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A free boundary inviscid model of flow-structure interaction

Mustafa Sencer Aydın, Igor Kukavica, Amjad Tuffaha

TL;DR

The paper studies local-in-time well-posedness for the 3D incompressible Euler equations coupled with an Euler–Bernoulli plate via a free boundary, in a channel geometry. It fixes the moving domain using an Arbitrary Lagrangian–Eulerian (ALE) transformation, producing a variable-coefficient Euler system with pressure $q$ and a plate equation $w_{tt}+\Delta_{h}^{2} w-\nu \Delta_{h} w_t = q$ on the interface, along with compatibility and zero-average pressure conditions. It proves local existence and uniqueness of strong solutions with minimal Sobolev regularity: initial fluid velocity in $H^{2.5+\delta}$ and plate velocity in $H^{1+\delta}$ (for any $\delta\in(0,1/2]$), yielding $v\in L^{\infty}([0,T];H^{2.5+\delta})$, $q\in L^{\infty}([0,T];H^{1.5+\delta})$, $w\in L^{\infty}([0,T];H^{4+\delta})$, and $w_t\in L^{\infty}([0,T];H^{2+\delta})$. The proof combines pressure elliptic estimates, tangential energy arguments, and vorticity bounds within a continuation framework, and establishes uniqueness via a Gronwall-type estimate for the solution differences. This advances the mathematical understanding of inviscid fluid–structure interaction with free boundaries by providing robust local well-posedness under minimal regularity assumptions.

Abstract

We address the existence and of solutions for the Euler-plate free-boundary system modeling an interaction of a three-dimensional inviscid fluid and an evolving plate. We prove the local existence and uniqueness of solutions for initial fluid and structural velocities belonging to $H^{2.5+}$ and $H^{2+}$, respectively. The results justify earlier a~priori estimates shown by two of the authors.

A free boundary inviscid model of flow-structure interaction

TL;DR

The paper studies local-in-time well-posedness for the 3D incompressible Euler equations coupled with an Euler–Bernoulli plate via a free boundary, in a channel geometry. It fixes the moving domain using an Arbitrary Lagrangian–Eulerian (ALE) transformation, producing a variable-coefficient Euler system with pressure and a plate equation on the interface, along with compatibility and zero-average pressure conditions. It proves local existence and uniqueness of strong solutions with minimal Sobolev regularity: initial fluid velocity in and plate velocity in (for any ), yielding , , , and . The proof combines pressure elliptic estimates, tangential energy arguments, and vorticity bounds within a continuation framework, and establishes uniqueness via a Gronwall-type estimate for the solution differences. This advances the mathematical understanding of inviscid fluid–structure interaction with free boundaries by providing robust local well-posedness under minimal regularity assumptions.

Abstract

We address the existence and of solutions for the Euler-plate free-boundary system modeling an interaction of a three-dimensional inviscid fluid and an evolving plate. We prove the local existence and uniqueness of solutions for initial fluid and structural velocities belonging to and , respectively. The results justify earlier a~priori estimates shown by two of the authors.
Paper Structure (10 sections, 7 theorems, 195 equations)

This paper contains 10 sections, 7 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. The Euler-plate system and the main results
  3. The ALE coordinates
  4. The main result
  5. Proof of Theorem \ref{['T01']}: Existence
  6. Pressure estimates
  7. Tangential estimates
  8. The vorticity estimate
  9. Conclusion of the proof of Proposition \ref{['P01']}
  10. Proof of Theorem \ref{['T01']}: Uniqueness

Key Result

Theorem 2.1

Let $\nu \in [0,1]$ and $(v_0,w_1) \in H^{2.5+\delta }\times H^{1+\delta}(\Gamma_1)$. Moreover, assume the compatibility conditions with Then, there exists a unique solution $(v,q,w)$ on $[0,T]$ to the Euler-plate system EQ14--EQ21 such that where $T=T(\Vert v_0\Vert_{H^{2.5+\delta}}, \Vert w_1\Vert_{H^{1+\delta}(\Gamma_1)})>0$.

Theorems & Definitions (12)

  • Theorem 2.1: Local existence
  • Theorem 2.2: A priori estimates for existence
  • Theorem 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['T01']}: Existence
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['L.Pre']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['L.Tan']}
  • Lemma 3.3
  • ...and 2 more