The Muskat problem with surface tension and equal viscosities in subcritical $L_p$-Sobolev spaces

Abstract

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces W s p (R), where p ∈ (1, 2] and s ∈ (1 + 1/p, 2). This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in W s−2 p (R), where s ∈ (1 + 1/p, s). Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.