As The Richards began to look forward to the future, the global mathematics community suddenly erupted without warning!
The initial cause was Tao Xuanzhi publishing a letter from Qiao Yu on his blog.
It’s quite common for successful mathematicians to frequently exchange emails discussing mathematical problems. The more adept the mathematician, the more frequent this occurs.
In fact, from an external perspective, both should have a shared language to so extent. For instance, they were both prodigies as children, and they didn’t waste their talents as adults.
Especially since both of them have extensive engagent in mathematics.
Moreover, Tao Xuanzhi had highly praised Qiao Yu, who was not yet widely recognized in the world of mathematics, on more than one occasion.
It’s evidence of this when he stood up for Qiao Yu multiple tis, and it was expected that they would have private contact.
The reason it caused a sensation in the mathematics community was the problem discussed in this letter—turbulence and the N-S equation!
After seven years, Qiao Yu finally re-entered the mathematical world.
The contents of this letter are as follows:
Mr. Tao Xuanzhi: Greetings upon reading your letter.
Recently, Elder Yuan calculated sothing, believing I have the potential to solve the fundantal problem of turbulence, so I have been contemplating the smoothness and uniqueness problem concerning turbulence and the N-S equation.
I must say, this is indeed a fascinating issue. Coincidentally, when researching this issue, I ca across your 2014 paper in the Arican Mathematical Society journal—"Finite-ti Blowup of the Averaged Solutions of the Three-dinsional N-S Equation."
Thus, I wrote this letter to discuss so of my recent ideas regarding the three-dinsional N-S equation.
The Euler bilinear operator constructed in your paper proved that a turbulent system with an initial value of u0 would blow up in finite ti.
I roughly understood it as robot A spilling a bottle of cola, hence it replicates itself as robot B to clean up the ss, then robot B replicates robot C to clean up, and so on.
In this perpetual cycle, until robot X directly releases explosive energy, the spilled cola is cleaned, and all the robots cease to exist.
I find it very interesting; your research terminates the possibility of proving one research direction concerning the N-S equation. It also gave great inspiration—the proof must distinguish between the original operator and the averaged operator.
This has given Qiao Algebraic Geotry another place to utilize.
Under the traditional analysis frawork, the original operator and the averaged operator form irreconcilable contradictions in Banach Space, much like the blowup chanism you revealed.
But if we project each velocity field unit u(x,t) into Modal Space (α, β), through modal projection N_α,β(u), we can construct a new bilinear form with the following characteristics:
B(u,v)=⊕_{γ∈Γ}[N_{α γ,β}(u)⊗_Ω∇_γ N_{α,β-γ}(v)]
Where Γ is the critical frequency interval defined in your paper. Now, let’s temporarily forget the boundaries of the Riemann surface and Euclidean space.
To appreciate the ingenuity of this construction!
I’m sure you noticed that as γ approaches the blowup threshold, the corresponding modal component N_{α γ,β}(u) vanishes automatically due to its self-sustaining requirent—essentially converting the explosion of robot X you discovered into a conservation law within Modal Space.
Now let’s recall the modal conservation theorem in Qiao Algebraic Geotry.
If we rewrite the initial condition u0 as N_α,β(u0)=⊕[φ_k⊗ψ_l], where each φ_k satisfies the modal unit stability condition ‖N_α,β(φ_k)‖≡1, then the energy transfer chain will inevitably appear as a directional reversal of the parater manifold M at step k l≤dimM.
For this, I constructed a special index class on the modal manifold M⁷ and proved that any solution leading to a finite-ti singularity must violate the modal unit theorem of N_α,β(1).
Of course, I’m sure you have already found the problem here!
My approach still has two fatal flaws that cannot be verified: one is how to embed the viscosity term Δu into the curvature tensor of Modal Space; the other is that I still cannot explain the asymptotic behavior of the blowup solution when modal paraters (α, β)→(0, π/2).
In fact, I have borrowed quantum simulation supercomputers for several singular vortex modal decompositions. But obviously, the current results do not directly prove evidence of smooth solutions and uniqueness.
Therefore, there must be sothing I’ve overlooked. If you’re not busy, maybe we can delve deeper into these two issues together.
If your team has free ti, you can also connect in calculation, let’s work together to resolve this unsolved mystery as soon as possible.
Moreover: Honestly, I wanted to take a break. But my teacher and Mr. Yuan think I’ve rested long enough! They have high hopes for , making unable to slack off.
So please do help think of a solution! I have a premonition that when we fully grasp the essence of turbulence, or say the mathematical essence, we’ll be able to open a new track in the aviation field, one that bears our nas.
...
After Tao Xuanzhi publicly released this letter on his blog, he also shared his insights.
"Although Qiao Yu has drawn a very large cake for , I find that with my limited knowledge reserves, I might not be able to complete the task he entrusted to independently.
So if anyone has better ideas, perhaps we can discuss together. Especially on the issue of embedding the viscosity term Δu into the curvature tensor of Modal Space."
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