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Stability and large-time behavior for the 2D Boussinesq system with vertical dissipation and horizontal thermal diffusion

Oussama Ben Said, Mona Ben Said

TL;DR

This work analyzes the stability and long-time behavior of perturbations near hydrostatic balance for the 2D anisotropic Boussinesq system on $\Omega=\mathbb{T}\times\mathbb{R}$ with vertical dissipation and horizontal diffusion. An energy-based framework combined with an orthogonal horizontal-average/oscillation decomposition is used to prove global $H^2(\Omega)$ stability for small data and to establish algebraic decay of the oscillatory part in $H^1$, reflecting stratification of buoyancy-driven flows. The temperature coupling provides an extra regularization that balances buoyancy and enables decay estimates despite degeneracy in horizontal dissipation. The results extend previous R^2 analyses to the cylinder-like domain, reveal the stabilizing role of temperature, and yield explicit decay rates and asymptotic 1D dynamics for the horizontal averages.

Abstract

This paper addresses the stability and large-time behavior problem on the perturbations near the hydrostatic balance of the two dimensional Boussinesq system, taking into account vertical dissipation and horizontal thermal diffusion. The spatial framework $Ω$ is defined as $ \mathbb{T}\times\mathbb{R}$, where $\mathbb{T}$ spans $[0, 1]$, representing the 1D periodic box, while $\mathbb{R}$ denotes the whole line. The results outlined in this article confirm the fact that the temperature can actually have a stabilizing effect on the buoyancy-driven fluids. The stability and long-time behavior issues discussed here are difficult due to the lack of the horizontal dissipation and vertical thermal diffusion. By formulating in the appropriate energy functional and implementing the orthogonal decomposition of the velocity and the temperature into their horizontal averages and oscillation parts, we are able to make up for the missing regularization and establish the nonlinear stability in the Sobolev space $H^2(Ω)$ and acheive the algebraic decay rates for the oscillation parts in the $H^1$-norm.

Stability and large-time behavior for the 2D Boussinesq system with vertical dissipation and horizontal thermal diffusion

TL;DR

This work analyzes the stability and long-time behavior of perturbations near hydrostatic balance for the 2D anisotropic Boussinesq system on with vertical dissipation and horizontal diffusion. An energy-based framework combined with an orthogonal horizontal-average/oscillation decomposition is used to prove global stability for small data and to establish algebraic decay of the oscillatory part in , reflecting stratification of buoyancy-driven flows. The temperature coupling provides an extra regularization that balances buoyancy and enables decay estimates despite degeneracy in horizontal dissipation. The results extend previous R^2 analyses to the cylinder-like domain, reveal the stabilizing role of temperature, and yield explicit decay rates and asymptotic 1D dynamics for the horizontal averages.

Abstract

This paper addresses the stability and large-time behavior problem on the perturbations near the hydrostatic balance of the two dimensional Boussinesq system, taking into account vertical dissipation and horizontal thermal diffusion. The spatial framework is defined as , where spans , representing the 1D periodic box, while denotes the whole line. The results outlined in this article confirm the fact that the temperature can actually have a stabilizing effect on the buoyancy-driven fluids. The stability and long-time behavior issues discussed here are difficult due to the lack of the horizontal dissipation and vertical thermal diffusion. By formulating in the appropriate energy functional and implementing the orthogonal decomposition of the velocity and the temperature into their horizontal averages and oscillation parts, we are able to make up for the missing regularization and establish the nonlinear stability in the Sobolev space and acheive the algebraic decay rates for the oscillation parts in the -norm.
Paper Structure (4 sections, 8 theorems, 271 equations)

This paper contains 4 sections, 8 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $H^2$ Nonlinear Stability
  4. Decay Rates Result

Key Result

Theorem 1.1

Let $\mathbb{T}=[0,1]$ be a 1D periodic box and let $\Omega=\mathbb{T}\times\mathbb{R}$. Assume $u_0,\theta_0\in H^2(\Omega)$ and $\nabla\cdot u_0 = 0.$ Then there exists $\varepsilon = \varepsilon(\nu,\eta)>0$ such that, if then (oussama) has a unique global solution $(u, \theta)$ that remains uniformly bounded for all time, for any $t\ge0$, where $C(\nu,\eta)$ and $C>0$ are constants.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • proof : Proof of Theorem \ref{['TH1']}