Navier-Stokes Subproblems

In 1934, Jean Leray had an idea. What if you relax the requirement that solutions be perfectly smooth? Drop that demand and, surprisingly, you can prove solutions always exist. Mathematicians call these relaxed solutions weak solutions.

Analogy: you can't find a perfect road between two cities, so you accept a dirt path with a few bumps instead. Leray showed the dirt path always exists. The Millennium Problem asks whether the perfect road does too, and after ninety years of effort nobody has managed to answer that question.

The catch? Uniqueness. We don't know if weak solutions are unique. Start with the same initial conditions and there might be several valid weak solutions, each satisfying the equations, and the equations may admit more than one admissible weak solution from the same initial data.

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. For contrast, Buckmaster-Vicol (2019) proved non-uniqueness of weak solutions in classes below the Leray-Hopf energy class (specifically in $C_t L^2_x$, without the $L^2_t \dot{H}^1_x$ control).
• Energy equality vs. inequality: weak solutions satisfy the inequality, but equality (as for smooth solutions) isn't guaranteed. Energy may be lost at singular times.
• Smoothness: if a Leray-Hopf solution is smooth, it's the unique classical solution. So regularity implies uniqueness.

We can't rule out singularities entirely. But 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 plain language: the singular set is extremely small in a parabolic measure-theoretic sense (one-dimensional parabolic Hausdorff measure zero). Singularities, if they exist, cannot form curves or surfaces in space-time, and they certainly cannot fill up any region.

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 $\frac{1}{r}\int_{Q_r} |\nabla u|^2$ 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 does it look like? Mathematicians classify potential blowups into two types:

• Type-I (self-similar): the blowup follows a specific rate, like a whirlpool that intensifies at a predictable pace. Better understood. Mostly ruled out already under various conditions.
• Type-II (non-self-similar): the blowup is faster or more irregular than the predicted rate, and it's far more mysterious and much harder to pin down with existing techniques.

Proving regularity means ruling out both types. That's why most modern approaches distinguish sharply between the Type-I and Type-II scenarios, even though some tools apply to both.

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

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

On the Type-I front, Seregin (2012) showed bounded $L^3$ norm at blowup time implies regularity, ruling out Type-I blowup in $L^3$. Gallagher-Koch-Planchon (2016) gave profile decompositions for Type-I scenarios, building on Nečas-Růžička-Šverák (1996) and Tsai (1998), who ruled out nontrivial backward self-similar solutions. Nečas-Růžička-Šverák (1996) established this, and Tsai (1998) extended it to locally self-similar settings under additional hypotheses.

Type-II remains open. It's where modern regularity programs focus.

There are specific measurements of a fluid solution that sit right at the boundary between controlled and uncontrolled behavior. Mathematicians call them critical norms. They're the dividing line.

Think of it like a tightrope. Several important critical norms have conditional regularity criteria: if they stay bounded, smoothness follows. The energy we can control sits below this tightrope, frustratingly out of reach, and the whole challenge is bridging that gap from the energy scale up to the critical threshold.

Which norms matter? The key ones measure velocity in $L^3$ (the cube of the speed, integrated over space) or related spaces. Recent work has confirmed something encouraging: if any of these critical quantities stays bounded, the solution stays smooth forever.

A norm $\|\cdot\|_X$ is critical if it's 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 concentrates. Some scale-critical part of the solution must concentrate in a way that prevents uniform control, and understanding exactly how that process works is the key to either ruling blowup out or proving it can happen.

Concentration-compactness gives mathematicians a way to study what happens when you zoom into a potential blowup point. Profile-decomposition arguments classify how a bad sequence could fail to stay compact: it may spread out, concentrate near one scale, or escape to larger distances. The goal is to rule out each possibility.

The strategy? Show every one of those scenarios leads to a contradiction, which forces regularity.

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" 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.

This article is part of Progress.

Each subproblem has its own arsenal. Explore the main approaches to Navier-Stokes regularity for the full picture, covering everything from harmonic analysis and energy methods to convex integration and concentration-compactness techniques that researchers are actively pushing forward today.

Why so stubborn? See Why It's Hard. The Clay formulation: The Millennium Problem. Turbulence: Reynolds Number and Turbulence.

This article is part of Progress.

For the analytical and geometric tools developed to address these subproblems, see Approaches. For the scaling and supercriticality obstacles, see Why It's Hard. For the Clay formulation, see The Millennium Problem.