The Vanishing Surface Tension Limit of the Muskat Problem

Abstract

The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where the latter is due to surface tension. To leading order, both the Muskat problems with and without surface tension effect are scaling invariant in the Sobolev space $H^{1+\frac{d}{2}}(\mathbb{R}^d)$, where $d$ is the dimension of the interface. We prove that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with surface tension $\frak{s}$ converge to the unique solution of the Muskat problem without surface tension locally in time with the rate $\sqrt{\frak{s}}$ when $\frak{s}\to 0$. This allows for initial interfaces that have unbounded or even not locally square integrable curvature. If in addition the initial curvature is square integrable, we obtain the convergence with optimal rate $\frak{s}$.