Global regularity in two dimensions (Ladyzhenskaya)

Write the vorticity (a scalar in 2D) as $\omega=\partial_x u_2-\partial_y u_1$ and substitute the momentum equation into $\partial_t\omega+u\cdot\nabla\omega$. Because $u$ is divergence-free and two-dimensional, direct computation shows every term coming from $(u\cdot\nabla)u$ that would stretch $\omega$ cancels, leaving the transport-diffusion equation $\omega_t+u\cdot\nabla\omega=\nu\Delta\omega$ with no source term.

A transport-diffusion equation with no source obeys a maximum principle: along the trajectory $\dot X(t)=u(X(t),t)$ of a fluid particle, $\frac{d}{dt}\omega(X(t),t)=\nu\Delta\omega(X(t),t)$, and diffusion cannot increase a maximum or decrease a minimum. Hence $\|\omega(\cdot,t)\|_{L^\infty}\le\|\omega_0\|_{L^\infty}$ for all $t\ge0$: the vorticity never blows up.

Once vorticity is bounded uniformly in time, elliptic regularity applied to the stream function (recovering $u$ from $\omega$) bounds all spatial derivatives of $u$ in terms of $\|\omega\|_{L^\infty}$, so $u$ stays in every Sobolev space it started in.

Bootstrapping this argument — bounded vorticity gives bounded velocity gradients, which control higher derivatives through the equation itself — shows the solution remains smooth for all $t>0$, with no finite-time blow-up possible; this is exactly the statement that fails to be known in three dimensions, where the vortex-stretching term $(\omega\cdot\nabla)u$ is present.