Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions

Abstract

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 nonlinear 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 (with respect to terms of highest order). The estimates are obtained by introducing new analytical tools that allow dissipative effects of the fluid to be exploited 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.