Backwards uniqueness of the $C_{0}$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system
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- by George Avalos and Roberto Triggiani PDF
- Trans. Amer. Math. Soc. 362 (2010), 3535-3561 Request permission
Abstract:
In this paper the âbackward-uniqueness propertyâ 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 LamĂ© 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-LamĂ© 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.References
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Additional Information
- George Avalos
- Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
- Roberto Triggiani
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
- Received by editor(s): February 15, 2008
- Published electronically: February 19, 2010
- Additional Notes: The research of the first author was partially supported by the NSF grant DMS-0606776.
The research of the second author was partially supported by the NSF grant DMS-0104305. - © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 3535-3561
- MSC (2010): Primary 35B99, 35M30
- DOI: https://doi.org/10.1090/S0002-9947-10-04851-8
- MathSciNet review: 2601599