Large-Time Behavior of Solutions to Darcy–Boussinesq Equations with Non-Vanishing Scalar Acceleration Coefficient

We study the large-time behavior of solutions to Darcy–Boussinesq equations with a non-vanishing scalar acceleration coefficient, which model buoyancy-driven flows in porous media with spatially varying gravity. First, we show that the system admits steady-state solutions of the form $(\mathbf{u},\rho ,p)=(\mathbf{0},{\rho }_s,p_s)$, where ${\rho }_s$ is characterised by the hydrostatic balance $\nabla p_s=-{\rho }_s\nabla \Psi$. Second, we prove that the steady-state solution satisfying $\nabla {\rho }_s=\delta (x,y)\nabla \Psi$ is linearly stable provided that $\delta (x,y)<{\delta }_0<0$, while the system exhibits Rayleigh–Taylor instability if $\Psi =gy$, ${\rho }_s={\delta }_{0}g$ and ${\delta }_0>0$. Finally, despite the inherent Rayleigh–Taylor instability that may trigger exponential growth in time, we prove that for any sufficiently regular initial data, the solutions of the system asymptotically converge towards the vicinity of a steady-state solution, where the velocity field is zero, and the new state is determined by hydrostatic balance. This work advances porous media modeling for geophysical and engineering applications, emphasizing the critical interplay of gravity, inertia, and boundary conditions.