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

Wei Zhang; Chengchun Hao; Jie Fu

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arXiv · 2508.08751

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

Wei Zhang·Chengchun Hao·Jie Fu

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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.

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Wei Zhang, Chengchun Hao, Jie Fu. 2025-08-12. Finite-Time Splash in Free Boundary Problem of 3D Neo-Hookean Elastodynamics. https://arxiv.org/abs/2508.08751

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