Subproblems

In 1934, Jean Leray made a crucial discovery: if you relax the requirement that solutions be perfectly smooth, you can prove that solutions always exist. These relaxed solutions are called weak solutions.

Think of it like this: if you can't find a perfect road between two cities, you might accept a dirt path with a few bumps. Leray showed the dirt path always exists. The Millennium Problem asks whether the perfect road does too.

The catch? We don't even know if weak solutions are unique. Given the same starting conditions, there might be multiple valid weak solutions — and we don't know which one the fluid "chooses."

Leray (1934) proved existence of global weak solutions $u \in L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$ satisfying the energy inequality

$$\frac{1}{2}\|u(t)\|_{L^2}^2 + \nu \int_s^t \|\nabla u(\tau)\|_{L^2}^2 \, d\tau \leq \frac{1}{2}\|u(s)\|_{L^2}^2$$

for a.e. $s \geq 0$ and all $t \geq s$. These are now called Leray-Hopf weak solutions. Key open questions:

• Uniqueness in the energy class is unknown (contrast: Buckmaster-Vicol (2019) proved non-uniqueness for weak solutions below the Lions exponent, i.e., in $L^p_t L^\infty_x$ for $p < 2$).
• Energy equality vs. inequality: weak solutions satisfy the inequality, but equality (as for smooth solutions) is not guaranteed — possible energy loss at singular times.
• Smoothness: if a Leray-Hopf solution is smooth, it is the unique classical solution. So regularity implies uniqueness.

Even though we can't rule out singularities entirely, we know they can't be too bad. The landmark result of Caffarelli, Kohn, and Nirenberg (1982) — the CKN theorem — proves that the set of points where a solution might blow up is incredibly small.

How small? In space-time, the set of possible singularities has "one-dimensional parabolic Hausdorff measure zero." In practical terms: singularities, if they exist, are isolated points that flash in and out of existence. They can't persist, they can't form lines or surfaces, and they certainly can't fill up any region of space.

This is remarkable: even without proving full smoothness, we know singularities are exceedingly rare.

The Caffarelli-Kohn-Nirenberg theorem (1982): for any suitable weak solution $(u,p)$ of the Navier-Stokes equations, the singular set $\Sigma$ satisfies

$$\mathcal{P}^1(\Sigma) = 0$$

where $\mathcal{P}^1$ is the one-dimensional parabolic Hausdorff measure. Equivalently, singularities cannot concentrate on curves in space-time.

The proof introduces suitable weak solutions satisfying the local energy inequality

$$\int |u|^2 \varphi(t) + 2\nu \int\!\!\int |\nabla u|^2 \varphi \leq \int\!\!\int |u|^2(\partial_t \varphi + \nu \Delta \varphi) + \int\!\!\int (|u|^2 + 2p)(u \cdot \nabla \varphi)$$

and uses an $\varepsilon$-regularity criterion: if the scale-invariant quantity $\frac{1}{r^2}\int_{Q_r} |\nabla u|^2$ is sufficiently small on a parabolic cylinder $Q_r$, then $u$ is regular at the center. The CKN bound then follows from a covering argument.

If a singularity exists, what might it look like? Mathematicians have classified potential blowups into two types:

• Type-I (self-similar): the blowup follows a specific rate — like a whirlpool that intensifies at a predictable pace. These are better understood and have been mostly ruled out under various conditions.
• Type-II (non-self-similar): the blowup is faster or more irregular than the predicted rate. These are far more mysterious and harder to analyze.

Proving regularity means ruling out both types. Most modern approaches treat them as separate problems, using different tools for each.

Suppose $T^* < \infty$ is a hypothetical first blowup time. The blowup is:

• Type-I: $\|u(t)\|_{L^\infty} \leq \frac{C}{\sqrt{T^* - t}}$ as $t \to T^*$. Equivalently, the rescaled solution $\lambda u(x_0 + \lambda x, T^* + \lambda^2 t)$ remains bounded. Type-I blowup is connected to self-similar solutions of the form $u(x,t) = \frac{1}{\sqrt{T^*-t}} U\left(\frac{x-x_0}{\sqrt{T^*-t}}\right)$.
• Type-II: $\limsup_{t \to T^*} \sqrt{T^* - t} \, \|u(t)\|_{L^\infty} = \infty$. The blowup rate exceeds the self-similar rate.

Significant Type-I results:

• Seregin (2012): bounded $L^3$ norm at blowup time implies regularity — ruling out Type-I blowup in $L^3$.
• Koch-Tataru-Gallagher (2016): Type-I blowup with the self-similar profile leads to contradiction via backward uniqueness and unique continuation.

Type-II blowup remains the primary open scenario and is the focus of modern regularity programs.

Mathematicians have identified specific measurements of a fluid solution that sit at the exact "boundary" between controlled and uncontrolled behavior. These are called critical norms.

Think of it like a tightrope: if you can show a critical norm stays bounded, the solution is smooth. If it blows up, the solution breaks down. The energy (which we can control) is too weak — it's below the tightrope. We need to reach the tightrope from below.

The key critical norms involve measuring the velocity in $L^3$ (the cube of the speed, integrated over space) or related spaces. Recent work has shown that if any of these critical quantities remains bounded, the solution stays smooth.

A norm $\|\cdot\|_X$ is critical if it is invariant under the Navier-Stokes scaling: $\|u_\lambda\|_X = \|u\|_X$. The main critical regularity criteria:

• Escauriaza–Seregin–Šverák (2003): $u \in L^\infty_t L^3_x$ near blowup $\Rightarrow$ regularity
• Ladyzhenskaya–Prodi–Serrin: $u \in L^p_t L^q_x$, $\frac{2}{p} + \frac{3}{q} = 1$, $q > 3$ $\Rightarrow$ regularity
• Beale–Kato–Majda: $\int_0^{T^*} \|\omega(t)\|_{L^\infty} dt < \infty$ $\Rightarrow$ regularity

The gap: the energy estimate gives $u \in L^{10/3}_{t,x}$ (by Sobolev embedding), but the critical Serrin condition requires $u \in L^5_{t,x}$. This gap from $10/3$ to $5$ is the heart of the supercriticality problem.

If a blowup happens, where does the energy go? It must concentrate — focusing into a smaller and smaller region of space. Understanding this concentration is key to either ruling it out or constructing it.

The tools of concentration-compactness let mathematicians study what happens in the limit as you zoom into a potential blowup point. Either the solution scatters (disperses to infinity), concentrates at a point (forming a "minimal blowup solution"), or escapes to spatial infinity.

Modern approaches try to show that each scenario leads to a contradiction — leaving regularity as the only option.

The concentration-compactness/profile decomposition approach (Kenig-Merle, 2006; adapted to Navier-Stokes by Gallagher-Koch-Planchon, Kenig-Koch, and others) proceeds as follows:

1. Critical element: If global regularity fails, there exists a "minimal blowup solution" — one with the smallest possible critical norm that still blows up.
2. Compactness: This minimal solution has a compactness property: modulo symmetries, its orbit $\{u(\cdot, t)\}_{t \in [0,T^*)}$ is precompact in the critical space.
3. Rigidity: Show that any compact-orbit solution must be zero (or globally regular), contradicting the assumption of blowup.

This program has been completed for energy-critical dispersive equations (NLS, NLW) but faces severe obstacles for Navier-Stokes due to the lack of a conserved critical quantity and the pressure nonlocality.

Mathematicians have developed powerful tools to attack each of these subproblems. Explore the main approaches to Navier-Stokes regularity, or learn about our proof.

For the analytical and geometric tools developed to address these subproblems, see Approaches. For an approach combining case partition with packing arguments, see Our Proof.