A new reformulation of the Muskat problem with surface tension

Abstract

Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density ϑ to the L2-adjoints of these operators evaluated at the density ϑ ′ 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 L2-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 (R) with p ∈ (1, ∞) and s ∈ (1 + 1/p, 2).