Exponential stability of a nondissipative, compressible flow–structure PDE model

Abstract

In this work, a result of exponential stability is obtained for solutions of a compressible flow-structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and its associated state equation for the associated pressure variable, each evolving within a three dimensional domain O, are coupled to a fourth order plate equation which holds on a flat portion Ω of the boundary ∂O. Moreover, since this coupled PDE model is the result of a linearization of the compressible Navier-Stokes equations about an arbitrary state, the flow PDE component contains a generally nonzero ambient flow profile U. By way of obtaining the aforesaid exponential stability, a "frequency domain" approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow-structure semigroup generator.

A time domain approach for the exponential stability of a linearized compressible flow‐structure PDE system

“…. We make use of the standard notation for the boundary trace of functions defined on , which are sufficiently smooth; for example, for a scalar function Φ ∈ H s (), 1 2 < s < 3 2 , (Φ) = Φ|  , which is a well-defined and surjective mapping on this range of s, owing to the Sobolev Trace Theorem on Lipschitz domains (see, e.g., Nečas 30 or Theorem 3.…”

“…In fact, in the case = 0, the fact that zero is an eigenvalue for the generator  0 was proved in Avalos and Geredeli. 1 Also an explicit characterization for the corresponding zero eigenspace was given there, again in the case = 0. However, in the presence of the unbounded-material derivative-term U • ∇w, it is not at all clear a priori that the case = 1 should give rise to the same zero eigenspace.…”

“…For further details, the reader is referred to previous studies. [1][2][3][4] The presence of the density (pressure) equations in compressible cases will tend to make the analysis quite different than that for incompressible flows; in particular, one must deal with the extra density (pressure) solution variable. Moreover, and intrinsic to the problem under consideration, linearization of the compressible Navier-Stokes equation occurs around a rest state that contains an arbitrary ambient vector field U. Qualitative properties of this model-that is, wellposedness and long-term analysis in the sense of global attractors (in the presence of the von Karman plate nonlinearity) were firstly analyzed in Chueshov,4 in the case U = 0.…”

“…1 By way of appropriately estimating U • ∇p in Avalos and Geredeli, 1 as a static (and not time-dependent term), the frequency domain approach allows for an appropriate decomposition of static Stokes flow, and an eventual invocation of the nonsmooth domain version of the Agmon-Douglis-Nirenberg Theorem; see p. 75 of Dauge. 5 In addition to dealing with unbounded term U • ∇p, a large part of the work in Avalos and Geredeli 1 is devoted to a spectral analysis of the compressible flow-structure generator, by way of ultimately invoking the well-known resolvent criteria for exponential decay in Huang 6 and Prüss. 7 Although the methodology set forth in Avalos and Geredeli 1 is effective in establishing exponential stability for solutions of the compressible flow-structure PDE models (2)-(4) below, with = 0, it's use seems limited when dealing with (2)-(4) when the physically relevant material derivative term is present (i.e., = 1; see Avalos et al 3 for the modeling aspects of this problem; also Dowell 8 ).…”

“…5 In addition to dealing with unbounded term U • ∇p, a large part of the work in Avalos and Geredeli 1 is devoted to a spectral analysis of the compressible flow-structure generator, by way of ultimately invoking the well-known resolvent criteria for exponential decay in Huang 6 and Prüss. 7 Although the methodology set forth in Avalos and Geredeli 1 is effective in establishing exponential stability for solutions of the compressible flow-structure PDE models (2)-(4) below, with = 0, it's use seems limited when dealing with (2)-(4) when the physically relevant material derivative term is present (i.e., = 1; see Avalos et al 3 for the modeling aspects of this problem; also Dowell 8 ). In particular, since the presence of the material derivative term U • ∇w on the boundary interface constitutes an unbounded perturbation of the compressible flow-structure PDE system, a necessary spectral analysis for = 1, analogous to that in Avalos and Geredeli, 1 is problematic.…”