On the free-boundary Incompressible Elastodynamics with and without surface tension

TL;DR

This paper establishes local well-posedness for a free-boundary incompressible elastodynamics problem with surface tension, extending previous methods and analyzing the zero-surface-tension limit.

Contribution

It adapts techniques from magnetohydrodynamics and water waves to prove well-posedness and uniform energy estimates in elastodynamics with surface tension.

Findings

Proved local well-posedness in Lagrangian coordinates.

Derived energy estimates uniform in surface tension coefficient.

Analyzed the zero-surface-tension limit under the Rayleigh-Taylor condition.

Abstract

We consider a free-boundary problem for the incompressible elastodynamics describing the motion of an elastic medium in a periodic domain with a moving boundary and a fixed bottom under the influence of surface tension. The local well-posedness in Lagrangian coordinates is proved by extending arXiv:2105.00596 on incompressible magnetohydrodynamics. We adapt the idea in arXiv:2211.03600 on compressible gravity-capillary water waves to obtain an energy estimate in graphic coordinates. The energy estimate is uniform in surface tension coefficient if the Rayleigh-Taylor sign condition holds and thus yields the zero-surface-tension limit.