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

TL;DR

This paper investigates a generalized 3D Muskat problem, establishing well-posedness, decay rates, and convergence of solutions in the stable regime, with results depending on the parameter alpha.

Contribution

It provides new existence, decay, and convergence results for a generalized 3D Muskat model, extending understanding of fluid interface dynamics in the stable regime.

Findings

Local-in-time well-posedness for alpha in [0,1)

Global existence for alpha in [0,0.5) with explicit initial data bounds

Maximum principles and decay rates for solutions and their gradients

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 $\alpha\to0^+$ are also proved.