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, $(\mathbf{u},\rho ,p)=(0,{\rho }_s,p_s)$, where the pressure gradient satisfies $\nabla p_s=-{\rho }_s\nabla f$. Moreover, the relation between ${\rho }_s$ and $f$ is constrained by $({\partial }_y{\rho }_s,-{\partial }_x{\rho }_s)\cdot ({\partial }_{x}f,{\partial }_{y}f)=0$, which allows us to write $\nabla {\rho }_s=h(x,y)\nabla f$ for some scalar function $h(x,y)$. Second, we prove that any hydrostatic equilibrium $(0,{\rho }_s,p_s)$ 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 $-\gamma f+\beta$ with constants $\gamma ,\beta >0$. This result underscores the system's inherent tendency to settle into a hydrostatic state, even in the presence of Rayleigh--Taylor instability.