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Multilayered fluid-structure interactions: existence of weak solutions for time-periodic and initial-value problems

Claudiu Mîndrilă, Arnab Roy

TL;DR

The paper tackles the problem of proving the existence of time-periodic weak solutions for a multilayered fluid-structure interaction model in a 3D/2D/3D setting, involving a 3D incompressible viscous fluid interacting with a 2D thin elastic shell and a 3D thick elastic solid. The authors develop a decoupled Galerkin framework augmented by a divergence-free extension operator to obtain robust a priori estimates, energy balances, and compactness. They show that, under small $L^2$-norm boundary data for the Bernoulli pressure, at least one time-periodic weak solution exists; the viscoelasticity of the thick solid is essential for diffusion estimates and energy stability, while the purely elastic thick-solid case yields an initial-value problem result. The work extends prior 2D/1D/2D analyses to a fully three-dimensional multilayered configuration and uses a set-valued fixed-point argument to couple the components, with a detailed limit passage to recover a weak solution of the original FSI system. The results have implications for physiological flows such as arterial blood dynamics and establish a rigorous foundation for time-periodic FSI in complex multilayered geometries.

Abstract

We study the interaction between incompressible viscous fluids and multilayered elastic structures in a 3D/2D/3D framework, where a 3D fluid interacts with a 2D thin elastic layer, coupled to a 3D thick elastic solid. The system is driven by time-periodic boundary conditions involving Bernoulli pressure. We prove the existence of at least one time-periodic weak solution when the boundary pressure has a sufficiently small $L^2-$ norm. A key feature of our analysis is the assumption of viscoelasticity in the thick solid, which is crucial for obtaining diffusion estimates and ensuring energy stability. Without this assumption, weak solutions are established for the initial-value problem. Our results extend prior work on 2D/1D/2D configurations to the more complex 3D/2D/3D setting, providing new insights into multilayered fluid-structure interactions.

Multilayered fluid-structure interactions: existence of weak solutions for time-periodic and initial-value problems

TL;DR

The paper tackles the problem of proving the existence of time-periodic weak solutions for a multilayered fluid-structure interaction model in a 3D/2D/3D setting, involving a 3D incompressible viscous fluid interacting with a 2D thin elastic shell and a 3D thick elastic solid. The authors develop a decoupled Galerkin framework augmented by a divergence-free extension operator to obtain robust a priori estimates, energy balances, and compactness. They show that, under small -norm boundary data for the Bernoulli pressure, at least one time-periodic weak solution exists; the viscoelasticity of the thick solid is essential for diffusion estimates and energy stability, while the purely elastic thick-solid case yields an initial-value problem result. The work extends prior 2D/1D/2D analyses to a fully three-dimensional multilayered configuration and uses a set-valued fixed-point argument to couple the components, with a detailed limit passage to recover a weak solution of the original FSI system. The results have implications for physiological flows such as arterial blood dynamics and establish a rigorous foundation for time-periodic FSI in complex multilayered geometries.

Abstract

We study the interaction between incompressible viscous fluids and multilayered elastic structures in a 3D/2D/3D framework, where a 3D fluid interacts with a 2D thin elastic layer, coupled to a 3D thick elastic solid. The system is driven by time-periodic boundary conditions involving Bernoulli pressure. We prove the existence of at least one time-periodic weak solution when the boundary pressure has a sufficiently small norm. A key feature of our analysis is the assumption of viscoelasticity in the thick solid, which is crucial for obtaining diffusion estimates and ensuring energy stability. Without this assumption, weak solutions are established for the initial-value problem. Our results extend prior work on 2D/1D/2D configurations to the more complex 3D/2D/3D setting, providing new insights into multilayered fluid-structure interactions.
Paper Structure (26 sections, 14 theorems, 123 equations, 1 figure)

This paper contains 26 sections, 14 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Weak formulation and main results
  4. Weak formulation
  5. Main results
  6. Formal a-priori estimates
  7. A divergence-free extension operator
  8. The energy balance
  9. Uniform energy estimates
  10. Compactness
  11. Proof of \ref{['eqn:compact-1']}
  12. Proof of \ref{['eqn:compact-2']}
  13. The main results
  14. Construction of the decoupled solution
  15. The Galerkin approximation
  16. ...and 11 more sections

Key Result

Theorem 2.6

If $P_{in/out}\in L^{2}_{per}\left(I\right)$ is time-periodic with $\left\Vert P_{in/out}\right\Vert _{L_{t}^{2}}\le C_0$ for a constant $C_0=C_0(\texttt{data)}$, then there exists at least one time-periodic weak solution $\left(\mathbf{u},\eta,\mathbf{d}\right)\in V_{soln}^{\eta}$ to eqn:FSI. Furth

Figures (1)

  • Figure 1: The fluid-structure domain: a 2D section

Theorems & Definitions (31)

  • Remark 1.17
  • Remark 1.18
  • Remark 1.19
  • Definition 2.3
  • Theorem 2.6
  • Remark 2.8
  • Corollary 2.10
  • Proposition 3.1
  • proof
  • Remark 3.5
  • ...and 21 more