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

TL;DR

This paper proves the existence of strong time-periodic solutions for a complex three-dimensional multilayered fluid-structure interaction system with nonlinear coupling, advancing understanding of moving-domain PDEs.

Contribution

It introduces a novel analytical framework combining maximal regularity, spectral analysis, and fixed point methods to handle nonlinear, multilayered FSI systems with moving interfaces.

Findings

First strong time-periodic existence result for multilayered FSI systems.

Developed new analytical tools including a refined lifting procedure and decoupling strategy.

Extended methods to nonlinear PDEs on moving domains with periodic forcing.

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.