Enhanced dissipation for the third component of 3D anisotropic Navier-Stokes equations

TL;DR

This paper investigates decay rates of solutions to 3D anisotropic Navier-Stokes equations, showing enhanced decay for the third velocity component due to anisotropic effects and divergence-free conditions.

Contribution

It establishes that the third component decays like 3D solutions, while horizontal components decay like 2D solutions, revealing anisotropic decay behavior.

Findings

Horizontal velocity components decay like 2D Navier-Stokes solutions.

Third velocity component decays like 3D Navier-Stokes solutions.

Enhanced decay rate for the third component due to anisotropy and divergence-free condition.

Abstract

In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equations.