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 $W_p^s(\mathbb{R})$, where $p\in (1,2]$ and $s\in (1+1/p,2)$. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $W_p^{\overline{s}-2}(\mathbb{R})$, where $\overline{s}\in (1+1/p,s)$. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

29 pages