Finite-Time Splash in Free Boundary Problem of 3D Neo-Hookean Elastodynamics

TL;DR

This paper proves that finite-time splash singularities can occur in 3D neo-Hookean elastodynamics with free boundaries, using a Lagrangian approach and specialized initial conditions to demonstrate boundary self-intersection.

Contribution

It introduces a novel analysis of splash singularity formation in 3D elastodynamics with free boundaries, extending previous fluid dynamics results to elastic materials.

Findings

Finite-time splash singularity occurs in 3D neo-Hookean elastodynamics.

Interface remains smooth until the moment of self-intersection.

Energy estimates ensure stability up to the singularity.

Abstract

This paper establishes finite-time splash singularity formation for 3D viscous incompressible neo-Hookean elastodynamics with free boundaries. The system features mixed stress-kinematic conditions where viscous-elastic stresses balance pressure forces at the evolving interface -- a configuration generating complex boundary integrals that distinguish it from Navier-Stokes or MHD systems. To address this challenge, we employ a Lagrangian framework inspired by Coutand and Shkoller (2019), developing specialized coordinate charts and constructing a sequence of shrinking initial domains with cylindrical necks connecting hemispherical regions to bases. Divergence-free initial velocity and deformation tensor fields are designed to satisfy exact mechanical compatibility. Uniform a priori estimates across the domain sequence demonstrate that interface evolution preserves local smoothness while developing finite-time self-intersection. Energy conservation provides foundational stability, while higher-order energy functionals yield scaling-invariant regularity control. The analysis proves inevitable splash singularity formation within explicitly bounded time, maintaining spatial smoothness near the singular point up to the intersection time.