Approaches

The most fundamental approach uses energy. A moving fluid has kinetic energy, and viscosity dissipates it — like friction slowing things down. The total energy can only decrease over time (assuming no external forcing).

This was Leray's key insight in 1934: use the energy bound to prove that some kind of solution must exist. His method constructs approximate solutions (with artificial smoothing), proves they all satisfy the energy bound, and then takes a limit.

The limitation: the energy bound is too coarse to guarantee smoothness. It tells you the fluid has finite total energy, but not that the velocity stays finite everywhere.

Paper links: The foundational Leray and Hopf papers predate arXiv, so there is no original arXiv link for this approach.

The Leray-Hopf construction proceeds via Galerkin approximation or mollification. Key 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 may be lost at irregular times). The gap between the energy class $L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$ and smoothness is precisely the regularity problem.

Paper links: The foundational Leray and Hopf papers predate arXiv, so there is no original arXiv link for this approach.

The Caffarelli-Kohn-Nirenberg approach (1982) doesn't try to prove full smoothness. Instead, it asks: how bad can the singularities be?

The answer: remarkably tame. Their $\varepsilon$-regularity theorem says that if the fluid's energy is sufficiently small in a small space-time region, then the solution is smooth there. Since the total energy is finite, there simply aren't enough "budget" for many singular points.

This is like proving that a wall might have cracks, but the total length of all cracks combined is zero — they can only be isolated points.

Paper links: The original Caffarelli-Kohn-Nirenberg paper predates arXiv. A modern arXiv-accessible reference in this line is Albritton-Barker-Prange, Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness.

The CKN $\varepsilon$-regularity theorem: there exists $\varepsilon_{\text{CKN}} > 0$ such that if $(u,p)$ is a suitable weak solution and

$$\frac{1}{r} \int_{Q_r(z_0)} |\nabla u|^2 \, dx \, dt < \varepsilon_{\text{CKN}}$$

then $u$ is regular (Hölder continuous) at $z_0 = (x_0, t_0)$. Here $Q_r(z_0) = B_r(x_0) \times (t_0 - r^2, t_0)$ is a parabolic cylinder.

The proof combines the local energy inequality with a Campanato-type iteration: if the scale-invariant energy is small, a boot-strap 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: The original Caffarelli-Kohn-Nirenberg paper predates arXiv. A modern arXiv-accessible reference in this line is Albritton-Barker-Prange, Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness.

In 1984, Beale, Kato, and Majda proved a clean criterion: the only way a solution can blow up is if the vorticity becomes infinite.

Vorticity measures how much the fluid is spinning locally. The BKM theorem says: if you can keep track of the maximum spin everywhere and show it doesn't become infinite, you've proved smoothness. All other quantities (velocity, pressure, etc.) automatically stay controlled.

This reduced the problem to a single quantity. Unfortunately, controlling that quantity has proved just as hard as the original problem.

Paper links: The foundational references cited in this section are classical journal papers rather than arXiv preprints, so there is no single original arXiv link to attach here.

The Beale-Kato-Majda criterion (1984): a smooth solution $u$ on $[0, T^*)$ extends beyond $T^*$ if and only if

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

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

• Kozono-Taniuchi (2000): $\|\omega\|_{L^\infty}$ can be replaced by $\|\omega\|_{\text{BMO}}$
• Planchon (2003): $\|\omega\|_{\dot{B}^0_{\infty,\infty}}$ suffices (Besov space criterion)
• Direction-restricted: Da Veiga (1995) showed $\|\nabla u\|_{L^p_t L^q_x}$ with $2/p + 3/q = 2$, $q > 3/2$ suffices — only the symmetric gradient, not full vorticity, is needed

The BKM criterion connects to the vortex stretching picture: blowup requires $\int_0^{T^*} \|\omega\|_{L^\infty} = \infty$, meaning vorticity must grow faster than $(T^* - t)^{-1}$.

Paper links: The foundational references cited in this section are classical journal papers rather than arXiv preprints, so there is no single original arXiv link to attach here.

A modern approach works with special mathematical spaces that sit right at the boundary of what the scaling symmetry allows. These are called critical spaces.

The idea: if you can show that a solution stays within certain critical-space bounds, smoothness follows automatically. Several teams have established this, creating a menu of "regularity criteria" — conditions that, if verified, guarantee smoothness.

The challenge remains going from what we can prove (subcritical bounds from energy) to what we need (critical bounds). This gap is narrow but has resisted all attempts to bridge it.

Paper links: Kenig-Koch, An alternative approach to 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.

Major programs in critical-space regularity:

• Koch-Tataru (2001): Global well-posedness for small data in $\text{BMO}^{-1}$, the largest 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 essentially optimal for perturbative methods.
• Gallagher-Koch-Planchon (2013): Profile decomposition for Navier-Stokes in $\dot{H}^{1/2}$. Any sequence of solutions with bounded critical norm has a subsequence decomposing into asymptotically decoupled profiles.
• Albritton-Barker (2018): Non-uniqueness of Leray-Hopf solutions would follow from the existence of certain unstable self-similar solutions — connecting regularity to the structure of the self-similar solution set.

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

Paper links: Kenig-Koch, An alternative approach to 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; Albritton-Barker, Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary.

Modern PDE theory uses tools from harmonic analysis — the mathematics of breaking functions into waves at different frequencies (like a musical chord into individual notes).

By decomposing the fluid velocity into components at different spatial scales and tracking how energy moves between scales, mathematicians can make the vague intuition of "energy cascade" precise. These techniques, called Littlewood-Paley decomposition, have produced the sharpest known results about regularity criteria and blowup rates.

Paper links: Deng-Yao, Ill-posedness of the incompressible Navier-Stokes equations in $\dot{F}^{-1,q}_{\infty}(\mathbb{R}^3)$; 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 $|\xi| \sim 2^j$. For Navier-Stokes:

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-Planchon: Global mild solutions for small data in $\dot{B}^{-1+3/p}_{p,\infty}$ for $p < \infty$.

Paper links: Deng-Yao, Ill-posedness of the incompressible Navier-Stokes equations in $\dot{F}^{-1,q}_{\infty}(\mathbb{R}^3)$; Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion.

A less traditional but increasingly powerful approach uses the geometry of the flow. Instead of tracking numbers (norms, energies), these methods study the shape of the solution — how vortex tubes bend, how regions of intense rotation are arranged in space.

The insight is that blowup isn't just about something getting big — it's about the fluid organizing itself in a very specific geometric configuration. If you can show that configuration is impossible (because it leads to a contradiction with, say, energy conservation or incompressibility), you've ruled out blowup.

This is the approach behind the proof presented on this site.

Paper links: A representative modern arXiv paper on localization and concentration near potential singularities is Albritton-Barker-Prange, Localized smoothing and concentration for the Navier-Stokes equations in the half space. The original Constantin vortex-line paper predates arXiv.

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

• Vortex line geometry: Constantin (1994) 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: by classifying each spatial region as belonging to one of finitely many scenarios (e.g., locally regular, Type-I-like, Type-II-like, densely packed) and showing each scenario either gives regularity or transfers the problem to a bounded counting argument, the blowup can be ruled out combinatorially.

This site's proof uses such an approach: a geometric-topological incompatibility argument formalized in Lean 4.

Paper links: A representative modern arXiv paper on localization and concentration near potential singularities is Albritton-Barker-Prange, Localized smoothing and concentration for the Navier-Stokes equations in the half space. The original Constantin vortex-line paper predates arXiv.

Each of these approaches has made progress, but none has solved the problem completely. See our proof for a new geometric approach that combines several of these techniques.

Or go back to the basics: what exactly is the Millennium Problem?

For a geometric-topological approach that combines CKN ε-regularity, BKM blowup criteria, and packing arguments into a unified contradiction, see Our Proof.