Mathematical Research Letters

Volume 30 (2023)

Number 4

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

Pages: 969 – 1010

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n4.a1

Authors

Aynur Bulut (Department of Mathematics, Louisiana State University, Baton Rouge, La., U.S.A.)

Manh Huynh Khang (Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

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 $\href{https://doi.org/10.1142/S0219199799000183}{[15]}$ 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.

Full Text (PDF format)

Received 23 April 2021

Received revised 14 June 2022

Accepted 16 November 2022

Published 3 April 2024