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

TL;DR

This paper investigates how the porous medium equation emerges as a limit of the thin film Muskat problem when the lighter fluid's density and viscosity vanish, providing convergence rates and limit behavior in low dimensions.

Contribution

It establishes the porous medium equation as a singular limit of the thin film Muskat problem and quantifies the convergence rate in dimensions up to four.

Findings

Height of the denser fluid converges to the porous medium equation solution.

Explicit convergence rate provided in dimensions d ≤ 4.

Limit of the lighter fluid's height determined in a specific regime.

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.