Backward uniqueness of the s.c. semigroup arising in parabolic–hyperbolic fluid–structure interaction

Abstract

A 2-d or 3-d fluid–structure interaction model in its linear form is considered, for which semigroup well-posedness (with explicit generator) was recently established in [G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid–structure interaction. Part I: Explicit semigroup generator and its spectral properties, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., 2007, pp. 15–55; G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid–structure interaction. Part II: Uniform stabilization with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., in press]. This is a system which couples at the interface the linear version of the Navier–Stokes equations with the equations of linear elasticity (wave-like). In this paper, we establish a backward uniqueness theorem for such a parabolic–hyperbolic coupled PDE system. If { e A t } t ⩾ 0 is the (contraction) s.c. semigroup describing its evolution on the finite energy space H , then e A T y 0 = 0 for some T > 0 and y 0 ∈ H , implies y 0 = 0 . This property has implications in establishing unique continuation and controllability properties, as in the case of thermoelastic equations [M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable coefficient, in: Marcel Dekker Lect. Notes Pure Appl. Math., vol. 216, February 2001, pp. 109–230, invited paper for the special volume entitled Shape Optimization and Optimal Designs, J. Cagnol, J.P. Zolesio (Eds). (Preliminary version is in invited paper in: A.V. Balakrishnan (Ed.), Semigroup of Operators and Applications, Birkhäuser, 2000, pp. 335–351.); M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and moment control, J. Math. Anal. Appl. 251 (2000) 452–478; M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and clamped controls, Discrete Contin. Dyn. Syst. 7 (2) (2001) 283–301].