Navier-Stokes Problem
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1. Overview
2. Step 1. Design a blow-up vortex
3. Step 2. Prove that the external force is smooth
4. Conclusion
1. Overview
⑴ [Navier-Stokes equation](https://jb243.github.io/pages/936#:-,%F
The equation of motion for a Newtonian fluid, commonly known as the Navier-Stokes equation, can be obtained by substituting the constitutive equation of a Newtonian fluid into Cauchy’s equation of motion. The result can be written using Gibbs’ symbolic notation as follows. Here, instead of the gravitational term $\rho \mathbf{g}$, a general external force $f$ may also be used.
⑵ Problem
Suppose initially $u(x,0)=u_0(x)$ is very smooth and has finite energy. In three dimensions, is $\left|u(x,t)\right|<\infty$ for all $t>0$, with all derivatives remaining finite? Or can something like $\left|\nabla u(x,t)\right|\to\infty$ as $t\to T$ occur at some finite time $T$? (singularity, blow-up).
🄐 Global existence and smoothness on $\mathbb{R}^3$claims that the solution is always smooth, $f=0$.
🄑 Global existence and smoothness on $\mathbb{T}^3$: claims that the solution is always smooth, $f=0$.
🄒 Breakdown on $\mathbb{R}^3$: for some initial conditions, a smooth solution does not exist globally; smooth $f$ is allowed.
🄓 Breakdown on $\mathbb{T}^3$: for some initial conditions, a smooth solution does not exist globally; smooth $f$ is allowed.
⑶ History
① 1822 ~ 1845: Formulation of the Navier-Stokes equations.
② 1934 ~ 1951: In 1934, Leray proved the existence of global weak solutions with finite energy (ref).
③ 1959: Through the work of Ladyzhenskaya and others, global existence, uniqueness, and smoothness were established for the 2D Navier-Stokes equations (ref).
④ 2000: Selected as one of the seven Millennium Prize Problems (Clay Mathematics Institute).
⑤ August 2026: Tristan Buckmaster (Anthropic) and Levent Alpöge claimed that singularities exist (ref) − 🄒, 🄓
⑥ September 8, 2026: OpenAI claimed, using its next-generation model known as BEL, that singularities exist (ref1, ref2 − 🄒, 🄓
2. Step 1. Design a blow-up vortex
⑴ Overview: Design a self-similar vortex constructed to blow up at $t=1$.
⑵ A specific functional form is given, but roughly speaking it has the following structure.
① By conservation of angular momentum, the flow rotates faster as it approaches the center, causing the velocity to diverge.
○ Radial inflow contributes to spin-up by carrying angular momentum toward smaller radii.
② Because mass cannot continue accumulating due to mass conservation and incompressibility, an axial flow escaping in the $z$ direction is also introduced.
Figure 1. A specific functional form
⑶ Energy perspective
① $E_{\mathrm{kinetic}}\sim \left|u\right|^2\times \mathrm{volume}$
② Although the velocity becomes infinite, the volume of the region in which such velocity exists converges to zero, so the energy of the vortex core converges to zero.
③ This is also related to Gabriel’s horn in geometry, which has finite volume but infinite surface area.
3. Step 2. Prove that the external force is smooth
⑴ Overview
① If we define $f:=R(u)=\partial_tu+(u\cdot\nabla)u-\nu\Delta u+\nabla p$, then the equation is automatically satisfied, but there is no reason whatsoever for such an $f$ to be smooth.
② Here, smooth means infinitely differentiable, i.e., $f\in C^\infty$.
⑵ 2-1. Construct the desired blow-up flow $u_0$.
① $u_0\to\infty$ (blow-up)
② The required external force $f=R(u_0)=R_0$ may also blow up, so it is inappropriate because $f\notin C^\infty$.
⑶ 2-2. Infinite iterative corrections
① Overview
○ Can we remove only the singular residual generated by the blow-up vortex while leaving the blow-up vortex itself almost untouched?
○ In other words, even if $\partial_tu$, $(u\cdot\nabla)u$, $-\nu\Delta u$, and $\nabla p$ each diverge, construct them so that the diverging parts cancel one another in the total residual $R(u)$.
② If an oscillation correction is introduced, then $u_1=u_0+w_1$ and $f_1=R(u_1)$.
○ Even if $w_1$ itself is small, it generates an effective momentum $\langle w_1\otimes w_1\rangle$.
○ $\left|R(u_1)\right|\approx\left|-\nabla\cdot T+\nabla\cdot(w_1\otimes w_1)\right|<\left|-\nabla\cdot T\right|\approx\left|R(u_0)\right|$.
○ $R_0(q)=q+q^2+q^3+\cdots ;;\boldsymbol{\to};; R_1(q)=q^2+q^3+\cdots$.
③ Applying another oscillation correction, $u_2=u_1+w_2$, $f_2=R(u_2)$
○ $R_1(q)=q^2+q^3+\cdots ;;\boldsymbol{\to};; R_2(q)=q^3+\cdots$
④ Iterative process
○ $(u_0,f_0)\to(u_1,f_1)\to(u_2,f_2)\to\cdots$
○ $R_0\xrightarrow{w_1}R_1\xrightarrow{w_2}R_2\xrightarrow{w_3}R_3\to\cdots$
○ Choose a parameter $\to$ verify an inequality $\to$ modify the profile $\to$ satisfy the moment condition $\to$ check that the modification does not violate the cone condition $\to$ obtain the next correction estimate $\to\cdots$.
⑤ Conclusion
○ As in mathematical induction, stage $j$ satisfies the required conditions $\Rightarrow$ the correction can be carried out $\Rightarrow$ stage $j+1$ also satisfies the same conditions.
○ Since the explicit form of $u$ is known, computation power was used to find a set of inductive conditions (a lot of brute-force work…)
○ Condition 1. $u$ blows up
○ Condition 2. $\left|u\right|_2$ is bounded
○ Condition 3. $\nabla\cdot u=0$
○ Condition 4. On the axis, the smooth wave generates stress in exactly the required direction
○ Condition 5. Each correction is smaller than the preceding correction
○ Condition 6. Keep the support compact
○ Condition 7. All derivatives of $f$ match continuously at $t=1$
⑷ 2-3. Finally remove the singular behavior of $f=R(u)$ at every derivative order: i.e., $R(u)\in C^\infty$.
4. Conclusion
⑴ 20% new design ideas + 80% painstaking technical work proving that the construction really closes.
① 20% new design idea: preserve the blow-up while removing only the singular forcing through oscillation.
② 80% painstaking technical work: 10,000 agents + 130 billion tokens
③ In fact, finding examples and counterexamples through brute-force trial and error is something AI is particularly good at, so this proof could be described as a rather AI-like proof.
⑵ Passed Lean verifier, but there are some critiques.
⑶ Considering that the downslope windstorm is also ultimately a phenomenon in which fluid velocity diverges, perhaps similar phenomena have repeatedly appeared in many different forms.
Entered: 2026.09.08 21:35