Logarithmic decay of the energy for an hyperbolic-parabolic coupled system
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 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 : https://doi.org/10.1051/cocv/2010026
Classification:  37L15,  35B37,  74F10,  93D20
Keywords: fluid-structure interaction, wave-heat model, stability, logarithmic decay
@article{COCV_2011__17_3_801_0,
author = {Fathallah, Ines Kamoun},
title = {Logarithmic decay of the energy for an hyperbolic-parabolic coupled system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {801-835},
doi = {10.1051/cocv/2010026},
zbl = {1223.37098},
mrnumber = {2826981},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_801_0}
}

Fathallah, Ines Kamoun. Logarithmic decay of the energy for an hyperbolic-parabolic coupled system. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 801-835. doi : 10.1051/cocv/2010026. http://www.numdam.org/item/COCV_2011__17_3_801_0/

[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 1178650 | Zbl 0786.93009

[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 1799852 | Zbl 0978.35077

[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 2011790 | Zbl 1137.35388

[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 2030088 | Zbl 1073.35036

[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 1618254 | Zbl 0918.35081

[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 2294103 | Zbl 1227.35062

[7] X Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34 (2009) 957-975. | MR 2560307 | Zbl 1180.35104

[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 2591978 | Zbl 1202.35336

[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 1385677 | Zbl 0863.58068

[10] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335-356. | MR 1312710 | Zbl 0819.35071

[11] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR 1432305 | Zbl 0884.58093

[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 2132724 | Zbl 1077.35030

[13] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115. | MR 1324385 | Zbl 0882.35015

[14] M.E. Taylor, Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 (1975) 457-478. | MR 509098 | Zbl 0332.35058

[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 2289863 | Zbl 1178.74075