The global regularity of solutions to the 3D Navier–Stokes equations is famously unknown, and the axisymmetric case with swirl is perhaps the simplest geometry in which blowup may occur. We study this setting on a periodic cylinder with no-slip boundary conditions and time-independent body forcing, where weak solutions are guaranteed to have a global attractor and computer-assisted proofs are uniquely positioned to give rigorous results in a non-perturbative regime. In this geometry the equations reduce to a system in a periodic axial variable and a radial variable. Our approach for developing computer-assisted proofs of steady states of the Navier-Stokes equations in a cylindrical domain uses a Fourier–Jacobi (Zernike) basis.

In any problem one wants the basis used to express solutions to have several desirable properties: easy to express differential operators, easy to compute products, and rapid convergence properties. Fourier series are best for periodic domains, but there is no uniformly best option for radial variables. A Fourier–Jacobi basis has its advantages  --- the differential operators act as sparse ladder and conversion  operators between Jacobi families; multiplication via interpolation may be computed accurately and relatively quickly; the spectral coefficients of solutions converge geometrically --- however laborious quantitative estimates are required for the computer-assisted proof. For example, the derivative nonlinearity requires bilinear estimates from multiplying different polynomial bases, and inverting the Stokes operator on this basis and recovering explicit decay estimates requires significant effort. With this infrastructure in place we prove the existence of non-perturbative steady states, and we expect the framework to apply more broadly to PDEs in cylindrical geometries, and to richer dynamics