The Vanishing Surface Tension Limit of the Muskat Problem

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