The porous medium equation as a singular limit of the thin film Muskat problem

Abstract

The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d $\le$ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.