Existence and Regularity Results for a Nonlinear Fluid-Structure Interaction Problem with Three-Dimensional Structural Displacement

TL;DR

This paper proves the existence and regularity of solutions for a complex 3D fluid-structure interaction model involving nonlinear coupling, fractional damping, and vector displacements, advancing the mathematical understanding of such systems.

Contribution

It introduces new regularity results and a constructive existence proof for a nonlinear 3D FSI problem with vector displacements and fractional damping.

Findings

Established hidden spatial regularity of structure displacement.

Proved temporal regularity for fluid and structure velocities.

Demonstrated existence of a local-in-time weak solution.

Abstract

In this paper we investigate a nonlinear fluid-structure interaction (FSI) problem involving the Navier-Stokes equations, which describe the flow of an incompressible, viscous fluid in a 3D domain interacting with a thin viscoelastic lateral wall. The wall's elastodynamics is modeled by a two-dimensional plate equation with fractional damping, accounting for displacement in all three directions. The system is nonlinearly coupled through kinematic and dynamic conditions imposed at the time-varying fluid-structure interface, whose location is not known a priori. We establish three key results, particularly significant for FSI problems that account for vector displacements of thin structures. Specifically, we first establish a hidden spatial regularity for the structure displacement, which forms the basis for proving that self-contact of the structure will not occur within a finite time interval. Secondly, we demonstrate temporal regularity for both the structure and fluid velocities, which enables a new compactness result for three-dimensional structural displacements. Finally, building on these regularity results, we prove the existence of a local-in-time weak solution to the FSI problem. This is done through a constructive proof using time discretization via the Lie operator splitting method. These results are significant because they address the well-known issues associated with the analysis of nonlinearly coupled FSI problems capturing vector displacements of elastic/viscoelastic structures in 3D, such as spatial and temporal regularity of weak solutions and their well-posedness.