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

Abstract

We study the two dimensional viscous Boussinesq equations, which model stratified flows in a circular domain under the influence of a general gravitational potential $f$. First, we show that the Boussinesq equations admit steady-state solutions only in the form of hydrostatic equilibria, $(\mathbf{u},ρ,p) = (0, ρ_s, p_s)$, where the pressure gradient satisfies $\nabla p_s = -ρ_s \nabla f$. Moreover, the relation between $ρ_s$ and $f$ is constrained by $(\partial_y ρ_s, -\partial_x ρ_s) \cdot (\partial_x f, \partial_y f) = 0$, which allows us to write $\nabla ρ_s = h(x,y) \nabla f$ for some scalar function $h(x,y)$. Second, we prove that any hydrostatic equilibrium $(0, ρ_s, p_s)$ is linearly unstable if $h(x_0, y_0) > 0$ at some point $(x, y) = (x_0, y_0)$. This instability coincides with the classical Rayleigh--Taylor instability. Third, by employing a series of regularity estimates, we reveal that although the presence of the Rayleigh--Taylor instability makes perturbations around the unstable equilibrium grow exponentially in time, the system ultimately converges to a state of hydrostatic equilibrium. The analysis is carried out for perturbations about an arbitrary hydrostatic equilibrium, covering both stable and unstable configurations. Finally, we derive a necessary and sufficient condition on the initial density perturbation under which the density converges to a profile of the form $-γf + β$ with constants $γ, β> 0$. This result underscores the system's inherent tendency to settle into a hydrostatic state, even in the presence of Rayleigh--Taylor instability.