Spectral Analysis and Asymptotic Decay of the Solutions to Multilayered Structure-Stokes Fluid Interaction PDE System

TL;DR

This paper investigates the spectral properties and long-term decay behavior of solutions to a complex PDE system modeling multilayered fluid-structure interactions, with applications in biological and vascular systems.

Contribution

It provides a spectral analysis of the coupled PDE system and establishes conditions for strong stability and asymptotic decay of solutions.

Findings

Spectrum of the semigroup generator has no parts on the imaginary axis.

Zero eigenvalue exists with a characterized one-dimensional eigenspace.

Solutions decay asymptotically to zero for initial data orthogonal to the zero eigenspace.

Abstract

In this work, the dynamics of a multilayered structure-fluid interaction (FSI) PDE system is considered. Here, the coupling of 3D Stokes and 3D elastic dynamics is realized via an additional 2D elastic equation on the boundary interface. Such modeling PDE systems appear in the mathematical modeling of eukaryotic cells and vascular blood flow in mammalian arteries. We analyze the long time behavior of solutions to such FSI coupled system in the sense of strong stability. Our proof is based on an analysis of the spectrum of the associated semigroup generator $\mathbf{A}$ which in particular entails the elimination of all three parts of the spectrum of $\mathbf{A}$ from the imaginary axis. In order to avoid steady states in our stability analysis, we firstly show that zero is an eigenvalue for the operator $\mathbf{A}$, and we provide a characterization of the (one dimensional) zero eigenspace $Null(\mathbf{A}).$ In turn, we address the issue of asymptotic decay of the solution to the zero state for any initial data taken from the orthogonal complement of the zero eigenspace $Null(\mathbf{A})^{\bot}.$