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Strong time-periodic solutions for a multilayered fluid-structure interaction system with nonlinear coupling

Felix Brandt, Claudiu Mîndrilă, Arnab Roy

Abstract

We investigate a time-periodic fully three-dimensional fluid-structure interaction system in which the Navier-Stokes equations for an incompressible viscous fluid are coupled with a multilayered elastic structure composed of a damped thin linear plate and a thick viscoelastic layer. The coupling is nonlinear, meaning that it is on a moving interface that is not known a priori, rendering the problem a moving-domain problem. We prove the existence of strong time-periodic solutions. The proof relies on a fixed point argument, combining sharp nonlinear estimates with a detailed analysis of the linearized system. The linearized problem is analyzed by employing the Arendt-Bu theorem on maximal periodic $\mathrm{L}^p$-regularity, which requires several new analytical ingredients including a refined lifting procedure, a decoupling strategy establishing $\mathcal{R}$-sectoriality of the coupled operator, a careful treatment of the thick structural layer, and a spectral analysis adapted to the multilayered setting. This provides the first strong time-periodic existence result for multilayered fluid-structure interaction systems, and the methods are expended to extend more broadly to nonlinear coupled PDEs on moving domains with periodic forcing.

Strong time-periodic solutions for a multilayered fluid-structure interaction system with nonlinear coupling

Abstract

We investigate a time-periodic fully three-dimensional fluid-structure interaction system in which the Navier-Stokes equations for an incompressible viscous fluid are coupled with a multilayered elastic structure composed of a damped thin linear plate and a thick viscoelastic layer. The coupling is nonlinear, meaning that it is on a moving interface that is not known a priori, rendering the problem a moving-domain problem. We prove the existence of strong time-periodic solutions. The proof relies on a fixed point argument, combining sharp nonlinear estimates with a detailed analysis of the linearized system. The linearized problem is analyzed by employing the Arendt-Bu theorem on maximal periodic -regularity, which requires several new analytical ingredients including a refined lifting procedure, a decoupling strategy establishing -sectoriality of the coupled operator, a careful treatment of the thick structural layer, and a spectral analysis adapted to the multilayered setting. This provides the first strong time-periodic existence result for multilayered fluid-structure interaction systems, and the methods are expended to extend more broadly to nonlinear coupled PDEs on moving domains with periodic forcing.

Paper Structure

This paper contains 14 sections, 26 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and function spaces
  3. The multilayered FSI problem and main result
  4. Transformation to a fixed domain
  5. Analysis of the linearized problem
  6. Preliminary considerations
  7. The thick structural layer
  8. Lifting arguments
  9. The multilayered fluid-structure operator
  10. Spectral theory
  11. Maximal periodic regularity of the multilayered fluid-structure operator
  12. Existence of a strong time-periodic solution
  13. Estimates of the nonlinear terms
  14. Proof of existence of the strong time-periodic solution

Key Result

Theorem 3.1

Let $p$, $q \in (1,\infty)$ be such that $\frac{1}{p} + \frac{3}{2q} < \frac{3}{2}$, and assume that $f \in \mathrm{L}^p(0,T;\mathrm{L}^q(\Omega_\mathrm{f})^3)$, $g \in \mathrm{L}^p(0,T;\mathrm{L}^q(\omega))$ and $h \in \mathrm{L}^p(0,T;\mathrm{L}^q(\Omega_\mathrm{s})^3)$. Then there is $R_1 > 0$ so then the multilayered FSI problem eq:multilayered fsi has a unique strong solution $(u^{\mathcal{F}

Figures (1)

  • Figure 1: Sketch of the geometry of the present multilayered FSI problem in the two-dimensional situation. The reference configuration with the fixed fluid and thick structure domains $\Omega_\mathrm{f}$ and $\Omega_\mathrm{s}$, respectively, and fixed interface $\Gamma_0 = \omega \times \{0\}$ is shown in (A), while (B) depicts the moving domain problem with time-dependent domains $\Omega_\mathrm{f}(t)$ and $\Omega_\mathrm{s}(t)$, and with moving interface $\Gamma_\eta(t) = \{(s,\eta(t,s)) : s \in \omega\}$. The rigid parts of the fluid and the thick structure boundaries are denoted by $\Gamma_\mathrm{f}$ and $\Gamma_\mathrm{s}$, respectively.

Theorems & Definitions (39)

  • Definition 1
  • Theorem 3.1
  • Remark 1
  • Theorem 4.1
  • Proposition 1: AB:02
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 29 more