Global smooth two-dimensional Navier–Stokes fields before the energy estimate

Let $\nu>0$ and let $u_0$ be smooth, divergence-free Schwartz initial data on $\mathbb R^2$. Then there exist velocity and pressure fields $u,p$ which are jointly smooth on $[0,\infty)\times\mathbb R^2$, satisfy the incompressible Navier–Stokes momentum equation for every positive time, retain the spatial Schwartz and divergence-free properties at every nonnegative time, and satisfy $u(0)=u_0$.

This theorem isolates the global two-dimensional PDE construction from the separate uniform-energy estimate. Analytically, the construction is based on the scalar vorticity equation $\partial_t\omega+u\cdot\nabla\omega=\nu\Delta\omega$: its maximum principle and parabolic estimates prevent the finite-time breakdown mechanism that remains uncontrolled in three dimensions. The theorem does not itself assert the uniform bounded-energy field required by IsSolution; that estimate is recorded separately so its integration-by-parts justification can be formalized independently.