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

TL;DR

This paper proves the existence and uniqueness of strong solutions for a class of two-phase free boundary problems involving non-Newtonian fluids with specific stress tensor structures, under small initial data conditions.

Contribution

It establishes the first rigorous results on strong solutions for two-phase non-Newtonian fluid free boundary problems with surface tension and gravity.

Findings

Unique strong solutions exist for small initial data.

Solutions are valid on a finite time interval (0,T).

The results depend on the specific form of the stress tensor and viscosity functions.

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.