Backwards uniqueness of the $C_{0}$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system

Abstract:

In this paper the “backward-uniqueness property” is ascertained for a two- or three-dimensional, fluid-structure interactive partial differential equation (PDE) system, for which an explicit $C_{0}$-semigroup formulation was recently given by Avalos and Triggiani (2007) on the natural finite energy space $\mathbf {H}$. (See also their Contemporary Mathematics article of 2007 for a preliminary, simplified, canonical model.) This system of coupled PDEs comprises the parabolic Stokes equations and the hyperbolic Lamé system of dynamic elasticity. Each dynamic evolves within its respective domain, while being coupled on the boundary interface between fluid and structure. In terms of said fluid-structure semigroup $\left \{ e^{\mathcal {A}t}\right \}$, posed on the associated finite energy space $\mathbf {H}$, the backward-uniqueness property can be stated in this way: If for given initial data $y_{0}\in \mathbf {H}$, $e^{\mathcal {A} T}y_{0}=0$ for some $T>0$, then necessarily $y_{0}=0$. The proof of this property hinges on establishing necessary PDE estimates for a certain static fluid-structure equation in order to invoke the abstract backward-uniqueness resolvent-based criterion by Lasiecka, Renardy, and Triggiani (2001). The backward-uniqueness property for the coupled Stokes-Lamé PDE is motivated by, and has positive implications to, the problem of exact controllability (in the hyperbolic state variables $\{w,w_t\}$) and, simultaneously, approximate controllability (in the parabolic state variable $u$) of the present coupled PDE model, under boundary control. A similar situation occurred for thermoelastic models as shown in papers by M. Eller, V. Isakov, I. Lasiecka, M. Renardy, and R. Triggiani.