Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation

TL;DR

This paper proves the existence and uniqueness of strong solutions for a coupled system involving a compressible, heat-conducting fluid and a damped elastic plate, using maximal regularity and fixed point methods.

Contribution

It establishes strong solution existence and uniqueness for a complex fluid-structure interaction system with compressible heat-conducting fluid and damped plate, extending mathematical understanding.

Findings

Existence of strong solutions for small time or data.

Use of maximal regularity and $ ext{R}$-sectoriality in proofs.

Applicability to coupled fluid-structure systems.

Abstract

The article is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier-Stokes-Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an $L^p-L^q$ setting for small time or for small data. Through a change of variables and a fixed point argument, the proof of the main results is mainly based on the maximal regularity property of the corresponding linear systems. For small time existence, this property is obtained by decoupling the linear system into several standard linear systems whereas for global existence and for small data, the maximal regularity property is proved by showing that the corresponding linear coupled {\em fluid-structure} operator is $\mathcal{R}-$sectorial.