$$L^{p}$$ Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate

Abstract

We consider a viscous incompressible fluid governed by the Navier–Stokes system written in a domain where a part of the boundary can deform. We assume that the corresponding displacement follows a damped beam equation. Our main results are the existence and uniqueness of strong solutions for the corresponding fluid-structure interaction system in an \(L^p\)-\(L^q\) setting for small times or for small data. An important ingredient of the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped plate equation. We show that this linear system possesses the maximal regularity property by proving the \({\mathcal {R}}\)-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.