Approaches to the Navier-Stokes Problem

The oldest approach starts with energy. A moving fluid carries kinetic energy, and viscosity eats it, like friction grinding things to a halt. Total energy can only decrease over time, assuming nothing's pumping energy in from outside.

Leray saw this in 1934 and made a key move: use the energy bound to prove that a global weak solution with finite kinetic energy has to exist. Build approximate solutions, artificially smoothed. Show they all obey the energy bound. Take a limit. Something must survive in that limit, and it does.

But here's the catch. Energy bounds are blunt instruments. They guarantee the fluid has finite total energy, sure, but they can't tell you the velocity stays finite at every single point in space and time. That gap between "finite energy" and "smooth everywhere" is exactly the regularity problem, and it's been open for ninety years.

Paper links: Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace (1934); Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen (1951).

The Leray-Hopf construction proceeds in two main variants: Galerkin approximation (projecting onto finite-dimensional subspaces) or Friedrichs mollification (smoothing the nonlinearity via convolution). Both share the same four-step skeleton. The standard strategy has four steps:

1. Approximate: Solve the mollified system $\partial_t u_\varepsilon + (J_\varepsilon u_\varepsilon \cdot \nabla) u_\varepsilon = \nu \Delta u_\varepsilon - \nabla p_\varepsilon$ on finite-dimensional subspaces.
2. Energy bound: The a priori estimate $\|u_\varepsilon(t)\|_{L^2}^2 + 2\nu \int_0^t \|\nabla u_\varepsilon\|_{L^2}^2 \leq \|u_0\|_{L^2}^2$ holds uniformly in $\varepsilon$.
3. Compactness: Extract a weakly convergent subsequence $u_\varepsilon \rightharpoonup u$ in $L^2_t \dot{H}^1_x$ using the Aubin-Lions lemma.
4. Pass to limit: The nonlinear term converges by strong $L^2_{\text{loc}}$ convergence of $u_\varepsilon$.

The resulting weak solution satisfies the energy inequality, not equality. Energy can be lost at irregular times. And that's the whole problem: the gap between the energy class $L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$ and actual smoothness is precisely what we can't close.

Paper links: Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace (1934); Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen (1951).

The Caffarelli-Kohn-Nirenberg approach (1982) doesn't try to prove full smoothness. It asks something else entirely: how bad can the singularities actually be?

Barely bad at all. Their $\varepsilon$-regularity theorem says that if certain scale-invariant local quantities are small enough in a small space-time region, the solution is automatically smooth there. And since total energy is finite, there simply isn't enough "budget" for many singular points to coexist.

Think of it this way. A wall might have cracks. But the total length of all those cracks combined is zero, meaning the singular set is extremely small in the parabolic measure-theoretic sense (one-dimensional parabolic Hausdorff measure zero).

Paper links: Caffarelli-Kohn-Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations (1982); Albritton-Barker-Prange, Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness.

The CKN $\varepsilon$-regularity theorem: for suitable weak solutions, smallness of certain scale-invariant local quantities (involving both $u$ and $p$ on a parabolic cylinder $Q_r(z_0) = B_r(x_0) \times (t_0 - r^2, t_0)$) implies regularity (Hölder continuity) near the point $z_0 = (x_0, t_0)$.

The proof combines the local energy inequality with a Campanato-type iteration: if the scale-invariant energy is small, a bootstrap argument shows $u$ is bounded, then Hölder, then smooth by classical Schauder theory.

The dimensional estimate $\mathcal{P}^1(\Sigma) = 0$ follows by a Vitali covering: if $\Sigma$ had positive $\mathcal{P}^1$ measure, infinitely many disjoint parabolic cylinders would each carry $\varepsilon_{\text{CKN}}$ energy, contradicting finite total energy.

Paper links: Caffarelli-Kohn-Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations (1982); Albritton-Barker-Prange, Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness.

Here's a sharp reduction of the whole problem. Beale, Kato, and Majda proved in 1984 that for the 3D Euler equations, blowup can only happen if vorticity control is lost. Analogous criteria were later established for Navier-Stokes. That's it. One condition.

Vorticity measures local spin. The BKM criterion says: keep the maximum spin bounded in the right norm, and the solution stays smooth. Everything else falls in line automatically.

One family of quantities to control. Unfortunately, actually controlling them has turned out to be exactly as hard as the original problem. The reduction is clean. The execution remains out of reach.

Paper links: Beale-Kato-Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations (1984); Kozono-Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations (2000).

The original Beale-Kato-Majda theorem (1984) is for the 3D Euler equations. For Navier-Stokes, analogous continuation criteria imply that a smooth solution $u$ on $[0, T^*)$ extends beyond $T^*$ whenever

$$\int_0^{T^*} \|\omega(\cdot, t)\|_{L^\infty} \, dt < \infty,$$

where $\omega = \nabla \times u$ is the vorticity. Refinements include:

• Kozono-Taniuchi (2000): $\|\omega\|_{L^\infty}$ can be replaced by $\|\omega\|_{\mathrm{BMO}}$
• Besov-space variants: critical or borderline Besov control can also serve as a continuation criterion
• Direction-restricted criteria: Serrin-type conditions on components of $\nabla u$ can also serve as continuation criteria (see e.g. Beirão da Veiga, 1995)

These criteria connect to the vortex-stretching picture: any finite-time singularity must force the vorticity to accumulate too quickly for the time integral above to stay finite.

Paper links: Beale-Kato-Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations (1984); Kozono-Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations (2000); Chemin-Planchon, Self-improving bounds for the Navier-Stokes equations (2012).

A more modern angle works with function spaces (like $L^3$ or $\dot{H}^{1/2}$) that sit right at the boundary of what the scaling symmetry allows. These are critical spaces, and they are where the sharpest regularity results live.

The logic is clean: if you can show a solution stays within certain critical-space bounds, smoothness follows automatically. Multiple teams have proved this, building a whole menu of regularity criteria (conditions that guarantee smoothness if you can verify them).

The problem is the gap. We can prove subcritical bounds from energy methods. We need critical bounds. That gap is narrow, sometimes a single derivative of regularity, but it has resisted every attempt to close it.

Paper links: Koch-Tataru, Well-posedness for the Navier-Stokes equations (2001); Kenig-Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces; Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion. For a detailed comparison of why energy criticality succeeds in 2D but fails in 3D, see Why 2D Is Easier Than 3D.

Major programs in critical-space regularity:

• Koch-Tataru (2001): Global well-posedness for small data in $\text{BMO}^{-1}$, a critical space where the bilinear estimate $\|\mathbb{P}\nabla \cdot (u \otimes v)\|_{\text{BMO}^{-1}} \lesssim \|u\|_{\text{BMO}^{-1}} \|v\|_{\text{BMO}^{-1}}$ holds. This is among the sharpest known results for perturbative methods.
• Gallagher-Koch-Planchon (2013): Profile decomposition approach to the $L^\infty_t L^3_x$ Navier-Stokes regularity criterion. Any sequence of solutions with bounded critical norm has a subsequence decomposing into asymptotically decoupled profiles.

The core obstruction: no known coercive functional is both controlled by the evolution and critical with respect to the Navier-Stokes scaling.

Paper links: Koch-Tataru, Well-posedness for the Navier-Stokes equations (2001); Kenig-Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces; Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion.

Modern PDE theory borrows heavily from harmonic analysis. The core idea: break a function into waves at different frequencies, the way you'd split a musical chord into individual notes. Except here, the "notes" are spatial oscillations of fluid velocity at wildly different scales.

Littlewood-Paley decomposition does exactly this. Chop the velocity field into scale-by-scale components. Track how energy flows between them. Suddenly the informal physical intuition of "energy cascade" becomes something you can actually prove theorems about, and the theorems are precise. These methods have produced many of the sharpest results on regularity criteria and blowup rates.

Paper links: Cannone-Meyer, Littlewood-Paley decomposition and Navier-Stokes equations (1995); Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion.

Littlewood-Paley theory decomposes $u = \sum_j \Delta_j u$ where $\Delta_j$ localizes to frequencies $|\xi| \sim 2^j$. Applied to the transport nonlinearity $(u \cdot \nabla)u$:

The paraproduct decomposition of the nonlinearity $(u \cdot \nabla)u$ splits into low-high, high-low, and high-high frequency interactions:

$$(u \cdot \nabla)u = T_u \nabla u + T_{\nabla u} u + R(u, \nabla u)$$

where $T$ is the paraproduct and $R$ the remainder. Each piece has different regularity properties in Besov spaces $\dot{B}^s_{p,q}$.

Key results using this machinery:

• Chemin-Lerner spaces: $\widetilde{L}^\rho_T \dot{B}^s_{p,q}$ provide the natural framework for critical well-posedness: the Navier-Stokes bilinear form maps $\widetilde{L}^\infty_T \dot{B}^{-1+3/p}_{p,q} \times \widetilde{L}^1_T \dot{B}^{1+3/p}_{p,q} \to \widetilde{L}^1_T \dot{B}^{-1+3/p}_{p,q}$.
• Cannone-Meyer: Littlewood-Paley methods give a clean wavelet/Besov formulation of the small-data theory.

Paper links: Cannone-Meyer, Littlewood-Paley decomposition and Navier-Stokes equations (1995); Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion.

Here's a different instinct entirely. Instead of tracking numbers (norms, energies), these methods study the shape of the solution: how vortex tubes bend, how regions of intense rotation arrange themselves in space.

The key insight is that blowup isn't just about something getting big. It's about the fluid organizing itself into a very specific geometric configuration. If you can show that configuration is impossible (because it contradicts the energy-dissipation structure, or incompressibility, or both), you've ruled out blowup without ever computing a norm.

This geometric viewpoint has grown into a viewpoint that has inspired several rigorous regularity criteria alongside purely analytic methods. And it feels different. It asks what shape does disaster take? instead of how big can this number get?

Paper links: Constantin-Fefferman, Geometric constraints on potentially singular solutions for the 3-D Euler equations (1993); Albritton-Barker-Prange, Localized smoothing and concentration for the Navier-Stokes equations in the half space.

Geometric-topological approaches exploit structural constraints that are invisible to purely analytical methods:

• Vortex line geometry: Constantin and Fefferman (1993) showed that if the vorticity direction field $\hat{\omega} = \omega/|\omega|$ is Lipschitz in regions of high vorticity, the solution is regular. Blowup requires the vorticity direction to develop a singularity simultaneously with the magnitude.
• Incompatibility arguments: if a blowup configuration is geometrically constrained (e.g., via packing bounds on the number of independent concentration regions that fit within energy and dissipation budgets), one can derive a contradiction without directly estimating critical norms.
• Case partition (speculative/programmatic): a proposed strategy would classify each spatial region as belonging to one of finitely many scenarios (e.g., locally regular, Type-I-like, Type-II-like, densely packed) and attempt to show each scenario either gives regularity or transfers the problem to a bounded counting argument. This remains a research program rather than an established result.

The proof page on this site explores arguments of this flavor; nothing here is presented as a completed formal proof of the Millennium problem.

Paper links: Constantin-Fefferman, Geometric constraints on potentially singular solutions for the 3-D Euler equations (1993); Albritton-Barker-Prange, Localized smoothing and concentration for the Navier-Stokes equations in the half space.

This one caught people off guard. The weak solutions from Leray's method (Section 1) turn out to be non-unique, at least when external forcing is present.

The weapon is convex integration, a technique originally built for geometry problems and adapted to fluid equations by De Lellis and Székelyhidi starting around 2009. The idea: construct "wild" solutions by iteratively piling on high-frequency corrections that collectively satisfy the equation but behave erratically.

For 3D Euler (Navier-Stokes without viscosity), Buckmaster and Vicol (2019) proved weak solutions aren't unique. Then in 2022, Albritton, Brué, and Colombo proved that even Leray-Hopf solutions of 3D Navier-Stokes are non-unique when external force is present. Whether non-uniqueness persists for the unforced Navier-Stokes equations remains open.

Why does this matter? Because "a weak solution exists" has been the headline result since 1934. Now we know it doesn't pin down a single answer. The question sharpens: which solution, if any, is the physically correct one?

Paper links: De Lellis-Székelyhidi, Dissipative continuous Euler flows (2013); Buckmaster-Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation (2019); Albritton-Brué-Colombo, Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2022).

Convex integration for fluid equations originates in the De Lellis-Székelyhidi program (2009–2013), adapting the Nash-Kuiper $C^1$ isometric embedding technique to construct weak solutions of the Euler equations that dissipate energy. Key stages:

• De Lellis-Székelyhidi (2013): existence of continuous ($C^0$) dissipative Euler flows on $\mathbb{T}^3$ (Hölder regularity below $1/5$ achieved in the subsequent Buckmaster-De Lellis-Isett-Székelyhidi 2015 work; later improved to $< 1/3$ by Isett, 2018, resolving the flexible side of the Onsager conjecture).
• Buckmaster-Vicol (2019): non-uniqueness of weak solutions to 3D Navier-Stokes in the class $C_t L^2_x \cap C_t W^{1,1+}_x$. The construction uses intermittent Beltrami flows as building blocks, adding oscillatory corrections at each iteration step while maintaining control of the Reynolds stress. This is below the Leray-Hopf energy class, so it doesn't directly contradict Leray uniqueness.
• Albritton-Brué-Colombo (2022): non-uniqueness of Leray-Hopf solutions for the forced 3D Navier-Stokes equations. The proof constructs a background unstable self-similar solution and uses an instability mechanism to branch into distinct Leray-Hopf solutions from the same initial data. This shows that the energy inequality alone doesn't select a unique solution when forcing is present.

The central open question is whether non-uniqueness persists for the unforced Navier-Stokes equations in the Leray-Hopf class. The forced result shows that the energy inequality isn't a sufficient selection principle, but it doesn't resolve whether the unforced equation has additional structure that restores uniqueness.

Paper links: De Lellis-Székelyhidi, Dissipative continuous Euler flows (2013); Buckmaster-Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation (2019); Albritton-Brué-Colombo, Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2022).

Can we at least rule out certain proof strategies? Terence Tao showed in 2016 that yes, we can. And the result is sobering.

Tao built a modified version of the Navier-Stokes equations, an "averaged" system, that keeps many key structural features of the real equations: the energy identity, the way enstrophy (a measure of vorticity intensity) grows, the scaling symmetry. But in this modified system, solutions blow up in finite time.

The implication rules out broad families of proof strategies. Any proof that global smoothness holds for the real equations must use some specific structural property of the true nonlinearity that the averaged system doesn't have. You can't prove regularity using only energy bounds, scaling, and enstrophy growth. Those tools alone are consistent with blowup.

This doesn't say the real equations blow up. It says entire families of proof strategies are dead ends. The eventual proof (if regularity holds) must be sharper than a generic energy argument. Much sharper.

Paper links: Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2016).

Tao (2016) considers a system of the form

$$\partial_t u + \tilde{B}(u, u) = \nu \Delta u - \nabla p, \quad \nabla \cdot u = 0,$$

where $\tilde{B}$ is a bilinear operator that agrees with the true Navier-Stokes nonlinearity $(u \cdot \nabla)u$ in the following senses:

• it is a Fourier multiplier of order 1, preserving the scaling $u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t)$
• it satisfies the same energy identity: $\langle \tilde{B}(u,u), u \rangle = 0$
• it reproduces the enstrophy growth structure

For this averaged system, Tao constructs smooth initial data whose solution blows up in finite time. The blowup mechanism programs a sequence of sharper and sharper enstrophy concentrations at smaller and smaller scales, with each stage doubling the enstrophy in a controlled cascade.

The obstruction this creates: any supercritical quantity that is (a) controlled by the evolution in the energy class and (b) invariant under the Navier-Stokes scaling can't by itself rule out blowup, because it would also be controlled in the averaged system, which does blow up. A regularity proof must exploit the specific algebraic cancellation structure of the true transport term $(u \cdot \nabla)u$ that the averaged operator $\tilde{B}$ doesn't share.

Paper links: Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2016).

This article is part of Progress.

From Leray's 1934 existence proof through convex integration and Tao's proof barriers, these are the main strategies people have thrown at the 3D Navier-Stokes problem. None has resolved the full 3D regularity problem. For context on how viscosity shapes the mathematics compared to the inviscid Euler equations, see Euler vs. Navier-Stokes. For the current status, see Is the Navier-Stokes Problem Solved? For the exact formal statement, return to The Millennium Problem.

This article is part of Progress.

The approaches above represent the major rigorous threads in the regularity and uniqueness literature through 2022. No combination has resolved the full 3D problem. The field keeps moving. Computer-assisted proof methods are an active area covered separately on this site.

For the comparison between viscous and inviscid systems (and why viscosity helps but not enough), see Euler vs. Navier-Stokes. For the subproblems these approaches target, see Subproblems. For the scaling obstacles, see Why It's Hard.