Leray's global weak solutions (1934)

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.