Local Well-posedness of the Free Boundary Incompressible Elastodynamics with Surface Tension

TL;DR

This paper proves the short-term well-posedness of a free boundary incompressible elastodynamics system with surface tension, using a vanishing viscosity approach and standard Sobolev spaces.

Contribution

It introduces a novel framework for analyzing free boundary elastodynamics solely in Sobolev spaces, avoiding co-normal spaces, and establishes the inviscid limit.

Findings

Well-posedness established for short time intervals

Uniform a priori estimates with respect to viscosity

Inviscid limit of incompressible viscoelasticity justified

Abstract

In this paper, we consider a free boundary problem of the incompressible elatodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with the surface tension, we establish its well-posedness theory on a short time interval. Our method is the vanishing viscosity limit by establishing a uniform a priori estimates with respect to the viscosity. As a by-product, the inviscid limit of the incompressible viscoelasticity (the system coupling with the Navier-Stokes equations) is also justified. We point out that based on a crucial new observation of the inherent structure of the elastic term on the free boundary, the framework here is established solely in standard Sobolev spaces, but not the co-normal ones used in \cite{MasRou}.