After sending the four students away, Xu Chuan once again stood in front of Professor Feverman's blackboard where he was writing math.

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 several equations.

For example, Navier was the first to propose the equation of motion for viscous fluids in 1827;

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

Or Saint-Venant and Stokes independently proposed in 1845 that the viscosity coefficient is a constant form, both of which are called OKES equations.

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, and its exact solution can only be obtained in some very simple special case flow problems before the solution ideas or techniques are further developed and breakthroughs.

Up to now, the advancement of the mathematical community has only been the existence of an overall smooth solution of the n·s equation under the assumption that a certain norm of a given initial value is appropriately small, or the region of fluid motion is appropriately small.

This is almost no advance on the overall NS equation.

After all, when the Reynolds number re≥1, the viscous force outside the boundary layer of the flowing object is much smaller than the inertial force, and the viscous term in the equation is almost negligible.

Ignoring the viscous term, the n-s equation can be reduced to Euler's equation in an ideal flow.

It is not difficult to solve Euler's equation simply.

However, it is obvious that this kind of solution does not meet Xu Chuan's requirements for the ns equation.

In the case of 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 the very least, he wanted to be able to determine the flow of fluids given certain initial and boundary conditions.

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

If this is not possible, the subsequent turbulence model and control system will not be considered.

And the equations that Fefferman asked the professor to list on the blackboard in front of him could give hope for advancing to this point.

If this isospectral problem can be solved, he and Fefferman will be able to take the ns equation a small step down.

At the very least, it is possible to determine the existence of a solution and smooth in surface space, given an initial condition and a boundary condition.

Don't underestimate it's a small step, but the mathematical community hasn't done it in 150 years.

So Xu Chuan desperately hopes to solve this problem.

.......

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

For the isospectral non-equidistant isomorphism conjecture, he has no idea for the time being, whether it is a Laplace operator, an elliptic operator, or a bounded connected region, he does not see any hope.

At least, these directions didn't give him any eye-catching ideas or ideas.

Shaking his head, Xu Chuan returned to his desk, temporarily gave up the breakthrough of the problem of falling out of the score, and began to sort out the communication with Fefferman during this time.

Maybe Feverman was right, maybe the inspiration came out of the self while organizing the data?

Sadly, the inspiration for this prophecy did not come to him until he had put his thoughts and ideas in order.

Fortunately, he is not an acute child, and his long-term scientific research experience has made Xu Chuan know that the more he faces this kind of world-class problem, the more he must calm down and stabilize his heart.

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

The best way to do this is to clear your mind and start with 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 pieces.

This method is known as "divide and conquer".

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

Additionally, dividing 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 approach does not work for all mathematical conjectures.

Because some mathematical conjectures cannot be split.

However, for the isospectral non-isometric isomorphism conjecture, it is not a problem that cannot be split, and its foundation is built on the mathematical problems of modern differential geometry, which integrates 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 spectrum theory and isospectral mathematics that he was most familiar with in his life, to improve and solve these problems little by little.

This approach is also common in the field of physics, where complex physical processes are generally made up of a number of simple "sub-processes".

Therefore, the most basic way to analyze physical processes is to hierarchize complex problems and resolve them into multiple interrelated "sub-processes" to study.

This method is not only useful for students in junior high school, high school, and university, but also for graduate and doctoral students, and can still be adapted to various fields of physics.

The method of splitting mathematics and the method of analysis of physics are similar.

Therefore, Xu Chuan is quite handy to use, and at least it takes a lot of time to learn a new mathematical research method.

......

For the next week or so, Xu Chuan was concentrating on trying to solve the isospectral nonisometric isomorphism conjecture with this method, and Princeton's weekly lectures were handed over to the older Roger Dean.

Roger Dean, who is now thirty-one this year, cares that the Politecnico di Milano is almost complete with a doctorate, and even his dissertation is ready, and he will come to Princeton for further study, and there is no problem in giving lectures to those undergraduates instead of him.

Of course, Xu Chuan did not prostitute other people's labor in vain, although according to the unspoken rules of academia, it did not matter if he prostituted for nothing, but he still applied for a position as an intern assistant in Princeton for this student.

With this position, Roger Dean can enjoy some of Princeton's grants, although not much, but 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 remuneration from Xu Chuan to this student, after all, he is not the kind of unscrupulous tutor who oppresses students, and he can't do anything to prostitute students' labor for nothing.

Of course, not everyone is like this, and for some doctoral supervisors, it is a matter of course to arrange for their own students to go to class instead of them.

Remuneration or something, I'm afraid they never thought about it.

There are even a very small number of tutors who are eager to occupy every achievement of the student's own independent research and development.

......

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

"Professor Feverman."

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

"Thank you." After taking the coffee from Amelia's hand, Feverman blew on the foam on it, took a small sip, and looked at Xu Chuan: "Xu, I may have a little idea about the question of equilibrium last time." ”

"You say."

Xu Chuan nodded, signaling that he was listening.

In fact, it is not only Professor Feverman who has ideas and inspiration, he has been splitting and studying isospectral non-isomorphic conjectures these days, and he also has some ideas in mind.

Feverman pondered for a moment, organized his thoughts, and then said, "Studying the spectrum of a manifold is a fundamental problem of Riemannian geometry. For compact Riemannian manifolds, all spectra are point spectra, i.e., all spectra of the Laplace operator are composed of eigenvalues with finite weights, while the situation is much more complicated for complete non-compact manifolds. ”

"Suppose Ω is an open region of , u is a smooth function defined on the Ω, and the hessian matrix of u is (?2u/?zj?zk) with eigenvalues λ1 and λ2...... λn, which defines the complex hessian operator as ......"

"By approximating the smooth function, the non-smooth function is also included in the pm. Called u∈ dm, if there is a regular Borel measure μ and a monotonically descending sequence of smooth functions {uj}? pm makes hm(uj) →μ and denoted as hm(u)=μ....."

“......”

"If we start from this aspect, we may be able to go deeper into the isospectral non-isometric isomorphism conjecture."

"I don't know what you think?"

After speaking his thoughts, Peverman looked at Xu Chuan expectantly.

Xu Chuan didn't answer immediately, his fingers tapped regularly on his desk, and he saw another path to the question of equivalence in Feverman's words.

A kind of Green's function of a second-order completely nonlinear partial differential equation, which is a path he has not thought of before.

But the path came out of Feverman's mouth, and he was keenly aware that it seemed just as 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 the Dirichlet function to study the isospectral problem is a direction that I have never thought of. ”

"But intuitively, this might be a viable path, and it's well worth trying."

Hearing this, a smile rose at the corner of Feverman's mouth: "Then let's go." ”

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

A hint of surprise crossed Fefferman's eyes, but he was quickly overshadowed 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 out of 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 the Ω, and Γ=Γ1uΓ2, Ω is a bounded non-empty open set in rn, or a general n-dimensional region with a limited Lebeig measure, △ is the laplace operator, and both t1 and t2 are non-empty. We define ......"

"Spectrum 6(p) is discrete and can be arranged into 0≤λ1≤λ2≤ according to the finite repleons of its eigenvalues... ≤λk≤… And when k→00, enter k→0 and define n(o,-λ,λ)=#{k∈n]ょ........

“......”

In the office, Xu Chuan is holding chalk and writing his thoughts and ideas on the blackboard, while Professor Feverman stands behind him and watches.

Mathematicians at their level do not need the reporter to go into too much detail about their ideas, which can be seen from the written formulas.

And as Xu Chuan wrote, Feverman's eyes gradually brightened, from curiosity at the beginning, to surprise, and then to astonishment.

Just as Xu Chuan saw a path to the problem of isospectral non-isomorphic conjecture from his account, he also saw a completely different path from Xu Chuan's writing.

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

No!

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

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

It's like one person pointing to an empty space and saying I'm going to build a house here, while another person has leveled the empty space with an excavator.

Both parties also build houses on vacant land, but the latter gives much more credibility than the former.

......

After restating the thoughts and thoughts that had been sorted out in his mind these days onto the blackboard in front of him, Xu Chuan turned to look at Feverman.

"That's my idea, by constructing a set of bounded open fields that do not intersect in pairs, and then using the Laplace operator to complete the construction of two mixed boundary value conditional isospectral non-equidistant isomorphic regions of R2 and R3."

"Maybe it's also a path that can lead to solving the isospectral problem."

"I don't know what you think?"

The idea proposed by Peverman and the idea he came up with were two completely different paths, but Xu Chuan did not think that Feverman was wrong.

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

In the same way, there are so many things involved in this top-level mathematical problem that there is no single solution to it.

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 bounded open field sets, using the Laplace operator to complete the construction of non-equidistant isomorphic regions, both are ways to solve the problem.

Although the differences between the two methods are very different.

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

Number Theory, Algebra, Geometry, Topology, Mathematical Analysis、..... Functional theory, ordinary differential equations, partial differential equations, and other mathematical classifications have long been you have me, I have you.

It is not uncommon for today's mathematics to start from a seemingly unrelated field and solve a major problem in another.

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

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

......