  We study the unsteady incompressible Navier-Stokes equations in three
dimensions interacting with a non-linear flexible shell of Koiter type. This
leads to a coupled system of non-linear PDEs where the moving part of the
boundary is an unknown of the problem. The known existence theory for weak
solutions is extended to non-linear Koiter shell models. We introduce a-priori
estimates that reveal higher regularity of the shell displacement beyond energy
estimates. These are essential for non-linear Koiter shell models, since such
shell models are non-convex (w.r.t.\ terms of highest order). The estimates are
obtained by introducing new analytical tools that allow to exploit dissipative
effects of the fluid for the (non-dissipative) solid. The regularity result
depends on the geometric constitution alone and is independent of the
approximation procedure; hence it holds for arbitrary weak solutions. The
developed tools are further used to introduce a generalized Aubin-Lions type
compactness result suitable for fluid-structure interactions.
