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Under the action of a time-periodic external forces we prove the existence ofat least one time-periodic weak solution for the interaction between athree-dimensional incompressible fluid, governed by the Navier- Stokes equationand a two dimensional elastic plate. The challenge is that the Eulerian domainfor the fluid changes in time and is a part of the solution. We introduce a twofixed-point methodology: First we construct a time-periodic solutions for agiven variable time-periodic geometry. Then in a second step a (set-valued)fixed point is performed w.r.t.\ the geometry of the domain. The existencerelies on newly developed a-priori estimates applicable for both coupled anduncoupled variable geometries. Due to the expected weak regularity of thesolutions such Eulerian estimates are unavoidable. Note in particular, thatonly the fluid is assumed to be dissipative. But the here produced a-prioriestimates show that its possible to exploit the dissipative effects of thefluid also for the solid deformation. The existence of periodic solutions for agiven geometry is valid for arbitrary large data. The existence of periodiccoupled solutions to the fluid-structure interaction is valid for all data thatexcludes a self-intersection a-priori.

Time-periodic weak solutions for an incompressible Newtonian fluid interacting with an elastic plate