Existence of solutions and stability analysis for a Darcy flow with extraction

摘要

In [CJ. van Duijn, G. Galiano, M.A. Peletier, A diffusion-convection problem with drainage arising in the ecology of mangroves, Interfaces Free Bound. 3 (2001) 15-44], a one-dimensional model describing the vertical movement of water and salt in a porous medium in which a continuous extraction of fresh water takes place was studied. Among other results, it was shown that for some range of parameter values, a heavier layer of water is formed above a lighter one in the transient state with a unique stable steady state. In this paper, we study the N-dimensional spatial model, for which Darcy's law must be introduced in the flow description. We prove the existence and uniqueness of weak solutions to the time evolution problem and perform a heuristic stability analysis in two ways: analytically, for a related problem, to find an approximation of the bifurcation curve in terms of the Rayleigh number, and numerically, to show the formation of instabilities in the original problem and their influence on the speed of convergence towards the stable steady state.