Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier-Stokes equations with mixed-boundary conditions in a channel

TL;DR

This paper investigates the mathematical properties of boundary control in a coupled fluid-structure system involving Navier-Stokes equations and linear elasticity, focusing on differentiability and boundary conditions in a channel.

Contribution

It extends previous analysis to nonlinear Navier-Stokes equations with mixed boundary conditions and corner domains, providing new insights into control-to-state mapping.

Findings

Existence results for the nonlinear coupled system

Analysis of control-to-state differentiability

Extension to mixed boundary conditions in corner domains

Abstract

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.