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

Abstract

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 $\alpha \to 0^{+}$ are also proved.