Rational Decay of A Multilayered Structure-Fluid PDE System

TL;DR

This paper analyzes a multilayered PDE system modeling blood transport, demonstrating how fluid dissipation stabilizes the entire coupled system through uniform decay rates.

Contribution

It derives uniform decay rates for a complex multilayered PDE system, revealing the stabilizing influence of fluid dissipation across interfaces.

Findings

Established uniform decay rates for the coupled PDE system.

Identified the stabilizing effect of fluid dissipation on structural components.

Developed a priori inequalities for static multilayered PDE models.

Abstract

In this work, we consider a certain multilayered (thick layer) wave--(thin layer) wave--heat (fluid) interactive PDE system. Such coupled PDE systems have been used in the literature to describe the blood transport process in mammalian vascular systems. In particular, the deformations of the boundary interface (thin layer) are described via the two dimensional elastic equation. The present work constitutes an investigation of the extent of the stabilizing effects of the underlying fluid dissipation -- across the boundary interface -- upon both the thick and thin structural components. (All three PDE components evolve on their respective geometries.) In this regard, our main result is the derivation of uniform decay rates for classical solutions of this multilayered PDE model. To obtain these estimates, necessary a priori inequalities for certain static multilayered PDE models are generated here to ultimately allow an application of a wellknown resolvent criterion for rational decay.