Schauder-type Estimates and Log-Critical Well-posedness for the Two-Phase Muskat Problem with Surface Tension · arXivDesk
2606.12388Jun 10, 2026This manuscript corresponds to one part of a study initially published on arXiv (arXiv:2407.05313). The original comprehensive preprint has been divided into a series of papers, each separately addressing the well-posedness of certain free boundary problems, the general case of the Muskat problem, and problems with fixed boundaries. The present article forms Part II of this series
Schauder-type Estimates and Log-Critical Well-posedness for the Two-Phase Muskat Problem with Surface Tension
We prove short-time well-posedness for the Muskat problem with surface tension in the full two-phase setting, allowing different viscosities, arbitrary density contrast, and rigid boundaries. In particular, no Rayleigh--Taylor sign condition on the density contrast is imposed. The interface is assumed to be a graph, uniformly separated from the fixed boundaries, and the initial data may be large in the log-critical class C˙1,logϰ∩H1
Nearby in the stack
, with
ϰ>1
. Thus the result reaches the natural Lipschitz threshold up to a logarithmic correction. The main difficulty is that, in the presence of viscosity jump and boundaries, the interface equation is not given by a closed explicit contour dynamics law. Instead, the normal velocity is recovered through an elliptic transmission problem in moving domains, and the resulting evolution is a genuinely nonlocal quasilinear equation. We derive sharp Schauder-type estimates, adapted to the log-critical scale, for the transmission operators generated by the bulk Darcy flow. These estimates identify the third-order parabolic mechanism produced by surface tension and control the nonlinear coupling between the interface geometry and the elliptic transmission structure. The proof builds on the Schauder framework developed in Part I of this series CHN1, but requires a new analysis of the Muskat transmission problem in moving domains. Combining this elliptic theory with the contour formulation and time-weighted Hölder estimates, we obtain existence, uniqueness, smoothing, and stability for large interfaces in arbitrary dimension.