Maximal regularity for the Stokes system coupled with a wave equation: application to the system of interaction between a viscous incompressible fluid and an elastic wall

We consider a viscous incompressible fluid interacting with an elastic structure located on a part of its boundary. The fluid motion is modeled by the bi-dimensional Navier-Stokes system, and the structure follows the linear wave equation in dimension 1 in space. In order to show the existence of strong solutions for the corresponding coupled system, we study the linearized system coupling the Stokes system with a wave equation and we show that the corresponding semigroup is analytic. This result can be compared to the case where the elastic displacement is governed by a beam equation for which the corresponding semigroup is only of Gevrey class.