Navier–Stokes Regularity: The Uniqueness of Weak Solutions
Table of Contents
The companion post on Navier–Stokes existence and smoothness asked whether smooth solutions can break down in finite time. This post asks the opposite question: when a solution is only weakly defined, satisfying the equations in an integral sense rather than pointwise, is it uniquely determined by its initial data? The answer, developed over the last two decades through a dramatic series of results, is a resounding no in many regimes. The frontier is now whether the physically natural class of Leray–Hopf weak solutions retains uniqueness.
Are Leray–Hopf weak solutions of the 3D incompressible Navier–Stokes equations $$\partial_t u + (u\cdot\nabla)u - \nu\Delta u + \nabla p = 0, \qquad \nabla\cdot u = 0$$ uniquely determined by their initial data $u_0 \in L^2(\mathbb{R}^3)$?
The question is one of the most urgent open problems in the PDE theory of fluid dynamics. It is logically independent of the blowup question: Leray–Hopf solutions exist globally for all time regardless of whether smooth solutions break down. What is not known is whether two Leray–Hopf solutions started from the same data must coincide.
Nash’s h-Principle: The Conceptual Ancestor #
The story begins not in fluid mechanics but in differential geometry. In 1954, John Nash proved that any Riemannian manifold admits a $C^1$ isometric embedding into Euclidean space, a result that contradicted the expectation, based on the rigid behaviour of $C^2$ embeddings (Cauchy), that the metric should impose strong constraints. The key insight is that $C^1$ embeddings are flexible: one can deform them by adding high-frequency oscillations that are invisible at the large scale but locally produce any prescribed metric tensor.
Gromov formulated this phenomenon as the h-principle: for certain underdetermined differential relations, the topological (homotopy-theoretic) obstructions are the only ones, and any formal solution can be deformed into an actual solution. The h-principle is a flexibility result: it says geometry is surprisingly unconstrained below a critical regularity threshold.
De Lellis and Székelyhidi recognised in the mid-2000s that the incompressible Euler equations are formally analogous to Nash’s embedding problem. The Euler system is underdetermined (more unknowns than equations), and one can attempt to construct wild solutions by adding high-frequency oscillations. The crucial observation is that the nonlinearity $u\otimes u$ in the Reynolds stress tensor plays the role of the metric tensor in Nash’s problem.
Wild Euler Solutions #
The first step was to show that the Euler equations possess infinitely many weak solutions for given initial data.
For any divergence-free $u _0 \in L^2(\mathbb{T}^3)$ and any prescribed energy profile $e(t) \in C^\infty([0,T])$ with $e(t) > |u _0| _{L^2}^2$ for all $t > 0$, there exist infinitely many weak solutions $u \in C_t^0 L_x^2$ of the 3D Euler equations with $u(\cdot,0) = u _0$ and $|u(\cdot,t)| _{L^2}^2 = e(t)$.
In particular, the Euler equations admit weak solutions that spontaneously gain or lose kinetic energy for no reason: wild solutions. The construction proceeds by convex integration: one builds the solution iteratively, at each stage adding a high-frequency perturbation (a Beltrami wave) that corrects the error in the momentum equation while staying nearly invisible in the velocity field.
Earlier, Scheffer (1993) and Shnirelman (1997) had shown the existence of weak Euler solutions with compact support in space-time: the fluid is at rest, then spontaneously moves, then returns to rest; but their constructions were indirect. De Lellis and Székelyhidi’s convex integration scheme gave the first systematic and quantitative approach.
Onsager’s Conjecture #
The De Lellis–Székelyhidi results raise an immediate question: at what regularity does the fluid behaviour transition from flexible (wild, non-unique) to rigid (energy-conserving, unique)? This is precisely what Lars Onsager conjectured in 1949.
For the 3D incompressible Euler equations, the threshold regularity for energy conservation is the Hölder exponent $1/3$:
- If $u \in C^{0,\alpha}$ with $\alpha > 1/3$, then every weak solution conserves kinetic energy.
- For every $\alpha < 1/3$, there exist weak solutions in $C^{0,\alpha}$ that dissipate energy.
The positive direction (conservation above $1/3$) was proved by Constantin–E–Titi (1994). The negative direction (dissipation possible below $1/3$) required much more work and was fully resolved only recently.
For every $\alpha < 1/3$ there exist weak solutions $u \in C^{0,\alpha}(\mathbb{T}^3\times[0,T])$ of the 3D Euler equations that fail to conserve kinetic energy.
Isett’s proof, published in the Annals of Mathematics in 2018, was the culmination of a decade of refinements of the De Lellis–Székelyhidi scheme. The key difficulty at regularity exactly $1/3$ is that the high-frequency perturbations must be sized to cancel the Reynolds stress error while staying in $C^{1/3-}$; this requires a delicate interplay of oscillation and concentration (intermittency). De Lellis, Székelyhidi, Buckmaster, and Vicol also obtained solutions attaining any prescribed energy profile in $C^{1/3-}$. Onsager’s conjecture is now a theorem.
Viscous Non-Uniqueness: Buckmaster–Vicol #
Adapting the convex integration scheme from Euler to Navier–Stokes requires overcoming the viscous term $\nu\Delta u$, which smooths out high-frequency oscillations. The intermittent Beltrami waves used by Isett concentrate energy at sparse spatial sets, reducing their interaction with the Laplacian. Buckmaster and Vicol exploited this idea to bring convex integration into the viscous setting.
There exist infinitely many weak solutions $u \in C_t^0 L_x^2(\mathbb{T}^3)$ of the 3D Navier–Stokes equations, belonging to the same regularity class as Leray–Hopf solutions, that do not satisfy the global energy inequality. In particular, weak solutions of 3D Navier–Stokes are not unique in the class $C_t^0 L_x^2$.
The Buckmaster–Vicol solutions, published in the Annals of Mathematics 189 (2019), 101–144, are weak in both the PDE sense and the energy sense: they satisfy the equations distributionally and have finite kinetic energy, but they can gain energy spontaneously, violating the natural dissipation law $\partial _t|u| _{L^2}^2 \leq -2\nu|\nabla u| _{L^2}^2$.
This non-uniqueness is striking but also limited: the Buckmaster–Vicol solutions are not Leray–Hopf solutions, because Leray–Hopf solutions are required to satisfy the energy inequality $|u(t)| _{L^2}^2 \leq |u _0| _{L^2}^2$. Whether this single additional constraint, that energy does not increase, suffices to restore uniqueness is the open question.
Crossing the Energy Barrier: Albritton–Brué–Colombo #
The energy inequality distinguishing Leray–Hopf solutions from Buckmaster–Vicol wild solutions seemed for a long time to be a genuine barrier to non-uniqueness. The following result crossed this barrier, but required introducing an external force.
There exists a body force $f \in L^1(0,T;, L^2(\mathbb{R}^3))$ and two distinct Leray–Hopf weak solutions of the forced 3D Navier–Stokes equations $\partial_t u + (u\cdot\nabla)u - \nu\Delta u + \nabla p = f$ with the same initial data $u_0 \equiv 0$ and the same force $f$.
Published in the Annals of Mathematics 196 (2022), 415–455, the proof uses a completely different mechanism from convex integration. The key ingredient is an unstable background solution: using Vishik’s construction of spectrally unstable steady states of the 2D Euler equations, Albritton–Brué–Colombo lift a 2D unstable vortex ring to an axisymmetric 3D solution and embed it into the Navier–Stokes flow via a self-similar change of variables. The force $f$ is chosen precisely to make this background exactly solve the forced equations; the instability then allows two different solutions to branch from the same initial data.
The force is singular; it belongs to $L^1_t L^2_x$ but is not smooth, and is concentrated near the initial time $t=0$. Whether the same non-uniqueness can be achieved with a smooth or zero force is the remaining open problem.
The Unforced Case: Current Frontier #
Non-uniqueness of Leray–Hopf solutions for the unforced Navier–Stokes equations remains open. The route to the unforced case requires finding a self-similar background profile that solves the unforced equations exactly and has an unstable eigenvalue, a far more demanding task than the forced case, where the profile can be any divergence-free function.
Do there exist two distinct Leray–Hopf solutions of the 3D Navier–Stokes equations with the same initial data and no external force?
Jia and Šverák (2013–2014) showed that non-uniqueness would follow from a spectral assumption: if there exists a forward self-similar Navier–Stokes solution whose linearised operator has an eigenvalue with positive real part, then Leray–Hopf solutions are non-unique. Guillod and Šverák (2017) provided compelling numerical evidence that such an unstable self-similar profile exists.
In September 2025, Giri and Kwon posted a preprint (arXiv:2509.25116) claiming a computer-assisted proof of the existence of an unstable self-similar profile for the unforced equations, which, via the Jia–Šverák mechanism, would establish non-uniqueness of Leray–Hopf solutions. The proof uses rigorous interval arithmetic to verify the existence of an unstable eigenvalue. As of this writing the preprint is under review by the community.
The Regularity Threshold #
The accumulated results suggest the following picture of the flexibility-rigidity dichotomy for the Euler and Navier–Stokes equations.
| Regularity class | Euler | Navier–Stokes |
|---|---|---|
| $C^{0,\alpha}$, $\alpha < 1/3$ | non-unique, dissipative (Isett 2018) | n/a |
| $C^{0,\alpha}$, $\alpha > 1/3$ | energy-conserving (Constantin–E–Titi 1994) | n/a |
| $L^2$ (global energy inequality) | non-unique | open (unforced); non-unique forced (ABC 2022) |
| $L^\infty_t L^3_x$ (LPS regularity) | n/a | unique and smooth (ESS 2003) |
The Leray–Hopf class sits precisely at the boundary where uniqueness is expected to break down but has not yet been proved to do so in the unforced case.
Research Directions #
1. Resolving the Jia–Šverák Spectral Condition #
The most direct path to unforced Leray–Hopf non-uniqueness is to rigorously confirm or refute the spectral condition of Jia–Šverák: find (or prove the nonexistence of) a forward self-similar Navier–Stokes profile with an unstable linearised eigenvalue. The 2025 Giri–Kwon computer-assisted preprint claims this is now done. If confirmed, the consequence is striking: Leray’s 1934 existence theorem cannot be supplemented by uniqueness, and the Navier–Stokes Cauchy problem is ill-posed in the Leray–Hopf class.
2. Selection Principles and Physical Solutions #
If Leray–Hopf solutions are indeed non-unique, a fundamental question becomes which solution is the physically correct one, the one observed in experiments and computed in simulations. Several selection criteria have been proposed: the vanishing viscosity limit of the Navier–Stokes solution as $\nu\to 0$ from above, entropy conditions analogous to those for hyperbolic conservation laws, and renormalisation group or statistical ensemble approaches motivated by turbulence theory. None of these has been rigorously validated as a selection criterion that distinguishes a unique Leray–Hopf solution from the others.
3. Sharp Regularity Thresholds for Navier–Stokes #
For Euler, Onsager’s conjecture identifies $C^{1/3}$ as the sharp regularity threshold for energy conservation. What is the analogous threshold for Navier–Stokes? The Buckmaster–Vicol solutions are in $C_t^0 L_x^2$ (very rough), while the Ladyzhenskaya–Prodi–Serrin class gives uniqueness. The precise exponent at which uniqueness breaks down, if it does, is not known. Determining the sharp Sobolev or Hölder regularity threshold for Navier–Stokes uniqueness, analogous to Onsager’s $1/3$, is a central open problem.
4. Uniqueness for Axisymmetric Initial Data #
A natural restricted problem is whether Leray–Hopf solutions with axisymmetric, swirl-free initial data are unique. Such data imposes a strong geometric constraint that eliminates most of the degrees of freedom available to convex integration. Partial results are known (e.g., global regularity for axisymmetric data without swirl is not proved but no counterexamples exist), but uniqueness in this class has not been established. If the Giri–Kwon instability is confirmed, understanding whether the instability mechanism survives axisymmetric perturbations is an immediate question.
5. Stochastic Regularisation #
There is a well-studied phenomenon, regularisation by noise, in which adding a stochastic forcing term to an ill-posed deterministic PDE restores well-posedness. For the Navier–Stokes equations, Hofmanová–Zhu–Zhu (2023) showed non-uniqueness persists even under multiplicative noise for certain body forces, by adapting the Albritton–Brué–Colombo construction. Whether a generic stochastic perturbation can restore uniqueness of Leray–Hopf solutions, and what the appropriate notion of “generic” should be, is a rich open direction combining convex integration with stochastic analysis.
References #
- Nash, J. (1954). $C^1$ isometric imbeddings. Annals of Mathematics, 60(3), 383–396.
- De Lellis, C. & Székelyhidi, L. (2009). The Euler equations as a differential inclusion. Annals of Mathematics, 170(3), 1417–1436.
- De Lellis, C. & Székelyhidi, L. (2013). Dissipative continuous Euler flows. Inventiones Mathematicae, 193(2), 377–407.
- Constantin, P., E, W., & Titi, E. S. (1994). Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Communications in Mathematical Physics, 165(1), 207–209.
- Isett, P. (2018). A proof of Onsager’s conjecture. Annals of Mathematics, 188(3), 871–963.
- Buckmaster, T. & Vicol, V. (2019). Nonuniqueness of weak solutions to the Navier–Stokes equation. Annals of Mathematics, 189(1), 101–144.
- Buckmaster, T. & Vicol, V. (2019). Convex integration and phenomenologies in turbulence. EMS Surveys in Mathematical Sciences, 6(1–2), 1–88.
- Albritton, D., Brué, E., & Colombo, M. (2022). Non-uniqueness of Leray solutions of the forced Navier–Stokes equations. Annals of Mathematics, 196(1), 415–455.
- Jia, H. & Šverák, V. (2014). Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions. Inventiones Mathematicae, 196(1), 233–265.
- Giri, V. & Kwon, H. (2025). Nonuniqueness of Leray–Hopf solutions to the unforced incompressible 3D Navier–Stokes equation. arXiv:2509.25116.