Fluid mechanics Incompressible viscous flow Theorem

Ladyzhenskaya's global regularity in two dimensions

Builds on Navier–Stokes equations Sobolev space No-slip condition

Statement

Let $\nu > 0$ and let $\Omega \subseteq \R^2$ be either the whole plane or a bounded open set with smooth boundary. Let $u_0 \in H^1(\Omega; \R^2)$ satisfy $\nabla \cdot u_0 = 0$ , and if $\Omega$ is bounded let $u_0 = 0$ on $\partial \Omega$ . Here $H^1$ is the Sobolev space of square-integrable fields with square-integrable first derivatives. Then there is a unique pair $(u, p)$ with the following properties.

It solves the Navier–Stokes equations $\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p$ , $\nabla \cdot u = 0$ in $\Omega \times (0, \infty)$ .
$u(\cdot, 0) = u_0$ , and if $\Omega$ is bounded, $u$ satisfies the no-slip condition $u = 0$ on $\partial \Omega$ .
$u \in C([0, \infty); H^1)$ and $u \in L^2(0, T; H^2)$ for every $T > 0$ .
$u$ and $\nabla p$ are smooth in $\Omega \times (0, \infty)$ .

So in two space dimensions no singularity forms in finite time.

Remarks

The energy identity gives a bound on $\norm{u(t)}_{L^2}$ for all time, as in three dimensions. In two dimensions the Ladyzhenskaya inequality $\norm{v}_{L^4}^2 \le C \norm{v}_{L^2} \norm{\nabla v}_{L^2}$ holds for $v \in H^1_0(\Omega)$ , with a constant $C$ independent of $v$ . With it the nonlinear term is controlled in the estimate for $\norm{\nabla u(t)}_{L^2}$ , and Gronwall's inequality gives a bound for all finite times. Higher derivatives then follow by the usual bootstrap, and uniqueness follows from an energy estimate for the difference of two solutions. On the whole plane one can also use the vorticity $\omega = \partial_1 u_2 - \partial_2 u_1$ , which solves $\partial_t \omega + u \cdot \nabla \omega = \nu \Delta \omega$ , so that $\norm{\omega(t)}_{L^2}$ never increases. In three dimensions the corresponding inequality is only $\norm{v}_{L^4}^2 \le C \norm{v}_{L^2}^{1/2} \norm{\nabla v}_{L^2}^{3/2}$ , the estimate no longer closes for large data, and the question is open.

Going deeper

For a student

The theorem is the two-dimensional counterpart of the Navier–Stokes existence and smoothness problem. The key difference is scaling: in two dimensions the energy $\norm{u}_{L^2}^2$ is invariant under the scaling of the equations, so the a priori bound is critical and can be closed. In three dimensions the energy is supercritical, and only Leray weak solutions are known to exist globally. A related argument, based on the transport of vorticity, gives global smooth solutions of the two-dimensional Euler equations with smooth data.

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Navier–Stokes equations concept
No-slip condition concept
Sobolev space concept
Ladyzhenskaya's global regularity in two dimensions statement

Tag	`9910P6BF51`
URI	`m2://9910P6BF51`
Kind	statement, theorem
Writing	W0 stub
MSC	76D03, 35Q30, 76D05
Status	This copy of the site has no build data. Run `m2 status 9910P6BF51`.
About	Navier–Stokes equations
Uses	Navier–Stokes equations, Sobolev space, No-slip condition
Changed	3 November 2026