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Logarithmic decay of the energyfor an hyperbolic-parabolic coupled system

Published online by Cambridge University Press:  06 August 2010

Ines Kamoun Fathallah
Ines Kamoun Fathallah*
Affiliation:
Laboratoire LMV, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue des États-Unis, Bâtiment Fermat, 78035 Versailles, France. [email protected]
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Abstract

This paper is devoted to the study of a coupled system which consists ofa wave equation and a heat equation coupled through a transmission conditionalong a steady interface. This system is a linearized model forfluid-structure interaction introduced by Rauch, Zhang and Zuazuafor a simple transmission condition and by Zhang and Zuazua for anatural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, wecomplete the results obtained by Zhang and Zuazua and by Duyckaerts.We prove, without a Geometric Control Condition, a logarithmic decayof the energy.

Keywords

Fluid-structure interactionwave-heatmodelstabilitylogarithmic decay
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 801 - 835
Copyright
© EDP Sciences, SMAI, 2010

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References

Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. CrossRef
Bellassoued, M., Distribution of resonances and decay rate of the local energy for the elastic wave equation. Comm. Math. Phys. 215 (2000) 375–408. CrossRef
Bellassoued, M., Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptot. Anal. 35 (2003) 257–279.
Bellassoued, M., Decay of solutions of the elastic wave equation with a localized dissipation. Ann. Fac. Sci. Toulouse Math. 12 (2003) 267–301. CrossRef
Burq, N., Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998) 1–29. CrossRef
Duyckaerts, T., Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface. Asymptot. Anal. 51 (2007) 17–45.
Logarithmic de, X Fucay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34 (2009) 957–975.
J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations. Arch. Ration. Mech. Anal. (to appear).
G. Lebeau, Équation des ondes amorties, in Algebraic and geometric methods in mathematical physics Kaciveli, 1993, Kluwer Acad. Publ., Dordrecht, Math. Phys. Stud. 19 (1996) 73–109.
Lebeau, G. and Robbiano, L., Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. CrossRef
Lebeau, G. and Robbiano, L., Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465–491. CrossRef
Rauch, J., Zhang, X. and Zuazua, E., Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl. 84 (2005) 407–470. CrossRef
Robbiano, L., Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95–115.
Taylor, M.E., Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 (1975) 457–478. CrossRef
Zhang, X. and Zuazua, E., Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal. 184 (2007) 49–120. CrossRef