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 −αΔu _tt ,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 u _t in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of u _t we use analyticity of the corresponding thermoelastic semigroup proved in [25].

The isothermal variant of this problem with additional mechanical damping term −εΔu _t , ε > 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.