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The purpose of this work is to study the existence of solutions for an unsteady fluid-structure interaction problem. We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with a flexible elastic plate located on one part of the fluid boundary. The fluid domain evolves according to the structure’s displacement, itself resulting from the fluid force. We prove the existence of at least one weak solution as long as the structure does not touch the fixed part of the fluid boundary. The same result holds also for a two-dimensional fluid interacting with a one-dimensional membrane.