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

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 $T_i=-πI+μ_i(|D(v)|^2)D(v)$ for $i=1,2$, respectively, and where the viscosity functions $μ_i$ satisfy $μ_i(s)\in C^3([0,\infty))$ and $μ_i(0)>0$ 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.

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