processes is to hierarchize complex problems and resolve them into multiple interrelated "sub-processes" for study.
This method is not only useful in students such as junior high school, high school and university, but can still be adapted to various fields of physics even if you are a graduate student or doctoral student.
The splitting method of mathematics and the analysis method of physics have the same purpose.
Therefore, Xu Chuan is quite comfortable in using it. At least it takes a lot of time to learn a new mathematical research method.
...
For more than a week, Xu Chuan concentrated on trying to use this method to solve the isometric isomorphism conjecture, and he handed over the weekly lectures at Princeton to the older Roger Dean
.
Roger Dean, who is already thirty-one this year, has almost completed his Ph.D. at the Politecnico di Milano in Italy, and has even prepared his graduation thesis. He came to Princeton for further study, so there is no problem in taking his place in lecturing to the undergraduates.
.
This chapter is not over yet, please click on the next page to continue reading! Of course, Xu Chuan did not use other people’s labor for nothing. Although according to the unspoken rules of academia, it didn’t matter if he did it for free, but he still applied for a job at Princeton for this student.
internship assistant position.
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 professorship at Princeton in the future.
This can be regarded as some remuneration given by Xu Chuan to this student. After all, he is not the kind of unscrupulous tutor who exploits students in various ways, and he cannot do anything like prostitute students' labor for free.
Of course, this is not the case for everyone. For some doctoral supervisors, it is a matter of course for their students to attend classes instead of themselves.
I'm afraid they have never thought about remuneration or anything like that.
There are even a handful of tutors who are eager to take advantage of every piece of independent research and development by their students.
...
In the office, Professor Fefferman, who had not 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 the foam on it, took a small sip, and looked at Xu Chuan: "Xu, regarding the equal spectrum issue last time
, I may have some ideas."
"you say."
Xu Chuan nodded, indicating that he was listening.
In fact, it is not only Professor Fefferman who has ideas and inspiration. He has been studying the isometric non-isometric isomorphism conjecture these days, and he also has some ideas in his mind.
Fefferman pondered for a moment, organized his thoughts and then said: "Studying the spectrum of a manifold is a basic problem of Riemannian geometry. For compact Riemannian manifolds, all spectra are point spectra, that is, Lapp
All spectra of the Rath operator are composed of eigenvalues with finite multiplicity, but for complete non-compact manifolds, the situation is much more complicated."
"Suppose Ω is an open area of , u is a smooth function defined on Ω, the Hessian matrix of u is (?2u/?zj?zk), and its eigenvalues are λ1, λ2...λn,
Define the complex hessian operator as..."
"Through smooth function approximation, pm also includes non-smooth functions. It is called u∈dm. If there is a regular borel measure μ and a monotonically decreasing smooth function sequence {uj}? pm makes hm(uj)→μ, and
Recorded as hm(u)=μ....."
"..."
"If we start from this aspect, we may hope to go deeper into the conjecture of isospectral non-isometric isomorphism."
"I wonder what you think?"
After expressing his thoughts, Fefferman looked at Xu Chuan expectantly.
Xu Chuan did not answer immediately. He tapped his fingers regularly on his desk. From Fefferman's words, he saw another path leading to the equal spectral problem.
The Green's function of a class of second-order completely nonlinear partial differential equations was a path that he had never 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 and using Dirichlet functions to study equal spectral problems, this direction is something I have never thought of.
"
"But from a purely intuitive point of view, this may be a feasible path and totally worth a try."
Hearing this, Fefferman raised a smile: "Then let's set off."
Xu Chuan smiled and said: "Don't worry, I also have some ideas about the conjecture of equispectral and non-isometric isomorphisms. Do you want to listen to 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 stood up, walked to the edge of the office, dragged the previously used blackboard out of the corner, picked up a piece of chalk, organized 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 non-empty open set in rn, or a general n-dimensional region with a finite Lebesgue measure, △ is the laplace operator, and both t1 and t2 are non-empty
.we define..."
"Spectrum 6(p) is discrete. According to the finite multiplicity of its eigenvalues, it can be arranged as 0≤λ1≤λ2≤...≤λk≤...and when k→00, enter k→0 and 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 hand, while Professor Fefferman stood behind him and watched.
Mathematicians at their level don't need the speaker to introduce their ideas in too much detail. It can be seen from the written formulas.
As Xu Chuan wrote, Fefferman's eyes gradually brightened, from curiosity at the beginning, to surprise, to astonishment.
Just as Xu Chuan saw a path leading to the conjecture problem of equispectral and non-isometric isomorphisms from his narrative, he also saw a completely different path from Xu Chuan's writings.
This line of thinking may also solve the difficulties that hinder their progress.
No!
If we look at the possibility alone, 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 had already pioneered another path.
This is like one person pointing to a vacant lot and saying I want to build a house here, while 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 much higher credibility than the former.
This chapter is not over, please click on the next page to continue reading!......
After restating the thoughts and ideas 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 set of bounded open fields that do not intersect each other, and then using the Laplacian operator to complete the isometric non-isometry 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 equal spectrum problem."
"I wonder what you think?"
The idea proposed by Fefferman and the way he thought of it were two completely different paths, but Xu Chuan did not think Fefferman was wrong.
Of course, he doesn't think his own idea is wrong.
Different approaches lead to the same goal. For this top-level mathematical problem, it involves many things, and there is no single way to solve the problem.
It is not like 1 1 = 2 which is always constant. Whether starting from the Dirichlet function and nonlinear partial differential equations, or constructing a bounded open domain set, the Laplace operator is used to complete the non-isometric isomorphic region.
Construction, both are methods of solving problems.
Although the differences between the two methods are quite different.
However, with the development of mathematics, the boundaries have long been molded into lakes.
Number theory, algebra, geometry, topology, mathematical analysis,... Function theory, ordinary differential equations, partial differential equations, these mathematical classifications have long been that you are in me, and I am in you.
In today's mathematics, it is no longer unusual to start from a seemingly unrelated field and solve a major problem 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 one after another wanted to complete the grand unification of algebra and geometry.
