Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

Abstract

We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem.A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −α∆utt, α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special ("local energy") topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of ut in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of ut we use analyticity of the corresponding thermoelastic semigroup proved in [25].The isothermal variant of this problem with additional mechanical damping term −ǫ∆ut, ǫ > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain. (2000). 35B40, 35D05, 35G25. Mathematics Subject Classification

Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures

“…Well-posedness results in the past literature deal mainly with the dynamics possessing some regularizing effects. This has been accomplished by either accounting for non-negligible rotational inertia [10,12,22] and strong damping of the form −α∆u t , or by incorporating helpful thermal effects into the structural model [57,58]. In the cases listed above, the natural structure of the dynamics dictate that the plate velocity has the property u t ∈ H 1 (Ω), which provides the needed regularity for the applicability of many standard tools in nonlinear analysis.…”

Feedback Stabilization of a Fluttering Panel in an Inviscid Subsonic Potential Flow

“…. This is given as Lemma 10 in [39, p. 472] (where it is proved) and is utilized in a critical way in [40] as well.…”

“…Our result implies that flutter (a periodic or chaotic end behavior) can be eliminated (in subsonic flows) with sufficient frictional damping in the structure. While such a result has been proved in the past for regularized plate models (with rotational inertia terms or thermal considerations [14,32,39,40]), this is the first treatment which does not incorporate smoothing effects for the structure.…”

“…(These treatments address von Karman plates, though the results also hold for Berger plates.) The treatments in [39,40] consider the plate with the addition of a heat equation in the plate dynamics. In this case no damping is needed, as the analytic smoothing and stability properties of thermoelasticity provide ultimate compactness of the plate dynamics and, furthermore, provide a stabilizing effect for the flow to equilibria (in subsonic flows).…”

“…The recent treatment [32] addresses (iii); it is shown that by considering "large" static damping and "large" viscous damping-of the form D · [u t + u] in the plate equation, D sufficiently large-flow-plate trajectories with finite energy initial data converge to the set of equilibria. The techniques in [32] are rooted in Lyapunov methods and utilize critically the prior work in [13,14,39,40]. The key point here is that for large static and viscous damping, exponential decay of plate velocities provides global-in-time Hadmard continuity which permits approximation by smooth data (for which the desired result holds-Theorem 4.9).The main goal of the present work is to obtain strong stability results ( strong convergence to a set of equilibria) without accounting for smoothing effects in plate dynamics and without without assuming large static damping imposed on the structure (plate).…”