A new reformulation of the Muskat problem with surface tension

TL;DR

This paper reformulates the Muskat problem with surface tension using adjoint operators, enabling a quasilinear parabolic analysis that proves local well-posedness in certain Sobolev spaces.

Contribution

It introduces a novel reformulation of the Muskat problem leveraging adjoint operators, facilitating the application of abstract quasilinear parabolic theory.

Findings

Established local well-posedness in Sobolev spaces $W^s_p(\

)$ for the reformulated problem.

Revealed advantages of expressing nonlinearities as derivatives in the new formulation.

Abstract

Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density $\vartheta$ to the $L_2$-adjoints of these operators evaluated at the density $\vartheta'$ are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the $L_2$-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces $W^s_p(\mathbb{R})$ with $p\in(1,\infty)$ and $s\in (1+1/p,2)$.