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

Longhui Xu

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.

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

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.

Paper Structure

This paper contains 24 sections, 10 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. History and Background
  3. Outline of the paper
  4. Reformulation of the system
  5. Lagrangian Coordinates
  6. Reformulation in Lagrangian Coordinates
  7. Local Well-posedness in Lagrangian Coordinates
  8. Graphic Coordinates
  9. Reformulation in Graphic Coordinates
  10. The main result
  11. The auxiliary results
  12. Preliminary Lemmas
  13. Elliptic Estimates
  14. Energy estimate
  15. Pressure Estimate
  16. ...and 9 more sections

Key Result

Theorem 2.1

(Local existence) Let $v_0,\eta_0\in H^{4.5}(\Omega)\cap H^5 (\Sigma)$ and $F_j^0\in H^{4.5}$ be divergence-free vector fields with $(F_j^0\cdot N)|_\Sigma =0$ and define the initial data $q_0$ of $q$ to satisfies the following elliptic equation Then there exists some $T>0$ depending on $\sigma, v_0, F_j^0$, such that the system (expsys)-(expbc) with initial data $(v_0,F_j^0,q_0)$ has a unique st

Theorems & Definitions (10)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Theorem 4.2