Nam Le

The Regularity Problem for the 3D Euler Equations

Le, Nhut Nam
Table of Contents

Leonhard Euler wrote down the equations governing the motion of an ideal incompressible fluid in 1757. Whether smooth solutions to these equations can develop a singularity in finite time, a point at which derivatives of the velocity blow up, has been an open problem ever since, and remains one of the central questions in mathematical fluid dynamics.

Problem (Euler Regularity)

Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth, divergence-free initial velocity field with sufficient decay at infinity. Does the unique local smooth solution $u(x,t)$ to the 3D incompressible Euler equations $$\partial_t u + (u \cdot \nabla)u + \nabla p = 0, \qquad \nabla \cdot u = 0, \qquad u(\cdot,0)=u_0$$ remain smooth for all time $t > 0$?

The problem is rated L4 on UnsolvedMath, reflecting its depth, and is closely related to the Clay Millennium Prize Problem on the Navier–Stokes equations. The two questions are linked through the zero-viscosity limit, but neither implies the other.

The Equations and What Regularity Means #

The Euler equations express conservation of momentum (first equation) and incompressibility (second equation) for an inviscid fluid. The unknowns are the velocity field $u(x,t) \in \mathbb{R}^3$ and pressure $p(x,t) \in \mathbb{R}$; the pressure is determined implicitly by incompressibility via an elliptic equation.

Vorticity. The central quantity for singularity analysis is the vorticity $\omega = \nabla \times u$, which satisfies the vorticity equation $$\partial_t \omega + (u \cdot \nabla)\omega = (\omega \cdot \nabla)u.$$ The right-hand side, the vortex stretching term, is the essential source of difficulty. It creates a quadratic feedback: large $\omega$ produces large $(\omega \cdot \nabla)u$, which can further amplify $\omega$.

Local well-posedness. For $u_0 \in H^s(\mathbb{R}^3)$ with $s > 5/2$, there exists a unique smooth solution on a time interval $[0, T^*)$ for some $T^* > 0$ depending on $|u _0| _{H^s}$ (Kato, 1972). The question is whether $T^*$ can be taken equal to $+\infty$.

Why 2D is easy, 3D is not. In two dimensions the vortex stretching term $(\omega \cdot \nabla)u$ vanishes identically by antisymmetry. The scalar vorticity $\omega = \partial_1 u_2 - \partial_2 u_1$ is then simply transported along fluid particle paths without amplification, and $|\omega|_{L^\infty}$ is conserved. Global regularity in 2D follows immediately. In 3D no such conservation holds, and the problem is genuinely open.

The Beale–Kato–Majda Criterion #

The first major structural result reduces the regularity problem to a single quantity.

Theorem (Beale–Kato–Majda, 1984)

A smooth solution $u$ of the 3D Euler equations loses regularity at time $T^*$ if and only if $$\int _0^{T^*} |\omega(\cdot,t)| _{L^\infty(\mathbb{R}^3)}, dt = +\infty.$$ In particular, if the vorticity remains bounded in $L^\infty$ on $[0,T]$ for every finite $T$, the solution remains smooth globally.

The BKM criterion redirects the problem: one must show that the vorticity magnitude $|\omega|_{L^\infty}$ cannot accumulate to infinity in finite time. Since $\omega$ satisfies a transport-stretching equation, this requires understanding the geometric structure of the vorticity field under its own evolution.

Geometric Conditions and Depletion of Stretching #

The vortex stretching term $(\omega \cdot \nabla)u$ can be decomposed as $$(\omega \cdot \nabla)u = |\omega|^2 (\hat\omega \cdot \nabla)\hat u,$$ where $\hat\omega = \omega/|\omega|$ is the unit vorticity direction. The key observation is that stretching is governed not only by the magnitude of $\omega$ but also by the geometry of the vorticity field.

Theorem (Constantin–Fefferman–Majda, 1996)

If the unit vorticity direction $\hat\omega = \omega/|\omega|$ is uniformly Lipschitz in a neighbourhood of the set ${|\omega| > \lambda}$ for all $t \in [0, T]$ and some $\lambda > 0$, then the solution remains smooth on $[0,T]$.

This result says that blowup, if it occurs, must be accompanied by violent geometric irregularity of vortex lines, not just large vorticity magnitude, but also loss of Lipschitz regularity of the vorticity direction. It has motivated a line of research on the geometric structure of vortex tubes near potential singularities.

Blowup for Less Regular Data #

Recent years have seen dramatic progress on singularity formation for initial data that is smooth except at isolated points.

Theorem (Elgindi, 2021)

There exist axisymmetric, swirl-free initial velocity fields $u_0 \in C^{1,\alpha}(\mathbb{R}^3)$ for sufficiently small $\alpha > 0$ such that the corresponding solution to the 3D Euler equations develops a finite-time singularity.

Elgindi’s proof, published in the Annals of Mathematics 194 (2021), 647–727, constructs a self-similar blowup profile and establishes its nonlinear stability using a dynamical rescaling formulation. The initial data is not smooth: it belongs to $C^{1,\alpha}$ but not to $C^2$. The singularity forms at the axis of symmetry $r=0$.

This was a breakthrough, but it left open the smooth case. Elgindi himself noted the next target: constructing blowup from initial data that is non-smooth only at a single point, or eventually from fully smooth data.

Extending Elgindi’s construction. Chen and Hou (2022) proved the same type of $C^{1,\alpha}$ blowup for the 3D axisymmetric Euler equations with boundary (inside a periodic cylinder), realising the Hou–Luo blowup scenario numerically proposed in 2014. Subsequent work by Córdoba, Martínez-Zoroa, and Zheng (2025, Annals of PDE) showed that the singularity can be formed from initial data in $C^\infty(\mathbb{R}^3 \setminus {0}) \cap C^{1,\alpha}$, with non-smoothness at a single point, a further step toward the smooth case.

The 2025 Breakthrough: Smooth Blowup with Boundary #

The most significant recent development is the following result, which provides a rigorous proof of finite-time singularity from smooth initial data.

Theorem (Chen–Hou, PNAS 2025)

There exists a family of smooth, finite-energy initial data for the 3D axisymmetric Euler equations in a smooth bounded domain (periodic cylinder) such that the corresponding solutions develop a finite-time singularity. The blowup is nearly self-similar and occurs at the intersection of the boundary $r=1$ and the symmetry plane $z=0$.

The paper, contributed by Thomas Hou and published in PNAS in June 2025 (reviewed by Caflisch, Gómez-Serrano, Sverak, and Tao), provides a computer-assisted proof. The strategy is to:

  1. construct a numerical approximate self-similar blowup profile via the dynamical rescaling formulation,
  2. prove rigorously that the true solution remains close to this profile using energy estimates with carefully verified error bounds (computed with interval arithmetic), and
  3. conclude nonlinear stability of the blowup via a bootstrap argument.

This resolves the problem affirmatively in the setting of smooth data and a smooth bounded domain. The boundary plays a crucial role: it creates an antisymmetric flow pattern driving azimuthal vorticity toward a critical ring, generating intense vortex stretching at a hyperbolic saddle point on the wall.

The remaining open case. The problem in $\mathbb{R}^3$ (or on the periodic torus $\mathbb{T}^3$) without boundary remains open. It is not known whether smooth initial data in free space can produce a singularity, or whether the absence of a boundary provides a genuine stabilising mechanism.

Research Directions #

1. Removing the Boundary #

The most pressing open question is whether the Chen–Hou construction can be extended to $\mathbb{R}^3$ or $\mathbb{T}^3$. The boundary in the 2025 result acts as a geometric catalyst: it enforces a no-flow condition that concentrates vorticity at a specific ring on the wall. Without a boundary, the antisymmetric flow structure that drives the singularity must be sustained entirely by the initial data and the nonlinear dynamics. Whether a comparable mechanism can persist in free space, without the reflective constraint of the wall, is the central open question.

2. Self-Similar Blowup in Full 3D #

All current singularity results are for axisymmetric flows, which reduce the problem from 3 spatial dimensions to 2 (the $rz$-plane). In full 3D, the angular variable $\theta$ is active, and perturbations in the azimuthal direction can either stabilise or destabilise the singularity. Elgindi, Ghoul, and Masmoudi (2021) proved stability of the $C^{1,\alpha}$ blowup under axisymmetric perturbations. Whether the singularity survives fully 3D (non-axisymmetric) perturbations, a question Elgindi posed as open, is crucial: a blowup that is destroyed by any non-symmetric perturbation has limited physical relevance.

3. Quantitative Vortex Stretching and the Role of Geometry #

The BKM criterion and the Constantin–Fefferman–Majda theorem both express the same idea from opposite directions: blowup is controlled by the magnitude and geometry of the vorticity. Current research asks whether a quantitative version can be made sharp. Specifically: if the vorticity direction $\hat\omega$ becomes Hölder-continuous but not Lipschitz, does blowup necessarily follow? Or is there a finer scale invariant quantity, perhaps involving the Hessian of the velocity or the curvature of vortex lines, that governs the problem?

4. Weak Solutions and Non-Uniqueness #

Separate from the question of whether smooth solutions blow up is the question of what happens after a potential singularity. De Lellis and Székelyhidi (2009–2013) proved that the Euler equations have infinitely many weak $L^\infty$ solutions for generic initial data, via convex integration. Isett (2018) proved that weak solutions can dissipate energy, confirming Onsager’s 1949 conjecture. These results show that the solution concept must be carefully chosen. After a smooth blowup, the system likely enters a regime of non-unique weak solutions, and identifying the physically relevant selection criterion, entropy conditions, vanishing viscosity, $h$-principle, is a major open problem.

5. Vanishing Viscosity and the Navier–Stokes Connection #

The Navier–Stokes equations add a viscous term $\nu \Delta u$ to the right-hand side. For any $\nu > 0$, global regularity of Navier–Stokes in 3D is itself open (the Clay Millennium Problem). For the zero-viscosity limit $\nu \to 0$, the central question is whether Navier–Stokes solutions converge to Euler solutions uniformly in time, a question tied to boundary layer behaviour (the Prandtl conjecture) and to the regularity of the Euler solution. If Euler develops a singularity at time $T^*$, the behaviour of Navier–Stokes solutions near $T^*$ as $\nu \to 0$ is completely unknown.

References #

  1. Euler, L. (1757). Principes généraux du mouvement des fluides. Mémoires de l’Académie des Sciences de Berlin, 11, 274–315.
  2. Beale, J. T., Kato, T., & Majda, A. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Communications in Mathematical Physics, 94(1), 61–66.
  3. Constantin, P., Fefferman, C., & Majda, A. J. (1996). Geometric constraints on potentially singular solutions for the 3-D Euler equations. Communications in Partial Differential Equations, 21(3–4), 559–571.
  4. Elgindi, T. M. (2021). Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$. Annals of Mathematics, 194(3), 647–727.
  5. Elgindi, T. M., Ghoul, T.-E., & Masmoudi, N. (2021). On the stability of self-similar blow-up for $C^{1,\alpha}$ solutions to the incompressible Euler equations. Cambridge Journal of Mathematics, 9(4), 1035–1075.
  6. Chen, J. & Hou, T. Y. (2023). Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ velocity and boundary. Communications in Mathematical Physics, 383, 4827–4890.
  7. Chen, J. & Hou, T. Y. (2025). Singularity formation in 3D Euler equations with smooth initial data and boundary. Proceedings of the National Academy of Sciences, 122(27). https://doi.org/10.1073/pnas.2500940122
  8. Córdoba, D., Martínez-Zoroa, L., & Zheng, F. (2025). Finite time singularities to the 3D incompressible Euler equations for solutions in $C^\infty(\mathbb{R}^3\setminus{0})\cap C^{1,\alpha}\cap L^2$. Annals of PDE. https://doi.org/10.1007/s40818-025-00214-2
  9. Isett, P. (2018). A proof of Onsager’s conjecture. Annals of Mathematics, 188(3), 871–963.
  10. Majda, A. J. & Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge University Press.
Tags:
Categories: