Beyond Lack of Compactness and Lack of Stability of a Coupled Parabolic-Hyperbolic Fluid-Structure System

In this paper we shall derive certain qualitative properties for a partial differential equation (PDE) system which comprises (parabolic) Stokes fluid flow and a (hyperbolic) elastic structure equation. The appearance of such coupled PDE models in the literature is well established, inasmuch as they mathematically govern many physical phenomena; e.g., the immersion of an elastic structure within a fluid. The coupling between the distinct hyperbolic and parabolic dynamics occurs at the boundary interface between the media. In [A-T.1] semigroup well-posedness on the associated space of finite energy was established for solution variables {w, w t , u}, say, where, [w, w t ] are the respective displacement and velocity of the structure, and u the velocity of the fluid (there is also an associated pressure term, p, say). One problem with this fluid-structure semigroup setup is that, due to the definition of the domain D(A) of the generator A, there is no immediate implication of smoothing in the w-variable (i.e., its resolvent R(λ, A) is not compact on this component space). Thus, one is presented with the basic question of whether smooth initial data (I.C.) will give rise to higher regularity of the solutions. Accordingly, one main result described here states that said mechanical displacement, fluid velocity, and pressure term do in fact enjoy a greater regularity if, in addition to the I.C. {w o , w 1 , w o } ∈ D(A), one also has w o in (H 2(Ω s )) d . A second problem of the model is the inherent lack of long time stability. In this connection, a second result described here provides for uniform stabilization of the fluid-structure dynamics, by means of the insertion of a damping term at the interface between the two media.