Large time behavior for solutions to the anisotropic Navier-Stokes equations in a 3D half-space

TL;DR

This paper studies the long-term decay of solutions to anisotropic Navier-Stokes equations in a 3D half-space, revealing enhanced dissipation and overcoming analytical challenges with Besov spaces and Littlewood-Paley techniques.

Contribution

It provides optimal decay estimates for nonlinear solutions and introduces new analytical methods to handle nonlocal operators in anisotropic half-space settings.

Findings

Optimal decay estimates for solutions in anisotropic norms

Identification of enhanced dissipation in the third velocity component

Development of Besov space framework for nonlocal operator analysis

Abstract

We consider the large time behavior of the solution to the anisotropic Navier--Stokes equations in a $3$D half-space. Investigating the precise anisotropic nature of linearized solutions, we obtain the optimal decay estimates for the nonlinear global solutions in anisotropic Lebesgue norms. In particular, we reveal the enhanced dissipation mechanism for the third component of velocity field. We notice that, in contrast to the whole space case, some difficulties arises on the $L^1(\mathbb{R}^3_+)$-estimates of the solution due to the nonlocal operators appearing in the linear solution formula. To overcome this, we introduce suitable Besov type spaces and employ the Littlewood--Paley analysis on the tangential space.