Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates

TL;DR

This paper proves the existence of time-periodic weak solutions for a fluid-structure interaction system coupling Navier-Stokes equations with a nonlinear Koiter plate, using a novel fixed point approach.

Contribution

It introduces a new method applying a single Leray-Schauder fixed point to a fully coupled Galerkin system for nonlinear elastic energy in fluid-structure interactions.

Findings

First to establish time-periodic weak solutions with nonlinear elastic energy.

Replaces two-stage fixed-point with a single fixed point for nonlinear systems.

Handles nonlinear Koiter energy destroying convexity of the solution map.

Abstract

We prove the existence of time-periodic weak solutions for a fluid-structure interaction system coupling the incompressible Navier-Stokes equations in a three-dimensional moving domain with a nonlinear Koiter plate equation on its upper boundary. The lateral boundary is space-periodic, a natural setting for flow in pipes and channels of periodic cross-section driven by a time-periodic pressure gradient, and the fluid satisfies a no-slip coupling condition at the moving interface. The elastic energy of the plate is governed by the nonlinear Koiter model, which yields an $H^2$-coercive operator and accounts for both membrane and bending effects. To the best of our knowledge, this is the first result on time-periodic weak solutions for a fluid-structure interaction system with a \emph{nonlinear} elastic energy. The main novelty, compared to our earlier works on the linear case -- a linear elastic plate and a linear Koiter shell respectively -- is the replacement of a two-stage fixed-point procedure -- a Leray-Schauder argument at the discrete level followed by a set-valued Kakutani-Glicksberg-Fan argument at the continuous level -- by a \emph{single} Leray-Schauder fixed point applied directly to the fully coupled Galerkin system. This reduction is not merely a simplification: the nonlinearity of the Koiter energy destroys the convexity of the solution map on which Kakutani-Fan relies, making the two-stage approach of~\cite{Claudiu22} unavailable and the single fixed point the only viable strategy.