On the global stability and large time behavior of solutions of the Boussinesq equations

TL;DR

The paper analyzes the 2D viscous Boussinesq equations in a circular domain with a general gravity potential, establishing that hydrostatic equilibria are the only steady states. It demonstrates that some hydrostatic profiles are linearly unstable via Rayleigh-Taylor mechanisms, yet nonlinear dynamics drive perturbations toward hydrostatic equilibria with vanishing velocity. Through a suite of regularity estimates and a novel energy framework, the authors prove global well-posedness and detailed large-time behavior, including convergence of density to a hydrostatic form and decay of velocity. The results extend to time-dependent potentials and various boundary conditions, providing a comprehensive view of stability and asymptotics for stratified, gravity-driven flows. The work advances understanding of how RT instabilities interact with global stabilization in bounded geometries and offers rigorous tools for analyzing similar coupled PDE systems.

Abstract

We study the two-dimensional viscous Boussinesq equations, which model the motion of stratified flows in a circular domain influenced by a general gravitational potential $f$. First, we demonstrate that the Boussinesq equations admit steady-state solutions only in the form of hydrostatic equilibria, given by $(\mathbf{u},ρ,p)=(0,ρ_s,p_s)$, where the pressure gradient satisfies $\nabla p_s=-ρ_s\nabla f$. Subsequently, we establish that any hydrostatic equilibrium satisfying the condition $\nabla ρ_s=δ(x,y)\nabla f$ is linearly unstable if $δ(x,y)$ is positive at some point $(x,y)=(x_0,y_0)$, This instability corresponds to the well-known Rayleigh-Taylor instability. Thirdly, by employing a series of regularity estimates, we reveal that although the presence of the Rayleigh-Taylor instability increases the velocity, the system ultimately converges to a state of hydrostatic equilibrium. This result is achieved by analyzing perturbations around any state of hydrostatic equilibrium, including both stable and unstable configurations. Specifically, the state of hydrostatic equilibrium can be expressed as $ρ=-γf+β$,where $γ$ and $β$ are positive constants, provided that the global perturbation satisfies additional conditions. This highlights the system's tendency to stabilize into a hydrostatic state despite the presence of instabilities.

Figures (2)

• Figure 1: Schematic representation of the domain $\Omega$
• Figure 2: $\mathbf{g}=-g\left(\frac{x}{r}, \frac{y}{r}\right)^T$

Theorems & Definitions (45)

• Lemma 1.1
• proof
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.1
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.2