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

TL;DR

This paper proves global existence and regularity of 2D Navier-Stokes equations on Riemannian manifolds using frequency decomposition, microlocal analysis, and multilinear estimates, extending previous flat-boundary results to curved geometries.

Contribution

It introduces a novel approach combining frequency analysis and microlocal techniques to establish regularity for Navier-Stokes on general 2D manifolds, extending prior flat-boundary methods.

Findings

Global regularity for 2D Navier-Stokes on manifolds established

New frequency projection and multilinear estimates developed

Extension of flat boundary techniques to curved geometries

Abstract

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 [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.