Abstract
In this paper we summarize some recent results on the asymptotic behavior of a linearized model arising in fluid-structure interaction, where a wave and a heat equation evolve in two bounded domains, with natural transmission conditions at the interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. First, we show the strong asymptotic stability of solutions. Next, based on the construction of ray-like solutions by means of Geometric Optics expansions and a careful analysis of the transfer of the energy at the interface, we show the lack of uniform decay of solutions in general domains. Finally, we obtain a polynomial decay result for smooth solutions under a suitable geometric assumption guaranteeing that the heat domain envelopes the wave one. The system under consideration may be viewed as an approximate model for the motion of an elastic body immersed in a fluid, which, in its most rigorous modeling should be a nonlinear free boundary problem, with the free boundary being the moving interface between the fluid and the elastic body.
The work is supported by the Grant MTM2005-00714 of the Spanish MEC, the DOMINO Project CIT-370200-2005-10 in the PROFIT program of the MEC (Spain), the SIMUMAT projet of the CAM (Spain), the EU TMR Project “Smart Systems”, and the NSF of China under grants 10371084 and 10525105.
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References
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024–1065.
M. Boulakia, Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid, C. R. Math. Acad. Sci. Paris, 336 (2003), 985–990.
T. Duyckaerts, Optimal decay rates of the solutions of hyperbolic-parabolic systems coupled by an interface, Preprint, 2005.
J.L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1: Contrôlabilité Exacte, RMA 8, Masson, Paris, 1988.
F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal., 32 (2002), 1–26.
J. Ralston, Gaussian beams and the propagation of singularitie, in Studies in Partial Differential Equations, Edited by W. Littman, MAA Studies in Mathematics 23, Washington, DC, 1982, 206–248.
J. Rauch, X. Zhang and E. Zuazua, Polynomial decay of a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407–470.
L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Anal., 10 (1995), 95–115.
M. Slemrod, Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, nonmonotone damping, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 87–97.
X. Zhang and E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system, C.R. Math. Acad. Sci. Paris, 336 (2003), 823–828.
X. Zhang and E. Zuazua, Polynomial decay and control of a 1—d hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380–438.
X. Zhang and E. Zuazua, Long time behavior of a coupled heat-wave system arising in fluid-structure interaction, in submission.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Zhang, X., Zuazua, E. (2006). Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_43
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DOI: https://doi.org/10.1007/978-3-7643-7719-9_43
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