Backwards uniqueness of the semigroup associated with a parabolichyperbolic StokesLamé partial differential equation system
Authors:
George Avalos and Roberto Triggiani
Journal:
Trans. Amer. Math. Soc. 362 (2010), 35353561
MSC (2010):
Primary 35B99, 35M30
Published electronically:
February 19, 2010
MathSciNet review:
2601599
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Abstract: In this paper the ``backwarduniqueness property'' is ascertained for a two or threedimensional, fluidstructure interactive partial differential equation (PDE) system, for which an explicit semigroup formulation was recently given by Avalos and Triggiani (2007) on the natural finite energy space . (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 fluidstructure semigroup , posed on the associated finite energy space , the backwarduniqueness property can be stated in this way: If for given initial data , for some , then necessarily . The proof of this property hinges on establishing necessary PDE estimates for a certain static fluidstructure equation in order to invoke the abstract backwarduniqueness resolventbased criterion by Lasiecka, Renardy, and Triggiani (2001). The backwarduniqueness property for the coupled StokesLamé PDE is motivated by, and has positive implications to, the problem of exact controllability (in the hyperbolic state variables ) and, simultaneously, approximate controllability (in the parabolic state variable ) 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.
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Additional Information
George Avalos
Affiliation:
Department of Mathematics, University of NebraskaLincoln, Lincoln, Nebraska 68588
Roberto Triggiani
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
DOI:
http://dx.doi.org/10.1090/S0002994710048518
PII:
S 00029947(10)048518
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 DMS0606776.
The research of the second author was partially supported by the NSF grant DMS0104305.
Article copyright:
© Copyright 2010 American Mathematical Society
