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

TL;DR

This paper proves the existence of weak solutions for multilayered fluid-structure interaction problems involving complex 3D/2D/3D configurations, under time-periodic or initial conditions, with a focus on viscoelastic materials and boundary-driven dynamics.

Contribution

It extends prior models to a more complex 3D/2D/3D setting and establishes existence results for weak solutions under small boundary pressure norms.

Findings

Existence of time-periodic weak solutions under small boundary pressure.

Weak solutions for initial-value problems with viscoelastic thick solids.

Extension of previous 2D/1D/2D results to 3D/2D/3D configurations.

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.