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 (FSI) 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> -regularity, which requires several new analytical ingredients including a refined lifting procedure, a decoupling strategy establishing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -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 FSI systems , and the methods are expended to extend more broadly to nonlinear coupled PDEs on moving domains with periodic forcing.