Global stability of large Fourier mode for 3-D anisotropic Navier-Stokes equations in cylindrical domain

TL;DR

This paper proves the global stability of large Fourier modes for 3-D anisotropic Navier-Stokes equations in cylindrical domains, extending previous results and improving decay rates of Fourier coefficients.

Contribution

It establishes global existence and stability for anisotropic Navier-Stokes equations with large Fourier mode data, and enhances decay estimates from polynomial to exponential.

Findings

Proved global stability for 3-D anisotropic Navier-Stokes in cylindrical domains.

Extended results from classical Navier-Stokes to anisotropic case.

Achieved exponential decay of Fourier coefficients, improving previous polynomial decay.

Abstract

In this paper, we first establish the global existence and stability of solutions to 3-D classical Navier-Stokes equations $(NS)$ in an infinite cylindrical domain with large Fourier mode initial data. Then we extend similar result for 3-D anisotropic Navier-Stokes equations $(ANS).$ We remark that due to the loss of vertical viscosity in $(ANS),$ the construction of the energy functionals for $(ANS)$ is much more subtle than that of $(NS).$ Compared with our previous paper for $(NS)$, we improve the polynomial decay in $k$ for the Fourier coefficients of the solution to be exponential decay in $k$ here.