Strong solutions for two-phase free boundary problems for a class of non-Newtonian fluids

Abstract

Consider the two-phase free boundary problem subject to surface tension and gravitational forces for a class of non-Newtonian fluids with stress tensors $T_i$ of the form $T_i=-\pi I+{\mu }_i(|D(v){|}^2)D(v)$ for $i=1,2$, respectively, and where the viscosity functions ${\mu }_i$ satisfy ${\mu }_i(s)\in C^3([0,\infty ))$ and ${\mu }_i(0)>0$ for $i=1,2$. It is shown that for given $T>0$ this problem admits a unique, strong solution on $(0,T)$ provided the initial data are sufficiently small in their natural norms.