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Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble

B. Ingimarson, I. Kukavica, W. S. Ożański

TL;DR

The paper proves global well-posedness and exponential decay for a curved-domain fluid–structure interaction model consisting of an incompressible Navier–Stokes fluid with a vacuum bubble coupled to a damped elastic body. By formulating a total weighted energy across multiple derivatives and using a tangential operator framework, the authors balance linear, nonlinear, and commutator effects to obtain a closed energy-dissipation inequality. A crucial ODE-type lemma then converts this inequality into exponential decay for small initial data, with a concrete choice of derivative weights and a small coupling parameter. The vacuum bubble boundary condition is shown to provide the necessary pressure control, enabling the energy method to work in curved geometries and extending prior flat-domain results to general curved domains with curved initial interfaces.

Abstract

We prove a priori estimates for the system of partial differential equations modeling the interaction between an elastic body and an incompressible fluid in a 3D curved domain. The fluid is governed by the incompressible Navier-Stokes equations and contains a bubble whose interior is a vacuum. The elastic body is described by a damped wave equation, and interaction with the fluid takes place along a free interface whose initial domain is curved. We show that the presence of the vacuum bubble stabilizes the system in the sense that it provides control of the average of the pressure function, and hence allows global existence and exponential decay of smooth solutions for small data.

Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble

TL;DR

The paper proves global well-posedness and exponential decay for a curved-domain fluid–structure interaction model consisting of an incompressible Navier–Stokes fluid with a vacuum bubble coupled to a damped elastic body. By formulating a total weighted energy across multiple derivatives and using a tangential operator framework, the authors balance linear, nonlinear, and commutator effects to obtain a closed energy-dissipation inequality. A crucial ODE-type lemma then converts this inequality into exponential decay for small initial data, with a concrete choice of derivative weights and a small coupling parameter. The vacuum bubble boundary condition is shown to provide the necessary pressure control, enabling the energy method to work in curved geometries and extending prior flat-domain results to general curved domains with curved initial interfaces.

Abstract

We prove a priori estimates for the system of partial differential equations modeling the interaction between an elastic body and an incompressible fluid in a 3D curved domain. The fluid is governed by the incompressible Navier-Stokes equations and contains a bubble whose interior is a vacuum. The elastic body is described by a damped wave equation, and interaction with the fluid takes place along a free interface whose initial domain is curved. We show that the presence of the vacuum bubble stabilizes the system in the sense that it provides control of the average of the pressure function, and hence allows global existence and exponential decay of smooth solutions for small data.
Paper Structure (10 sections, 6 theorems, 201 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 201 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Sketch of the proof of Theorem \ref{['T01']}.
  4. Preliminaries
  5. Combined energy estimates
  6. The total energy estimate
  7. The linear terms $L^S$
  8. The nonlinear terms $N_S$
  9. The commutator terms $C_S$
  10. An ODE-type Lemma

Key Result

Theorem 2.1

Let $(v,w,q,\eta,a)$ be a smooth solution to our system on some time interval $[0,T)$, and set Then there exists $C \geq 1$ and $\varepsilon > 0$, independent of $T$, such that if $Y(0) \leq \varepsilon$, then for $t \in [0,T)$.

Figures (2)

  • Figure 1: The sketch of the model at time $t=0$.
  • Figure 2: A local mapping

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • proof : Proof of Lemma \ref{['L02']}
  • Lemma 4.4
  • proof : Proof of Lemma \ref{['L01']}
  • proof : Proof of \ref{['EQ94']}.
  • Remark 6.1
  • Lemma 7.1
  • ...and 1 more