Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channel

In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent collisions of the obstacle with the boundary of the channel, we introduce a strong force in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.