Logarithmic decay of the energy for an hyperbolic-parabolic coupled system
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 801-835.

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

DOI : 10.1051/cocv/2010026
Classification : 37L15, 35B37, 74F10, 93D20
Mots-clés : fluid-structure interaction, wave-heat model, stability, logarithmic decay 
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     title = {Logarithmic decay of the energy for an hyperbolic-parabolic coupled system},
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Fathallah, Ines Kamoun. Logarithmic decay of the energy for an hyperbolic-parabolic coupled system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 801-835. doi : 10.1051/cocv/2010026. http://www.numdam.org/articles/10.1051/cocv/2010026/

[1] 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. | MR | Zbl

[2] M. Bellassoued, Distribution of resonances and decay rate of the local energy for the elastic wave equation. Comm. Math. Phys. 215 (2000) 375-408. | MR | Zbl

[3] M. Bellassoued, Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptot. Anal. 35 (2003) 257-279. | MR | Zbl

[4] M. Bellassoued, Decay of solutions of the elastic wave equation with a localized dissipation. Ann. Fac. Sci. Toulouse Math. 12 (2003) 267-301. | Numdam | MR | Zbl

[5] N. Burq, 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. | MR | Zbl

[6] T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface. Asymptot. Anal. 51 (2007) 17-45. | MR | Zbl

[7] X Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34 (2009) 957-975. | MR | Zbl

[8] 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). | MR | Zbl

[9] 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. | MR | Zbl

[10] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335-356. | MR | Zbl

[11] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR | Zbl

[12] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl. 84 (2005) 407-470. | MR | Zbl

[13] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115. | MR | Zbl

[14] M.E. Taylor, Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 (1975) 457-478. | MR | Zbl

[15] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal. 184 (2007) 49-120. | MR | Zbl

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