I think it is worth for reviewing known results on the Navier-Stokes equations. I am writing a lecture note which contains my own digestion of OpenAI paper [OAI26b] trying to discover the missing link between our classical theory to the construction. It is worth to figure out how to view this construction in human’s perspective.

AI usage: All writing is done by myself. AI was used to convert the original latex file in the part of my lecture note into blog form. Also, Gemini helps me to figure out Leonardo da Vinci’s contribution on fluids.

The story to understand fluids: Existence and Uniqueness

For over two thousand years, humanity has sought to understand the chaotic behavior of fluids born out of survival and raw terror. To ancient seafarers, the ocean’s unpredictable fury was so overwhelming that it was depicted through mythology, personified as a wild, unruly god like Poseidon. It took centuries for human thought to shift from mythic fear to rational observation. Around 1510, in the high Renaissance, Leonardo da Vinci (1452–1519) observed an infinite cascade of turbulence and captured it in a drawing (see the wonderful article in IAS). While his observation was crucial, da Vinci could not express his observation in appropriate mathematical language since he was way ahead of Galileo Galilei (1564–1642) and Issac Newton (1643–1727).

After the invention of Calculus by Newton and Leibniz in 16841, many physicists have studied the properties of fluids and tried to find a governing equation for fluids. In the middle of the 17th century, Torricelli and Pascal observed atmospheric pressure and developed the notion of pressure. Later in 1738, Daniel Bernoulli found the relation between the pressure and the velocity of the fluid, which is now called Bernoulli’s theorem.

A substantial leap was due to Leonhard Euler, who derived the Euler equation in 1755, which is written in the modern form

\begin{equation} \partial_t u+(u\cdot \nabla)u+\nabla p=0,\quad \mathrm{div}\, u=0 \end{equation}

where he can describe the motion of ideal fluids by connecting the density and pressure of the fluid. While Newton defined viscosity in 1687, the incompressible Euler equation does not assume the presence of any friction, which is not possible in fluids in nature. This leads to the d’Alembert paradox, which says that if the fluid is governed by the Euler equation, then the object in the fluid has no friction.

This paradox was first resolved by Claude-Louis Navier (1822), who introduced an additional dissipation term from a molecular point of view2, but he did not understand the physical meaning of this term. Later, Cauchy, Poisson, and Saint-Venant tried to derive this new equation by continuum mechanics. Finally, Stokes (1845) rigorously derived the Navier-Stokes equation by extending Newton’s law of viscosity in 3D:

\begin{equation} \left\{ \begin{aligned} \partial_t u-\nu\Delta u+(u\cdot\nabla)u+\nabla p&=f,\\ \mathrm{div}\, u&=0 \end{aligned} \right. \end{equation}

While the equation was successfully derived, only special solutions were found, and many people tried to describe the fluid in special situations. Poiseuille flow and Couette flow are fundamental examples, and Oseen also introduced Oseen flow to linearize the problem in 1910 to overcome the issue of Stokes flow. Later, Prantl introduced boundary layer theory by similarity analysis and was successful in describing the behavior of real fluids in short time regimes.

However, these solutions were found or approximated under certain circumstances and the existence of the solutions to (2) was an outstanding open problem since the derivation of the equation. In 1934, Jean Leray [L34] introduced the turbulent solution, which is now called a weak solution, and proved the existence of weak solutions in 2D and 3D on the whole space case and periodic domain case satisfying

\begin{equation} \sup_{0<t<T} \Vert u(t)\Vert_{L^{2}}^2+2\nu\int_0^t \Vert\nabla u(\tau)\Vert_{L^{2}}^2 d\tau \leq C(T)\left(\Vert u_0\Vert_{L^{2}}^2+C\int_0^T \Vert f(\tau)\Vert_{L^{2}}^2d\tau\right). \end{equation}

Later, this construction was extended to bounded domains by Hopf [H51]. In the same paper, he also proved the short-time existence and uniqueness of strong solutions to the Navier-Stokes equations. Furthermore, Leray also obtained a lower bound on the lifespan of the strong solutions.

Since the fundamental work by Leray, who proved global existence of weak solutions and short-time existence of solutions, many mathematicians have devoted themselves to determining whether the 3D Navier-Stokes equation admits finite-time blow-up or is globally regular. In this seminal paper, he suggested that a backward self-similar solution of the form could be a good candidate to predict a blow-up solution to the Navier-Stokes equation. Such a case was ruled out by Nečas, Růžička, and Šverák [NRS96], who proved that if there is a solution of the form $u(t,x)=\frac{\sqrt{\nu}}{\sqrt{T^\ast-t}}U(x/\sqrt{T^\ast-t})$ and $u$ satisfies the global energy inequality, then $u$ is identically zero. On the other hand, one can guess that a self-similar singularity is still possible for a solution satisfying the local energy inequality. This scenario was also ruled out by Tsai [T98].

Another important observation in answering the global regularity problem is to seek a regularity criterion. If

\begin{equation} u\in L^{p}(0,T;L^{q}),\quad \frac{2}{p}+\frac{3}{q}\leq 1, \quad 3<q\leq \infty. \end{equation}

then Prodi [P59], Serrin [S62], and Ladyzhenskaya [L67] proved that weak solutions become strong solutions. Hence the Leray-Hopf weak solution is unique in this class. The endpoint case ($p=\infty$ and $q=3$) was later proved by Escauriaza, Seregin, and Sverak [ESS03].

On the other hand, after the pioneering work of Jean Leray [L34], many researchers have investigated the size of the singular set $\mathcal{S}$, the set of all points $(t,x)$ for which $u$ is not locally bounded in any neighborhood of $(t,x)$. In that paper, Leray proved that the singular set of time has Lebesgue measure zero. Later, in a series of works, Scheffer [S77], [S80] proved that the parabolic Hausdorff dimension of $\mathcal{S}$ is less than $5/3$. Later, Caffarelli, Kohn and Nirenberg [CKN82] proved that the singular set has one-dimensional parabolic Hausdorff measure zero. This means that the singularity of the solution to the Navier-Stokes equation does not form a curve. A more simplified proof was given by Lin [L98].

In the past 15 years, there have been several impressive advances on the construction of weak solutions to the Euler equations that lose uniqueness. In this direction, De Lellis and Székelyhidi [DLS09] introduced a convex integration scheme to the incompressible Euler equations and proved the Onsager conjecture for a certain regime [DLS13], which was extended by Buckmaster, De Lellis, Isett, and Székelyhidi [BDLIS15]. The endpoint case was solved by Isett [I18]. The strong version was proved by Giri, Kwon3, and Novack [GKN26].

This program was successfully applied to many problems in fluid mechanics. For the Navier-Stokes equation without force, Buckmaster and Vicol [BV19] proved that there are infinitely many solutions to the Navier-Stokes equations. Later, Cheskidov and Luo [CL22] proved nonuniqueness result $u\in L^{p}_tL^{\infty}_x$, $p<2$, which is sharp in view of the Ladyzhenskaya-Prodi-Serrin criteria. However, these solutions may not satisfy (3).

In this regard, Albritton, Brué and Colombo [ABC22] proved the nonuniqueness of Leray-Hopf solutions when they allow forcing. Their idea comes from the instability of vortex rings in the 2D Euler equation due to Vishik [V18], [V18b] (see [ABCDGJK23]) and the program of Jia and Sverak [JS15] after the construction of the global forward self-similar solution to the Navier-Stokes equation [JS14].

However, the aforementioned results do not talk about any finite-time singularity formation of the fluid nor ill-posedness of the problem, which has become another concurrent trend in the past 13 years.

Singularity formation of fluids: before AI comes in

An interesting breakthrough in seeking singularity started from the numerical evidence by Luo and Hou [LH14]. In this paper, they exhibited numerical evidence for the incompressible Euler equation with a boundary showing finite-time singularity formation. The idea is to design a hyperbolic point at the boundary by oppositely rotating two cylinders. This scenario was carried out rigorously by Chen and Hou [CH21], who proved a finite-time blowup construction with $C^{1, \alpha}$ initial data and [CH22] for smooth initial data.

Inspired by the numerical scenario of Luo and Hou, Kiselev and Sverak [KS14] constructed an example for the 2D Euler equation with a boundary whose gradient of the vorticity has double exponential growth. Since then, many research works have been studied extensively. Finite-time singularity formation for the axisymmetric Euler equation in a domain was constructed by Elgindi and Jeong [EJ19]. We also note that this program provides several important advances on the finite-time singularity formation, including Jeong and Kim [JK24], which is a simplified proof of strong ill-posedness of the 3D Euler equation by Bourgain and Li [BL15].

Building upon these ideas on studying nonlocal controls of fluids, Elgindi [E21] proved the finite-time singularity of the 3D axisymmetric Euler equation without swirl in the whole space for $C^{1,\alpha}$-initial data. Later, many relevant works were carried out by Elgindi and Pasqualotto [EP23], [EP25]. See an excellent survey on this topic written by Drivas and Elgindi [DE23].

For the Navier-Stokes equation, Hou and Huang [HH22] and Hou [H23] suggested numerical evidence for potential self-similar blow-up to the axisymmetric Navier-Stokes equations. More precisely, they provided numerical evidence that 3D Navier-Stokes equations develop nearly self-similar blow-up from initial data having finite energy. This scenario will be revisited later.

The aforementioned finite-time blow-up construction highly relies on the self-similar blow-up scenario. While the blow-up scenario in [CH21], [CH22] is not exactly a self-similar solution, philosophically, it highly relies on the self-similar blow-up paradigm. Starting in 2023, a completely new mechanism was discovered by Córdoba and Martínez-Zoroa and their collaborators [CL25], [CMZ26b], [CMZ26], [CMZ25], [CLM25b], [CMO25], [CMO24], [CM22], [CM24], [CLM25]. Their idea can be described as infinitely many scales stretching. In particular, Córdoba, Martínez-Zoroa, and Zheng [CMZ25] proved finite-time blow-up of the 3D incompressible Euler equation with $C^{1,\alpha}\cap L^{2}$ but it is smooth outside the origin. They also constructed finite-time blow-up for the hypodissipative Navier-Stokes equation with forcing.

Finite-time blow-up construction by AI-assisted and by OpenAI

Seeking a finite-time singularity was a long-standing question, and there were several attempts to construct a blow-up profile by using physically informed neural networks (PINNs). In this direction, Wang, Lai, Gómez-Serrano, and Buckmaster [WLGB23] suggested a way to construct a smooth self-similar blow-up profile for the 2D Boussinesq and the 3D Euler equations. In this direction, Wang and others [WBM25] found another family of unstable singularities by using physical neural networks and applied them to CCF, IPM, and Boussinesq equations.

With large language models, Alpöge, Buckmaster, and Coiculescu [ABC26] constructed finite-time blow-up for the IPM equation with a uniformly spacetime smooth force by refining the argument in [CL25]. Later, OpenAI announced the proof of finite-time blow-up for the incompressible Euler equation with smooth initial data [OAI26a] and the incompressible Navier-Stokes equation with smooth forcing without initial data [OAI26b]. The construction of the blow-up solutions for the Euler equation is closely connected to [CMZ25], [CMZ26], the infinite layer stretching. However, these two papers are impossible to read by human experts for several reasons. Unlike good research papers that human experts wrote, these constructions do not explain which part is new. Also, the writing style is not concise and insufficient to understand the materials by human experts. Also, the notation does not follow conventions in the field. Finally, these products were not monitored by experts in fluid mechanics. It is regarded as a standard for more than two centuries that the author takes full responsibility to explain the material and write a readable paper. However, no one from OpenAI takes full responsibility to explain this chaotic situation in mathematics. While they achieved an impressive result, this irresponsible practice should be criticized at large. Not surprisingly, many previous works on self-similar solutions should be cited, but they were not cited in the OpenAI paper.

A detailed human-digested proof for the Euler equation is given by Zheng [Z26]. However, for the Navier-Stokes equations, while Lei and Ren [LR26] wrote an exposition for [OAI26b], it only explains a technical aspect of the construction, and it is not that easy to figure out each of the motivations to consider the delicate issue of constructing blow-up solutions to the Navier-Stokes equations.

References

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[ABC22] Dallas Albritton, Elia Brué, and Maria Colombo, Non-uniqueness of Leray solutions of the forced Navier-Stokes equations, Ann. of Math. (2) 196 (2022), no. 1, 415–455.

[ABC26] L. Alpöge, T. Buckmaster, and M. P. Coiculescu, Extending the Córdoba-Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing, 2026.

[BL15] Jean Bourgain and Dong Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math. 201 (2015), no. 1, 97–157.

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[CMO24] Diego Córdoba, Luis Martínez-Zoroa, and Wojciech S. Ożański, Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations, Duke Math. J. 173 (2024), no. 10, 1931–1971.

[CMO25] Diego Córdoba, Luis Martínez-Zoroa, and Wojciech S. Ożański, Instantaneous continuous loss of regularity for the SQG equation, Adv. Math. 481 (2025), Paper No. 110553, 49.

[CMZ25] Diego Córdoba, Luis Martinez-Zoroa, and Fan Zheng, Finite time singularities to the 3D incompressible Euler equations for solutions in $C^\infty (\mathbb{R}^3 \setminus {0})\cap C^{1,\alpha}\cap L^2$, Ann. PDE 11 (2025), no. 2, Paper No. 19, 56.

[CMZ26b] Diego Córdoba, Luis Martínez-Zoroa, and Fan Zheng, Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,\epsilon }\cap L^\infty_{t} L^{2}_{x}$, Arch. Ration. Mech. Anal. 250 (2026), no. 3, Paper No. 38, 45.

[CMZ26] Diego Córdoba, Luis Martínez-Zoroa, and Fan Zheng, Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,\epsilon }\cap L^\infty_{t} L^{2}_{x}$, Arch. Ration. Mech. Anal. 250 (2026), no. 3, Paper No. 38, 45.

[DLS09] Camillo De Lellis and László Székelyhidi, Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417–1436.

[DLS13] Camillo De Lellis and László Székelyhidi, Jr., Dissipative continuous Euler flows, Invent. Math. 193 (2013), no. 2, 377–407.

[DE23] Theodore D. Drivas and Tarek M. Elgindi, Singularity formation in the incompressible Euler equation in finite and infinite time, EMS Surv. Math. Sci. 10 (2023), no. 1, 1–100.

[EP23] T. M. Elgindi and F. Pasqualotto, From instability to singularity formation in incompressible fluids, 2023.

[E21] Tarek Elgindi, Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$, Ann. of Math. (2) 194 (2021), no. 3, 647–727.

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[EP25] Tarek M. Elgindi and Federico Pasqualotto, Invertibility of a linearized Boussinesq flow: a symbolic approach, Comm. Math. Phys. 406 (2025), no. 11, Paper No. 261, 36.

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[GKN26] Vikram Giri, Hyunju Kwon, and Matthew Novack, The $L^3$-based strong Onsager theorem, Ann. of Math. (2) 204 (2026), no. 1, 265–421.

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[OAI26b] OpenAI, Finite-time blowup for Navier-Stokes, 2026.

[OAI26a] OpenAI, Finite-time blowup for the Euler equation, 2026.

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[V18b] M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018.

[WLGB23] Y. Wang, C.-Y. Lai, J. Gómez-Serrano, and T. Buckmaster, Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks, Phys. Rev. Lett. 130 (2023), 244002.

[WBM25] Yongji Wang, Mehdi Bennani, James Martens, Sébastien Racanière, Sam Blackwell, Alex Matthews, Stanislav Nikolov, Gonzalo Cao-Labora, Daniel S. Park, Martin Arjovsky, et al., Discovery of unstable singularities, arXiv:2509.14185, 2025.

[Z26] F. Zheng, Finite-time blowup of the Euler equation without forcing, 2026.

  1. Newton first found the method of fluxions in 1665, and Leibniz independently invented calculus in 1673–1676. ↩

  2. Now we describe this as a kinetic point of view, but there was no kinetic theory at that time. ↩

  3. I was frequently asked whether Hyunju Kwon is my sister since we worked on fluid mechanics and shared the same initial. She is not my biological sister. ↩