Regularity results in 2D fluid-structure interaction

TL;DR

This paper proves the existence of a unique global strong solution for 2D fluid-structure interaction with a general geometric setup, using novel maximal regularity estimates for the steady Stokes system in domains with minimal boundary regularity.

Contribution

It introduces a new maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity, enabling analysis of fluid-structure interaction with flexible geometries.

Findings

Existence of a unique global strong solution in 2D fluid-structure interaction.

Development of a maximal regularity estimate for the steady Stokes system in irregular domains.

Application of the estimate to control the velocity in $W^{2,2}$ in moving boundary problems.

Abstract

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, were the structure can even occupy the complete boundary. Our main tool -- being of independent interest -- is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity in $W^{2,2}$ in terms of a forcing in $L^2$ provided the boundary belongs roughly to $W^{3/2,2}$. This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class $W^{2,2}$, known results do not apply here as they require a $C^2$-boundary.