Local Well-posedness of the Free-boundary Problem in Incompressible Elastodynamics with Surface Tension

TL;DR

This paper establishes the local well-posedness of a 3D free-boundary incompressible elastodynamics problem with surface tension, using an approximation scheme with artificial viscosity to handle boundary regularity without loss.

Contribution

It introduces a novel approach adapting previous ideas to prove well-posedness for elastodynamics with surface tension, ensuring no regularity loss in the process.

Findings

Proved local well-posedness of the 3D free-boundary elastodynamics system.

Developed an approximation scheme with artificial viscosity for boundary regularity.

Achieved energy estimates that prevent regularity loss.

Abstract

We prove the local well-posedness of the 3D free-boundary incompressible elastodynamics with surface tension describing the motion of an elastic medium in a periodic domain with a moving graphical surface. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. We adapt the idea in arXiv:2312.11254 to generate an approximate problem with artificial viscosity indexed by $\kappa > 0$ to boost the boundary regularity, which recovers the original system as $\kappa\to 0$, and the energy estimates yield no regularity loss.