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

Wangrong Yang, Aibin Zang

TL;DR

The paper analyzes perturbations of the hydrostatic balance for the 3D incompressible anisotropic Boussinesq equations with horizontal dissipation in a half-space. It combines a transform-based linear analysis with a nonlinear bootstrapping framework to establish global stability in $H^3(\mathbb{R}^3_+)$ for small initial data and to derive explicit decay rates for the velocity, temperature, and their derivatives. The linear problem yields precise decay via a mixed Fourier transform, while the nonlinear problem is controlled through Duhamel representations and intricate energy estimates, ensuring optimal decay under anisotropic dissipation. These results advance understanding of stability and long-time behavior in geophysical fluid models near hydrostatic balance with boundary effects and partial dissipation.

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.

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

TL;DR

The paper analyzes perturbations of the hydrostatic balance for the 3D incompressible anisotropic Boussinesq equations with horizontal dissipation in a half-space. It combines a transform-based linear analysis with a nonlinear bootstrapping framework to establish global stability in for small initial data and to derive explicit decay rates for the velocity, temperature, and their derivatives. The linear problem yields precise decay via a mixed Fourier transform, while the nonlinear problem is controlled through Duhamel representations and intricate energy estimates, ensuring optimal decay under anisotropic dissipation. These results advance understanding of stability and long-time behavior in geophysical fluid models near hydrostatic balance with boundary effects and partial dissipation.

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 . After taking a Fourier transform in and Fourier cosine and sine transforms in for the system, we obtain the decay rates for the global solution itself as well as its derivatives.
Paper Structure (7 sections, 11 theorems, 177 equations)

This paper contains 7 sections, 11 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Nonlinear stability
  3. Solution formula for the linearized problem
  4. Decays of solutions to the nonlinear system
  5. Estimates of $\|u\|_{L^2},\|\theta\|_{L^2}$ and verification of \ref{['I4.11']}
  6. Estimates of $\|\partial_3u\|_{L^2},\|\partial_3\theta\|_{L^2}$ and verification of \ref{['I4.12']}
  7. Estimates of $\|\nabla_h u\|_{L^2}$ and $\|\nabla_h\theta\|_{L^2}$ and verification of \ref{['I4.13']}

Key Result

Theorem 1.1

(Global stability). Assume that initial data $(u_0,\theta_0)\in H^3(\mathbf R_+^3)$ with boundary conditions N1.5 and N1.6, satisfies $\nabla\cdot u_0=0.$ In addition, let $\partial_3^2\theta_0=0, ~\hbox{on}~ x_3=0.$ Then there exists $\varepsilon>0$ such that, if then the system E1.9 with boundary conditions N1.5 and N1.6 has a unique global solution $(u,\theta)\in L^\infty(0,\infty;(H^3(\mathbf

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 8 more