Bounded Semigroup Wellposedness for a Linearized Compressible Flow Structure PDE Interaction with Material Derivative

TL;DR

This paper proves well-posedness and boundedness of a linearized compressible flow-structure PDE system with material derivative effects, using a specialized inner product and PDE estimates.

Contribution

It introduces a novel approach to establish well-posedness and uniform boundedness for a compressible FSI system with non-dissipative features, on a carefully chosen subspace.

Findings

System is well-posed on a subspace of finite energy space.

Associated semigroup is uniformly bounded in time.

Maximal dissipativity achieved via a special inner product.

Abstract

We consider a compressible flow structure interaction (FSI) PDE system which is linearized about some reference rest state. The deformable interface is under the effect of an ambient field generated by the underlying and unbounded material derivative term which further contributes to the non-dissipativity of the FSI system, with respect to the standard energy inner product. In this work we show that, on an appropriate subspace, only one dimension less than the entire finite energy space, the FSI system is wellposed, and is moreover associated with a continuous semigroup which is \emph{uniformly bounded} in time. Our approach involves establishing maximal dissipativity with respect to a special inner product which is equivalent to the standard inner product for the given finite energy space. Among other technical features, the necesssary PDE estimates require the invocation of a multiplier which is intrinsic to the given compressible FSI system.