Large time behavior of solutions to the $3$D anisotropic Navier-Stokes equation

TL;DR

This paper analyzes the large time decay behavior of solutions to the 3D anisotropic Navier-Stokes equations, revealing different decay rates for horizontal and vertical velocity components and their asymptotic expansions.

Contribution

It provides a detailed analysis of the decay rates and asymptotic behavior of solutions to the 3D anisotropic Navier-Stokes equations with horizontal viscosity, highlighting the influence of nonlinear terms.

Findings

Horizontal velocity components decay like the 2D heat kernel.

Vertical component decays like the 3D heat kernel.

Nonlinear terms influence the leading asymptotic behavior of horizontal components.

Abstract

We consider the large time behavior of the solution to the $3$D Navier-Stokes equation with horizontal viscosity $\Delta_{\rm h} u=\partial_1^2 u+\partial_2^2 u$ and show that the $L^p$ decay rate of the horizontal components of the velocity field coinsides to that of the $2$D heat kernel, while the vertical component decays like the $3$D heat kernel. Moreover, we consider the asymptotic expansion of the solution and find that a portion of the nonlinear term affect the leading term of the horizontal components of the velocity field, whereas the leading term of the vertical component is given by only the linear solution.