Asymptotic Stability of a Compressible Oseen-Structure Interaction via a Pointwise Resolvent Criterion

TL;DR

This paper establishes the asymptotic stability of a linearized compressible flow-structure interaction model with non-dissipative properties, using a pointwise resolvent criterion to simplify the analysis of long-term dynamics.

Contribution

It introduces a novel application of the pointwise resolvent condition to analyze the asymptotic stability of a complex coupled PDE system with non-dissipative features.

Findings

Proves asymptotic stability in a reduced invariant subspace.

Demonstrates the effectiveness of the pointwise resolvent criterion.

Provides a clear and concise proof avoiding technical complexities.

Abstract

In this study, we consider a linearized compressible flow structure interaction PDE model for which the interaction interface is under the effect of material derivative term. While the linearization takes place around a constant pressure and density components in structure equation, the flow linearization is taken with respect to a non-zero, fixed, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms which causes the coupled system to be nondissipative. We analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one dimensional less) of the entire state space where the continuous semigroup is "\textit{uniformly bounded}". For this, we appeal to the pointwise resolvent condition introduced in \cite% {CT} which avoids many technical complexity and provides a very clean, short and easy-to-follow proof.