Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction

Archive for Rational Mechanics and Analysis

Volumn 184, Issue 1, 2007, Pages 49-120

  Zhang, Xu ^a,b   Zuazua, Enrique ^b  

^a SICHUAN UNIVERSITY   (China)

^b UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33847335677     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-006-0020-x     Document Type: Article

References (57)

1. AMMARI, K., TUCSNAK, M.: Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39, 1160-1181 (2000)
2. BARDOS, C., LEBEAU, G., RAUCH, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024-1065 (1992)
3. BEIRÃO DA VEIGA H.: On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mech. 6, 21-52 (2004)
4. BLOM, F.J.: A monolithical fluid-structure interaction algorithm applied to the piston problem. Comput. Methods Appl. Mech. Engrg. 167, 369-391 (1998)
5. BOULAKIA, M.: Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid. C. R. Math. Acad. Sci. Paris 340, 113-118 (2005)
6. BURQ, N.: Contrôlabilité exacte des ondes dans des ouverts peu réguliers. Asymptot. Anal. 14, 157-191 (1997)
7. CAUSIN, P., GERBEAU, J.P., NOBILE, F.: Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 194, 4506-4527 (2005)
8. CHAMBOLLE, A., DESJARDINS, B., ESTEBAN, M.J., GRANDMONT, C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7, 368-404 (2005)
9. CONCA, C., SAN MARTÍN, J., TUCSNAK, M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Partial Differential Equations. 25, 1019-1042 (2000)
10. COUTAND, D., SHKOLLER, S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Ration. Mech. Anal. 176, 25-102 (2005)
11. DAUTRAY, R., LIONS, J.-L.: 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 Séries], Masson, Paris, 1985, With the collaboration of M. Artola, C. Bardos, M. Cessenat, A. Kavenoky, H. Lanchon, P. Lascaux, B. Mercier, O. Pironneau, B. Scheurer and R. Sentis.
12. DESJARDINS, B., ESTEBAN, M.J.: On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Partial Differential Equations 25, 1399-1413 (2000)
13. DESJARDINS, B., ESTEBAN, M.J., GRANDMONT, C., LE TALLEC, P.: Weak solutions for a fluid-elastic structure interaction model. Rev. Mat. Complut. 14, 523-538 (2001)
14. DU, Q., GUNZBURGER, M.D., Hou, L.S., LEE, J.: Semidiscrete finite element approximations of a linear fluid-structure interaction problem. SIAM J. Numer. Anal. 42, 1-29 (2004)
15. DUYCKAERTS, T.: Optimal decay rates of the solutions of hyperbolic-parabolic systems coupled by an interface. Preprint, 2005
16. DUYCKAERTS, T.: private communication
17. FEIREISL, E.: On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167, 281-308 (2003)
18. GERBEAU, J.F., VIDRASCU, M.: A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. M2AN Math. Model. Numer. Anal. 37, 631-647 (2003)
19. GRANDMONT, C., MADAY, Y.: Existence for an unsteady fluid-structure interaction problem. M2AN Math. Model. Numer. Anal. 34, 609-636 (2000)
20. GUNZBURGER, M., LEE, H.-C., SEREGIN, G.A.: Global existence of weak solutions for viscous incompressible flows around amoving rigid body in three dimensions. J. Math. Fluid Mech. 2, 219-266 (2000)
21. HEIL, M.: An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems. Comput. Methods Appl. Mech. Engrg. 193, 1-23 (2004)
22. HOFFMANN, K.-H., STAROVOITOV, V. N.: On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9, 633-648 (1999)
23. HÖRMANDER, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin, 1985
24. HOWE, M.S.: Acoustics of Fluid-Structure Interactions. Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998
25. IMANUVILOV, O.YU., YAMAMOTO, M.: Remarks on Carleman estimates and exact controllability of the Lamé system. Journées "Équations aux Dérivées Partielles" (Forgesles-Eaux, 2002), Exp. No. V, 19 pp., Univ. Nantes, Nantes, 2002
26. ISHIHARA, D., YOSHIMURA, S.: A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation. Internat J. Numer. Methods Engng. 64, 167-203 (2005)
27. LE TALLEC, P., MANI, S.: Numerical analysis of a linearised fluid-structure interaction problem. Numer. Math. 87, 317-354 (2000)
28. LEBEAU, G.: Contrôle analytique I: Estimations a priori. Duke Math. J. 68, 1-30 (1992)
29. LI, T.T., ZHENG, S.M., TAN, Y.J., SHEN, W.X.: Boundary value problems with equivalued surface boundary conditions for self-adjoint elliptic differential equations (I). J. Fudan Univ. Nat. Sci., 1, 61-71 (In Chinese) (1976)
30. LIONS, J.L.: Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1: Contrôlabilité Exacte, RMA, 8, Masson, Paris, 1988
31. LIONS, J.L., MAGENES, E.: Non-Homogeneous Boundary Value Problems and Applications (Vol. I). Springer-Verlag, New York, 1972
32. MACIÀ, F., ZUAZUA, E.: On the lack of observability for wave equations: a Gaussian beam approach. Asymptot. Anal. 32, 1-26 (2002)
33. MATHEWS, I.C.: Numerical techniques for three-dimensional steady-state fluid-structure interaction. J. Acoust. Soc. Amer. 79, 1317-1325 (1986)
34. MORAND, H., OHAYON, R.: Fluid Structure Interaction: Applied Numerical Methods. Wiley, New York, 1995
35. OHAYON, R., FELIPPA, C., eds.: Special Issue on Fluid-Structure Interactions. Comput. Methods Appl. Mech. Engrg. 190, 2977-3292 (2001)
36. PAZY, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983
37. PHUNG, K.D.: Quantification of unique continuation for the wave equation with Dirichlet boundary condition. Unpublished manuscript, 2004
38. RALSTON, J.: Gaussian Beams and the Propagation of Singularities. Studies in Partial Differential Equations (Edited by W. LITTMAN), MAA Studies in Mathematics, 23, Washington, DC, 1982, 206-248
39. RAUCH, J.: Qualitative behavior of dissipative wave equations on bounded domains. Arch. Ration. Mech. Anal. 62, 77-85 (1976)
40. RAUCH, J., ZHANG, X., ZUAZUA, E.: Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl. 84, 407-470 (2005)
41. ROBBIANO, L.: Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10, 95-115 (1995)
42. ROGERS, E.H.: A minimax theory for overdamped systems. Arch. Ration. Mech. Anal. 16, 89-96 (1964)
43. SAN MARTÍN, J., STAROVOITOV, V., TUCSNAK, M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161, 113-147 (2002)
44. SERRE, D.: Chute libre d'un solide dans un fluide visqueux incompressible. Existence. Japan J. Appl. Math. 4, 99-110 (1987)
45. SLEMROD, M.: Weak asymptotic decay via a "relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping. Proc. Roy. Soc. Edinburgh Sect. A 113, 87-97 (1989)
46. STEIN, K. TEZDUYAR, T.E., SATHE, S., BENNEY, R., CHARLES, R.: Fluid-structure interaction modelling of parachute soft-landing dynamics. Internat. J. Numer. Methods Fluids 47, 619-631 (2005)
47. TABACHNIKOV, S.: Billiards. Panor. Synth. No. 1, 1995
48. TAKAHASHI, T.: Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differential Equations 8, 1499-1532 (2003)
49. TAKAHASHI, T., TUCSNAK, M.: Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6, 53-77 (2004)
50. TAYLOR, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981
51. VAN LOON, R., ANDERSON, P.D., DE HART, J., BAAIJENS, P.P.T.: A combined fictitious domain/adaptive meshing method for fluid-structure interaction in heart valves. Internat. J. Numer. Methods Fluids 46, 533-544 (2004)
52. VÁZQUEZ, J.-L., ZUAZUA, E.: Large time behavior for a simplified ID model of fluid-solid interaction. Comm. Partial Differential Equations 28, 1705-1738 (2003)
53. XIA, D., LI, S., YAN, S., SHU, W.: Theory of Real Functions and Functional Analysis, Vol. 2 (Second version). Gaodeng Jiaoyu Chubanshe (Higher Education Press), Beijing, 1985 (In Chinese)
54. ZHANG, X., Li, F.: Existence, uniqueness and limit behavior of solutions to a nonlinear boundary-value problem with equivalued surface. Nonlinear Anal. 34, 525-536 (1998)
55. ZHANG, X., ZUAZUA, E.: Control, observation and polynomial decay for a coupled heat-wave system. C. R. Math. Acad. Sci. Paris 336, 823-828 (2003)
56. ZIEMER, W.P.: Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120, Springer-Verlag, Berlin, 1989
57. ZUAZUA, E.: Controllability and Observability of Partial Differential Equations: Some Results and Open Problems. Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, Elsevier Science, to appear