Global weak solutions to a time-periodic body-liquid interaction problem

Abstract

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier–Stokes fluid interacting with a rigid body of finite size, subject to an undamped linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field $\mathbf{V}$ far from the body. We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of $\mathbf{V}$ and, rather remarkably, the frequency of $\mathbf{V}$ is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical “invading domains” technique, our approach requires several new ideas. Indeed, due to the lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincaré map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.