Global well-posedness and exponential decay for a fluid-structure model with small data

Abstract

We address the system of partial differential equations modeling motion of an elastic body interacting with an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by a damped wave equation $w_{tt}-\Delta w + \alpha w_t =0$, where $\alpha >0$. We prove the global existence and exponential decay of strong solutions for small initial data in a suitable Sobolev space. We show that the elastic velocity $w_t$ and the acceleration $w_{tt}$ can be controlled by the $H^2$ elliptic estimates and by the dissipation of the fluid via the free interface. We also find that, even though the vanishing of the final displacement $w$ appears invisible in the energy method, it can be deduced from the preservation of total volume. Our approach allows for any superlinear perturbation of the wave equation.