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

TL;DR

This paper proves well-posedness and instant smoothness of the Muskat problem with surface tension and equal viscosities in certain subcritical Sobolev spaces, and offers a criterion for global solutions.

Contribution

It establishes the well-posedness and regularity of the Muskat problem in subcritical Sobolev spaces with surface tension and equal viscosities, extending previous results.

Findings

Solutions become instantly smooth.

Global existence criterion provided.

Well-posedness in subcritical Sobolev spaces proven.

Abstract

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $W^s_p(\mathbb{R})$, 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 $W^{\overline{s}-2}_p(\mathbb{R})$, 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.