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

Longhui Xu

TL;DR

This work proves local well-posedness for the 3D free-boundary incompressible elastodynamics with surface tension in a periodic setting by reformulating the problem on a fixed domain using graphical coordinates. A κ-approximate system with artificial viscosity is introduced to boost boundary regularity, and then uniform energy estimates are derived via L2 energy, div-curl (Hodge) theory, and Alinhac’s good unknowns, ensuring no regularity loss as κ→0. The nonlinear problem is solved by a Picard iteration built from a linearized, κ-fixed system; higher-order div-curl and tangential estimates guarantee contraction and convergence to a solution of the nonlinear approximate system, from which a ξ-limit is taken to recover the original system. The main result gives a local-in-time solution with an explicit energy bound E(t) ≤ C(σ^{-1}) P(E(0)), validating a robust, κ-independent energy framework for this free-boundary elastic problem with surface tension. Overall, the paper advances the mathematical understanding of elastodynamic free boundaries by delivering a rigorous, lossless energy method in graphical coordinates and establishing a concrete path to removing the artificial viscosity.

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 $κ> 0$ to boost the boundary regularity, which recovers the original system as $κ\to 0$, and the energy estimates yield no regularity loss.

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

TL;DR

This work proves local well-posedness for the 3D free-boundary incompressible elastodynamics with surface tension in a periodic setting by reformulating the problem on a fixed domain using graphical coordinates. A κ-approximate system with artificial viscosity is introduced to boost boundary regularity, and then uniform energy estimates are derived via L2 energy, div-curl (Hodge) theory, and Alinhac’s good unknowns, ensuring no regularity loss as κ→0. The nonlinear problem is solved by a Picard iteration built from a linearized, κ-fixed system; higher-order div-curl and tangential estimates guarantee contraction and convergence to a solution of the nonlinear approximate system, from which a ξ-limit is taken to recover the original system. The main result gives a local-in-time solution with an explicit energy bound E(t) ≤ C(σ^{-1}) P(E(0)), validating a robust, κ-independent energy framework for this free-boundary elastic problem with surface tension. Overall, the paper advances the mathematical understanding of elastodynamic free boundaries by delivering a rigorous, lossless energy method in graphical coordinates and establishing a concrete path to removing the artificial viscosity.

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 to boost the boundary regularity, which recovers the original system as , and the energy estimates yield no regularity loss.

Paper Structure

This paper contains 20 sections, 11 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Reformulation in flattened domains
  3. History and background
  4. Main result
  5. Uniform Estimates of the nonlinear approximate system
  6. $L^2$ energy conservation
  7. Div-curl estimate
  8. Tangential estimate
  9. Reformulation in Alinhac's good unknown
  10. Tangential estimates with full spatial derivatives
  11. Tangential estimates with full time derivatives
  12. Well-posedness of the nonlinear approximate system
  13. Linearized approximate system
  14. Well-posedness of the linearised approximate system
  15. Higher-order estimates of the linearized approximate system
  16. ...and 5 more sections

Key Result

Theorem 1.1

Fix $\sigma>0$. Let $(q_0,v_0,F_k^0)\in H^4(\Omega)$ and $\psi_0\in H^{5.5}(\Sigma)$ be initial data of the system (phisys) that verifies the compatibility condition up to $3$-th order, $E(0)\leq M$ for some constant $M>0$, and the initial constraints Then there exists $T>0$ depending only on the initial data and $\sigma$ such that (phisys) admits a unique solution $(v(t),F_k(t),\psi(t))$ satisf

Theorems & Definitions (16)

  • Remark 1.1: Initial constraints for the deformation tensor
  • Theorem 1.1
  • Remark 2.1: Reduction of pressure
  • Proposition 2.1
  • Lemma 2.1: Integration by parts
  • Lemma 2.2: Transport Theorem
  • Proposition 2.2
  • Lemma 2.3: The Hodge-type elliptic estimate
  • Proposition 2.3
  • Proposition 2.4
  • ...and 6 more