The Navier–Stokes equations

Even though the nonlinear term makes the equations too hard to solve exactly, viscosity keeps steadily draining kinetic energy; that single quantitative fact is enough to build an approximate solution and pass to a limit, even without knowing whether the limit is unique or smooth.

Approximate the equations by a Galerkin projection: pick the eigenfunctions of the Stokes operator as a basis, project the equations onto the span of the first $n$ of them, and solve the resulting finite-dimensional ODE system for an approximate solution $u_n$; standard ODE theory gives existence on a short time interval.

Test the Galerkin equation against $u_n$ itself. The nonlinear and pressure terms drop out (they are orthogonal to $u_n$ once the divergence-free constraint is used), leaving the energy identity $\frac{1}{2}\frac{d}{dt}\|u_n\|_{L^2}^{2}+\nu\|\nabla u_n\|_{L^2}^{2}=0$. Integrating in time shows $u_n$ stays bounded in $L^{\infty}(L^2)\cap L^2(H^1)$ uniformly in $n$, and in particular the short-time solution extends to all time.

These uniform bounds let us extract a subsequence of $u_n$ converging weakly-* in $L^{\infty}(L^2)$ and weakly in $L^2(H^1)$ to some limit $u$. The Aubin–Lions compactness lemma upgrades this to strong convergence in $L^2$ on bounded time-space regions, which is exactly what is needed to pass to the limit in the quadratic nonlinear term $(u_n\cdot\nabla)u_n\to(u\cdot\nabla)u$.

Passing to the limit in the weak formulation of the Galerkin equations shows $u$ satisfies the Navier–Stokes equations in the sense of distributions, and the weak lower semicontinuity of the norm under the limit preserves the energy inequality. This $u$ is Leray's global weak solution; whether it is unique and smooth is exactly the open Millennium Prize question.

Vorticity in 3D obeys $\omega_t+u\cdot\nabla\omega=(\omega\cdot\nabla)u+\nu\Delta\omega$, and the term $(\omega\cdot\nabla)u$ can amplify vorticity through stretching. In 2D vorticity is a scalar perpendicular to the plane of motion, so that stretching term vanishes identically, leaving only transport and diffusion — nothing left to amplify it without bound.

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.