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 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.

References in corpus (3)

• On the interaction between quasilinear elastodynamics and the Navier-Stokes equations
• Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls
• Weak-strong uniqueness for an elastic plate interacting with the Navier Stokes equation

Cited by in corpus (3)

• Construction of a right inverse for the divergence in non-cylindrical time dependent domains
• Strong time-periodic solutions for a multilayered fluid-structure interaction system with nonlinear coupling
• Weak solutions and singular limits for a compressible fluid-structure interaction problem with slip boundary conditions