Regularity of boundary traces for a fluid-solid interaction model

TL;DR

This paper develops a regularity theory for boundary traces in a fluid-solid interaction model, enabling better control and analysis of the coupled PDE system involving fluid dynamics and elasticity.

Contribution

It introduces a novel boundary trace regularity framework for the fluid component, facilitating solvability of control problems and operator Riccati equations in fluid-solid interactions.

Findings

Established boundary trace regularity for fluid velocity at the interface.

Proved solvability of associated optimal control problems.

Demonstrated well-posedness of operator Riccati equations.

Abstract

We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system -- of interest in its own right -- establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations. These results complement the recent advances in the PDE analysis and control of the Stokes-Lam\'e system.