On decay of solutions to the anisotropic Boussinesq equations near the hydrostatic balance in half space $\mathbb{R}_+^3$

TL;DR

This paper investigates the stability and decay of solutions to the 3D anisotropic Boussinesq equations near hydrostatic balance in a half-space, demonstrating global stability and decay rates using energy and Fourier analysis methods.

Contribution

It establishes the global stability and decay rates of solutions to the anisotropic Boussinesq system with boundary conditions, a novel analysis near hydrostatic balance.

Findings

Proves global stability in Sobolev space $H^3( ^3_+)$.

Derives decay rates for solutions and derivatives.

Utilizes Fourier transform techniques for analysis.

Abstract

The system of the Boussinesq equations is one of the most important models for geophysical fluids. This paper focuses on the initial-boundary problem of the 3D incompressible anisotropic Boussinesq system with horizontal dissipation. The goal here is to assess the stability property and large-time behavior of perturbations near the hydrostatic balance. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space $H^3(\mathbb{R}^3_+)$. After taking a Fourier transform in $x_h = (x_1, x_2)$ and Fourier cosine and sine transforms in $x_3$ for the system, we obtain the decay rates for the global solution itself as well as its derivatives.