A linearized viscous, compressible flow-plate interaction with non-dissipative coupling

Abstract

We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile U. In contrast to model in [6], we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a C0-semigroup e At t≥0 on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator A, yielding the flow-structure dynamics.

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.…”

“…Theorem (See Theorem 3.1 of Avalos et al 2 . and Theorem 5.1 of Avalos et al 3 .) Assume that ambient vector field (when κ = 0 and κ = 1.)…”

“…In contrast to incompressible fluid flows, wherein the fluid density is assumed to be a constant, compressible gas flow models will contain an additional fluid density variable and involve other state spaces. For further details, the reader is referred to previous studies 1‐4 . 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.…”

“…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.…”

“…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.…”