Partial regularity (Caffarelli-Kohn-Nirenberg) and criticality: 2D versus 3D

Scaling sorts every quantity into three classes. Under \(\mathbf{u}_\lambda(\mathbf{x},t) = \lambda\mathbf{u}(\lambda\mathbf{x},\lambda^2t)\) in \(d\) space dimensions, \(\|\mathbf{u}_\lambda\|_{L^q} = \lambda^{1-d/q}\|\mathbf{u}\|_{L^q}\) and \(\|\mathbf{u}_\lambda\|_{\dot H^s} = \lambda^{s+1-d/2}\|\mathbf{u}\|_{\dot H^s}\). A quantity is critical if it scales like \(\lambda^0\), subcritical if a bound on it controls the solution better at small scales than a critical one does, and supercritical if worse. In 3D the critical norms include \(L^3\), \(\dot H^{1/2}\), \(L^\infty\) weighted by \(|\mathbf{x}|\), and (Koch and Tataru, 2001) \(BMO^{-1}\). The energy \(\|\mathbf{u}\|^2_{L^2}\) scales like \(\lambda^{-1}\), and so does the total dissipation \(\nu\int\!\!\int|\nabla\mathbf{u}|^2\): both are supercritical. In 2D, \(\|\mathbf{u}\|_{L^2}\) scales like \(\lambda^0\): the energy is critical, and the enstrophy bound is even better.

That single computation explains the dichotomy. In 2D the controlled quantities are critical or better, and it is proved that for every smooth (or merely finite-energy) datum there is a unique global smooth solution of Navier-Stokes; for Euler, smooth data give global smooth solutions (Wolibner and Hölder, 1933), and bounded integrable vorticity already gives a unique global weak solution (Yudovich, 1963). In 3D the controlled quantities are supercritical: a hypothetical blow-up concentrating at small scales could have tiny energy while its critical norms explode. Tao (2016) made this barrier precise: an averaged version of the Navier-Stokes equations, with the same energy identity and the same harmonic analysis estimates, does blow up in finite time, so any proof of global regularity must use finer structure of the true nonlinearity.

What can be proved in 3D is that singularities, if any, are rare. A suitable weak solution (Scheffer, 1977; Caffarelli, Kohn and Nirenberg, 1982) is a weak solution with pressure that also satisfies a local energy inequality in every space-time cylinder. The Caffarelli-Kohn-Nirenberg theorem (1982) states: for a suitable weak solution, the singular set \(\Sigma\) (points where \(\mathbf{u}\) is not locally bounded) has one-dimensional parabolic Hausdorff measure zero, \(\mathcal{P}^1(\Sigma) = 0\). Parabolic measure uses cylinders \(Q_r = B_r\times(t-r^2,t)\), respecting the scaling, so a set that persists at one point for an interval of time has parabolic dimension 2 and is excluded, and so is any curve of singular points at a single time. The key is an \(\varepsilon\)-regularity statement: there is an absolute \(\varepsilon \gt 0\) such that if \[ \limsup_{r\to0}\ \frac{1}{r}\int\!\!\int_{Q_r(\mathbf{x},t)}|\nabla\mathbf{u}|^2\,dy\,ds \lt \varepsilon, \] then \((\mathbf{x},t)\) is a regular point. The quantity on the left is scale-invariant, and a covering argument using the finiteness of the total dissipation gives the measure bound. Lin (1998) gave a simpler proof.

Special blow-up shapes have been excluded too. Leray asked whether self-similar solutions \(\mathbf{u}(\mathbf{x},t) = (T-t)^{-1/2}\mathbf{U}(\mathbf{x}/\sqrt{T-t})\) could blow up; Nečas, Růžička and Šverák (1996) showed none exist with \(\mathbf{U} \in L^3\), and Tsai (1998) extended the result. For the 3D Euler equations with smooth finite-energy data and no external force the question of finite-time blow-up is also open, though singularities have been constructed for less regular data (Elgindi, 2021) and in the presence of boundaries with computer assistance (Chen and Hou, 2022 onwards).

A summary for 3D Navier-Stokes. Proved: global Leray-Hopf weak solutions for all finite-energy data; local smooth solutions for smooth data; global smooth solutions for small critical data; conditional regularity (LPS, ESS, BKM); partial regularity (CKN). Open: global smoothness for large smooth data without forcing, and uniqueness of Leray-Hopf solutions without forcing. In September 2026 preprints claimed finite-time blow-up when a suitably chosen smooth external force is added, for Navier-Stokes (OpenAI) and for 3D Euler (Buckmaster, Alpöge and Coiculescu); these claims are not yet refereed, and they do not settle the unforced problems.

Picture it: zoom into a flow by a factor \(\lambda\); critical norms look the same, the energy shrinks away to nothing, and that is where a singularity could hide from it. Think it: the whole subject is a contest between what the equations control (supercritical) and what regularity requires (critical); partial regularity measures how small the losing set must be.