  We show weak-strong uniqueness and stability results for the motion of a two
or three dimensional fluid governed by the Navier-Stokes equation interacting
with a flexible, elastic plate of Koiter type. The plate is situated at the top
of the fluid and as such determines the variable part of a time changing domain
(that is hence a part of the solution) containing the fluid. The uniqueness
result is a consequence of a stability estimate where the difference of two
solutions is estimated by the distance of the initial values and outer forces.
For that we introduce a methodology that overcomes the problem that the two
(variable in time) domains of the fluid velocities and pressures are not the
same. The estimate holds under the assumption that one of the two weak
solutions possesses some additional higher regularity. The additional
regularity is exclusively requested for the velocity of one of the solutions
resembling the celebrated Ladyzhenskaya-Prodi-Serrin conditions in the given
framework.
