Weak-Strong Uniqueness for an Elastic Plate Interacting with the Navier--Stokes Equation
Sebastian Schwarzacher, Matthias Sroczinski
Abstract
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.