Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime

We address a generalised three-dimensional $\alpha$-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when $\alpha\in[0,1)$ and also prove global-in-time existence for strong solutions when $\alpha\in[0,\frac{1}{2})$ with initial data controlled by explicit constants. We obtain maximum principles for the $L^{\infty}$-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these $L^{\infty}$-norms. Finally, some convergence results for strong solutions as $\alpha\to0^+$ are also proved.