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Stable Singularity of the Euler Equations on $\mathbb{R}^3$

This paper presents evidence for a stable finite-time singularity in the 3D Euler equations on $\mathbb{R}^3$ by utilizing physics-informed neural networks to identify a self-similar blowup profile and establishing a framework to certify its nonlinear stability through linear damping and explicit estimates.

Original authors: Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar

Published 2026-09-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluids are the great movers of our world, shaping everything from the weather patterns that bring rain to the blood flowing through our veins. For over a century, mathematicians and physicists have relied on a set of equations known as the Euler equations to describe how ideal fluids move when they have no internal friction. These equations are a cornerstone of fluid dynamics, yet they hold a deep and persistent mystery: whether a smooth, calm flow can suddenly and violently tear itself apart in a finite amount of time. This event, known as a singularity or "blowup," would mean the fluid's speed or spin becomes infinite at a specific point, a physical impossibility that signals the equations have broken down. Solving this puzzle is one of the most significant open challenges in mathematics, with implications for understanding everything from ocean currents to the behavior of gases in stars.

For decades, researchers have searched for a scenario where such a blowup occurs, but the mathematics is notoriously difficult. The equations are sensitive, and finding a solution that starts smooth and ends in a singularity is like trying to balance a pencil on its tip; even the slightest error in the calculation causes the solution to collapse into nonsense. While singularities have been proven to happen in simpler, modified versions of these equations, or in fluids confined by walls, no one had ever found a stable, self-similar blowup in the full, unbounded three-dimensional Euler equations. A self-similar blowup is a specific type of collapse where the fluid's shape remains the same as it shrinks, merely getting smaller and faster at a precise rate, much like a zooming camera that keeps the image centered and clear as it approaches the subject.

In a new study, a team of researchers has provided the first strong evidence that such a stable singularity does exist in the three-dimensional Euler equations on an unbounded domain. They did not find this by hand-calculating or by traditional computer simulations alone. Instead, they used a modern approach called a physics-informed neural network. Imagine a computer program that acts like a student learning a subject not just by memorizing answers, but by constantly checking its work against the fundamental laws of physics. The researchers trained this program to find a specific shape of fluid flow that satisfies the Euler equations while shrinking toward a singularity. The program had to navigate a vast, complex landscape of possibilities, avoiding dead ends where the math would fail, until it discovered a profile that fit the equations with extraordinary precision.

The researchers found a candidate profile that shrinks at a critical rate of 0.5. This specific rate is not arbitrary; it is a "mathematically distinguished" value that theoretical work had suggested was the only place where such a singularity might survive. The profile they found moves along a central axis, maintaining a consistent shape as it accelerates toward the moment of blowup. To ensure this was not just a numerical trick, the team converted the computer-generated shape into a precise mathematical description made of smooth curves. They then subjected this shape to a rigorous stability test. In the world of fluids, a singularity is only interesting if it is stable, meaning that if you start with a flow that is slightly different from the perfect singularity, it will still evolve into that same singularity rather than drifting away. If the singularity were unstable, it would require infinitely precise initial conditions to occur, making it physically irrelevant.

The team's analysis showed that the candidate profile is indeed stable. They demonstrated that the fluid flow around the singularity acts like a stabilizing force, pushing any small disturbances away from the core and preventing them from growing out of control. This damping effect was verified across the entire domain, including in tricky regions where the flow is slow. The researchers used a combination of advanced computer optimization and rigorous mathematical proofs to certify that the linear part of the equations provides a strong restoring force. While the full proof of nonlinear stability is still being finalized in a companion paper, the evidence for this linear stability is robust and suggests that the entire mechanism holds together.

This work does not claim to have solved the Millennium Prize problem of fluid dynamics, nor does it prove that a singularity will definitely happen in the real world. Instead, it provides a concrete, mathematically verified example of a stable blowup scenario that was previously only a theoretical possibility. By using artificial intelligence to discover the shape and then using classical mathematics to prove its stability, the researchers have bridged a gap between numerical discovery and rigorous proof. They have shown that the Euler equations can support a finite-time singularity that is not a fluke of bad math, but a stable, self-similar structure. This finding narrows the search for a definitive answer to one of the oldest questions in fluid mechanics, suggesting that the equations do, in fact, allow for a violent, finite-time breakdown of smooth flow.

Technical Summary: Stable Singularity of the Euler Equations on $\mathbb{R}^3$

Problem Statement
The paper addresses the central open problem in fluid dynamics regarding whether smooth solutions to the 3D incompressible Euler equations can develop a finite-time singularity (blowup) on the unbounded domain $\mathbb{R}^3$ . While finite-time blowup has been established for Euler equations with boundaries or under non-smooth initial conditions, the existence of a stable singularity from smooth initial data in the unbounded domain remains unresolved. The authors focus specifically on the critical scaling exponent $\lambda = 0.5$ , identified by Constantin et al. as the "mathematically distinguished" regime for potential self-similar blowup, where previous theoretical results had ruled out stable singularities for $\lambda < 0.5$ and $\lambda > 0.5$ under specific ansatz assumptions.

Methodology
The authors employ a hybrid approach combining physics-informed machine learning with rigorous computer-assisted proofs. The methodology proceeds in three distinct stages:

Numerical Discovery via PINNs:
- Ansatz: A traveling-wave self-similar ansatz is adopted, allowing the singularity center to drift along the symmetry axis ( $z$ -axis) while maintaining a fixed shape in rescaled coordinates. This formulation introduces the drift speed $C$ as a free parameter, offering greater optimization flexibility than stationary ansatzes.
- Optimization: Physics-Informed Neural Networks (PINNs) are trained to minimize the residual of the steady-state profile equations. To achieve the high precision required for subsequent certification, the authors utilize:
  - 64-bit floating-point arithmetic (FP64) to reduce round-off errors.
  - Adaptive loss weighting and residual-based resampling to concentrate training points in critical regions.
  - A "boosting" strategy where sequential networks correct residual errors.
  - Self-scaled curvature-aware optimizers (SS-eSOAP and SS-Broyden) to navigate the stiff loss landscape, reducing the Mean Squared Error (MSE) of the profile residual to $O(10^{-10})$ .
- Domain Handling: A $\sinh$ transformation maps the unbounded physical domain to a finite computational box, ensuring high resolution near the origin (where blowup occurs) while covering the far field.
Analytic Representation:
- The high-precision PINN solution is converted into a piecewise polynomial spline representation. This step is crucial for rigorous certification, as it allows for exact differentiation and evaluation of profile-dependent quantities using interval arithmetic (via the Arb library), avoiding the difficulties of certifying neural network evaluations directly.
Stability Analysis and Certification:
- Linearization: The dynamic rescaling equations are linearized around the spline profile. Perturbations are analyzed in scale-consistent variables ( $\delta v = (\delta \omega, \nabla \delta u)$ ).
- Weighted Energy Estimates: A stability framework is constructed using a full energy functional $E_k^2 = E_0^2 + \mu_k H_k^2$ , combining low-order and high-order weighted norms. The weights are optimized numerically to maximize linear damping.
- Damping Mechanisms:
  - Low-order damping is established throughout most of the domain, with the exception of two neighborhoods around the meridional fixed points.
  - High-order damping is utilized to control these specific "low-transport" regions near meridional fixed points where low-order damping fails.
- Computer-Assisted Proof: The stability argument is reduced to a finite collection of explicit inequalities involving computable constants. These constants (residuals, matrix eigenvalues, interpolation bounds) are rigorously enclosed using interval arithmetic. The proof structure is designed to be formalizable in the Lean proof assistant (LeanPDE).

Key Results

Approximate Profile: The authors successfully discovered a highly accurate approximate self-similar profile for the 3D axisymmetric Euler equations at the critical scaling $\lambda = 0.5$ . The profile exhibits polynomial spatial decay, consistent with non-existence theorems that require super-algebraic decay to rule out blowup.
Certified Linear Damping: The paper provides the first evidence of certified linear damping for the Euler system at $\lambda = 0.5$ . The analysis demonstrates that the linearized operator possesses a damping margin through energy estimates, establishing a stabilizing mechanism that combines low-order damping on most of the domain with high-order damping in the problematic off-axis regions.
Framework for Nonlinear Stability: A complete framework is established to prove nonlinear stability. The argument reduces the existence of a finite-time singularity to the rigorous certification of a finite set of explicit estimates.
Convection-Varied Systems: The PINN formulation successfully finds profiles for the convection-varied Euler system across the parameter range $\epsilon \in [0, 1]$ , suggesting that the disappearance of dynamically observed blowup scenarios at higher convection levels does not imply the non-existence of singular profiles.

Significance and Claims
The paper claims to provide the first evidence that the 3D Euler equations admit a stable, self-similar finite-time blowup on $\mathbb{R}^3$ at the critical exponent $\lambda = 0.5$ .

Modest Claims: The authors explicitly state that the work provides evidence and a framework rather than a completed, unconditional proof of blowup. The final step—rigorously certifying all remaining constants and margins to close the nonlinear stability argument—is identified as the remaining computational work.
Methodological Contribution: The paper highlights the efficacy of physics-informed optimization in discovering candidate singularity structures that are difficult to construct manually. It demonstrates that AI can serve as a discovery tool for specific PDE instances, while human-designed mathematical frameworks and formal verification (Lean) ensure the rigor of the proof.
Stability vs. Instability: Unlike previous works that relied on global outgoing properties (which do not hold for Euler with self-similar ansatz), this work establishes stability via a weaker local outgoing property and a delicate balance of low-order and high-order damping mechanisms.

In summary, the paper bridges numerical discovery and rigorous analysis, offering a concrete candidate profile and a verified linear stability mechanism that, if fully certified, would resolve the finite-time blowup problem for the 3D Euler equations.

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