A geometric trapping approach to global regularity for 2D Navier–Stokes on manifolds

In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier–Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai $\href{https://doi.org/10.1142/S0219199799000183}{[15]}$ which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schr¨odinger equation, and ideas from microlocal analysis.