Fluid mechanics Incompressible viscous flow Theorem

Caffarelli–Kohn–Nirenberg partial regularity

Builds on Navier–Stokes equations Weak solution Hausdorff measure

Statement

Let $\nu > 0$ and let $D \subseteq \R^3 \times \R$ be an open set of space-time points $(x, t)$ . Let $u \colon D \to \R^3$ and $p \colon D \to \R$ be a suitable weak solution of the Navier–Stokes equations in $D$ with no external force. This means the following.

$u$ is in $L^\infty_t L^2_x$ , $\nabla u$ is in $L^2$ and $p$ is in $L^{5/4}$ on every compact subset of $D$ .
The pair $(u, p)$ satisfies $\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p$ and $\nabla \cdot u = 0$ in $D$ in the sense of distributions.
The local energy inequality holds: for every smooth function $\varphi \ge 0$ with compact support in $D$ , $2 \nu \iint \abs{\nabla u}^2 \varphi \, dx \, dt \le \iint \Big( \abs{u}^2 (\partial_t \varphi + \nu \Delta \varphi) + (\abs{u}^2 + 2p) \, u \cdot \nabla \varphi \Big) dx \, dt.$

A point of $D$ is regular if $u$ is essentially bounded on some neighbourhood of it, and singular otherwise. Let $S$ be the set of singular points. Then $\mathcal{P}^1(S) = 0$ . Here $\mathcal{P}^1$ is the one-dimensional parabolic Hausdorff measure, built like the Hausdorff measure but with covers by parabolic cylinders $B_r(x) \times (t - r^2, t + r^2)$ of size $r$ .

Remarks

The proof rests on an $\varepsilon$ -regularity criterion. There is an absolute constant $\varepsilon > 0$ such that if $\limsup_{r \to 0} r^{-1} \iint_{Q_r(x, t)} \abs{\nabla u}^2 \, dy \, ds < \varepsilon$ , with $Q_r(x, t)$ the parabolic cylinder of size $r$ around $(x, t)$ , then $(x, t)$ is regular. A covering argument then shows that the set where the criterion fails has $\mathcal{P}^1$ -measure zero, since $\abs{\nabla u}^2$ is integrable. Caffarelli, Kohn and Nirenberg also proved that suitable weak solutions exist for every divergence-free initial velocity of finite energy, so the theorem applies to a class of Leray–Hopf weak solutions. In particular, the singular set of such a solution cannot contain a curve in space-time. Its projection on the time axis has $\tfrac{1}{2}$ -dimensional Hausdorff measure zero, since a parabolic cylinder of size $r$ projects to a time interval of length $2r^2$ . The local energy inequality is needed: the argument controls $u$ near a point only through it.

Going deeper

For a student

The theorem is the best known partial result on the regularity of weak solutions of the three-dimensional Navier–Stokes equations. It uses the natural scaling $u_\lambda(x, t) = \lambda u(\lambda x, \lambda^2 t)$ of the equations, under which the quantity $r^{-1} \iint_{Q_r} \abs{\nabla u}^2$ is invariant. This is why the parabolic dimension $1$ appears. Later work simplified the proof and adapted the $\varepsilon$ -regularity method to other equations. Full regularity would settle the Navier–Stokes existence and smoothness problem.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Hausdorff measure concept
Navier–Stokes equations concept
Weak solution concept
Caffarelli–Kohn–Nirenberg partial regularity statement

Tag	`2SP210VK9P`
URI	`m2://2SP210VK9P`
Kind	statement, theorem
Writing	W0 stub
MSC	76D03, 35Q30, 35B65
Status	This copy of the site has no build data. Run `m2 status 2SP210VK9P`.
About	Navier–Stokes equations
Uses	Navier–Stokes equations, Weak solution, Hausdorff measure
Changed	3 November 2026