Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface

Asymptotic Analysis

Volumn 51, Issue 1, 2007, Pages 17-45

  Duyckaerts, Thomas ^a  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

Asymptotic stability; Coupling; Fluid structure interaction; Hyperbolic equations; Non uniform stability; Observability; Parabolic equations

Indexed keywords

COMPUTATIONAL GEOMETRY; INTERFACES (MATERIALS); LINEAR SYSTEMS; OPTIMIZATION; POLYNOMIAL APPROXIMATION; WAVE EQUATIONS;

CARLEMAN INEQUALITY; HYPERBOLIC-PARABOLIC SYSTEM; POLYNOMIAL DECAY; TRANSMISSION CONDITIONS;

FLUID STRUCTURE INTERACTION;

EID: 33846165814     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

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(5) (1992), 1024-1065.
2. M. Boulakia, Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid, C. R. Math Acad. Sci. Paris 340(2) (2005), 113-118.
3. N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptotic Anal. 14(2) (1997), 157-191.
4. 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(1) (1998), 1-29.
5. N. Burq and G. Lebeau, Mesures de défaut de compacité, application au système de Lamé, Ann. Sci. École Norm. Sup. (4) 34(6) (2001), 817-870.
6. C. Conca, H.J. San Martín and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25(5-6) (2000), 1019-1042.
7. R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3, Collection du Commissariat à l'Énergie Atomique: Série Scientifique [Collection of the Atomic Energy Commission: Science Series]. Masson, Paris, 1985. With the collaboration of Michel Artola, Claude Bardos, Michel Cessenat, Alain Kavenoky, Hélène Lanchon, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer and Rémi Sentis.
8. B. Desjardins and M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Rational Mech. Anal 146(1) (1999), 59-71.
9. B. Desjardins and M.J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations 25(7-8) (2000), 1399-1413.
10. T. Duyckaerts, Stabilization of the linear system of magneto-elasticity. Preprint. Available at: http://www.u-cergy.fr/rech/pages/duyckaerts, 2004.
11. A.V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Transl. Math. Monogr., Vol. 187, Amer. Math. Soc., Providence, RI, 2000. Translated from the 1999 Russian original by Tamara Rozhkovskaya.
12. D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Comm. Partial Differential Equations 27(7-8) (2002), 1283-1299.
13. L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], Vol. 274, Springer, Berlin, 1994. Pseudo-differential operators, Corrected reprint of the 1985 original.
14. B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered piezoelectric 3-d body by boundary dissipation, Preprint, 2004.
15. H. Koch and E. Zuazua, A multi-body system coupling wave equations in n and n - 1 space dimensions, Preprint, 2005.
16. G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations 20(1-2) (1995), 335-356.
17. G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J. 86(3) (1997), 465-491.
18. G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal. 148(3) (1999), 179-231.
19. J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30(1) (1988), 1-68.
20. L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9) 79(3) (2000), 227-269.
21. I.E. Muñoz Rivera and M. De Lima Santos, Polynomial stability to three-dimensional magnetoelastic waves, Acta Appl. Math. 76(3) (2003), 265-281.
22. J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., to appear (2005).
23. D.L. Russell, Decay rates for weakly damped systems in Hubert spaces obtained by control theoretic methods, J. Differential Equations 19 (1975), 344-370.
24. D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math. 4(1) (1987), 99-110.
25. J.L. Vázquez and E. Zuazua, Large time behavior for a simplified ID model of fluid-solid interaction, Comm. Partial Differential Equations 28(9-10) (2003), 1705-1738.
26. X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math. 5(1) (2003), 25-83.
27. J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl. (9) 84(4) (2005), 407-470.