Mathematical analysis of PDE systems which govern fluid-structure interactive phenomena

Boletim da Sociedade Paranaense de Matematica

Volumn 25, Issue 1-2, 2007, Pages 17-36

  Avalos, George ^a   Triggiani, Roberto ^b  

^a UNIVERSITY OF NEBRASKA LINCOLN   (United States)

^b UNIVERSITY OF VIRGINIA   (United States)

Author keywords

Backward uniqueness; Fluid structure interaction; Partial differential equations; Semi groups wellposedness; Stabilization

Indexed keywords

EID: 70350620525     PISSN: 00378712     EISSN: 21751188     Source Type: Journal    
DOI: 10.5269/bspm.v25i1-2.7422     Document Type: Review

References (42)

1. F. Alabau and V. Komornik, "Boundary observability, controllability and stabilization of linear elastodynamic systems", in Contemporary Mathematics 209: Optimization Methods in Partial Differential Equations, Amer. Math. Soc., Providence, RI (1997), pp. 1-8.
2. W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one parameter semigroups, Trans. Amer. Math. Soc. 306 No. 8 (1988), pp. 837-852.
3. G. Avalos, The Exponential Stability of a Coupled Hyperbolic/Parabolic System Arising in Structural Acoustics, Abstract and Applied Analysis, Vol. 1, no. 2 (1996), pp. 203-217.
4. G. Avalos and M. Dvorak, "A divergence free FEM formulation for fluid-structure interac- tions", preprint (2007).
5. G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyper- bolic/parabolic system, Semigroup Forum, Vol. 57 (1998), pp. 278-292.
6. G. Avalos and R. Triggiani, "The coupled PDE system arising in fluid/structure interaction, Part I: explicit semigroup generator and its spectral properties", Contemporary Mathematics 440: Fluids and Waves, American Mathematical Society, Providence Rhode Island (2007), pp. 15-54.
7. G. Avalos and R. Triggiani, "Uniform stabilization of a coupled PDE system arising in fluid- structure interaction with boundary dissipation at the interface", to appear in Discrete and Continuous Dynamical Systems.
8. G. Avalos and R. Triggiani, "Wellposedness and stability analysis of a coupled Stokes-Lame system, as a PDE model of certain fluid-structure interactions, preprint (2007).
9. G. Avalos and R. Triggiani, "The backwards uniqueness property for the C0- semigroup associated with a Stokes-Lame partial differential equation system, preprint (2007).
10. G. Avalos and R. Triggiani, "concerning the interactive boundary feedback stabilization of a coupled Stokes-Lame PDE System", preprint (2007).
11. G. Avalos and R. Triggiani, "Backwards uniqueness of the S.C. semigroup arising in parabolic- hyperbolic fluid-structure interaction, to appear in Journal of Differential Equations.
12. A. V. Balakrishnan, Applied Functional Analysis and Applications, Second Edition, Springer- Verlag, New York (1981).
13. J. M. Ball, "Strongly continuous semigroups, weak solutions and the variation of constants formula", Proceedings of American Math Society, 63, pp. 370-373 (1977).
14. V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, "Weak and strong solutions in nonlinear fluid-structure interactions", Contemporary Mathematics 440: Fluids and Waves, American Mathematical Society, Providence Rhode Island (2007), pp. 55-82.
15. Assia Benabdallah and Irena Lasiecka, "Exponential decay rates for a full von Karman sys- temn of dynamic thermoelasticity", Journal of Differential Equations, 160 (2000), pp. 51-93.
16. K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung. 30 (1995), pp. 162-182.
17. G. Chen, "A note on the boundary stabilization of the wave equation", SIAM J. Control Optim., 19 (1981), pp. 106-113.
18. R. Chill and Y. Tomilov, Stability of operator semigroups: ideas and results, preprint (2005).
19. P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago (1989).
20. C. R. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York (1995).
21. M. Eller, V. Isakov, G. Nakamura, and D. Tataru, "Uniqueness and stability in the Cauchy Problem for Maxwell's and elasticity systems", Nonlinear partial differential equations and their applications, College de France, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., 31, North-Holland, Amsterdam (2002), pp. 329-349.
22. M. A. Horn, "Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity", Jornal of Mathematical Analysis and Applications, 223 (1998), pp. 126-150.
23. M. A. Horn, "Sharp trace regularity for the solutions of the equations of dynamic elasticity", J. Math. Systems, Estim. Control 8 (2), 217-219.
24. V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York (1998).
25. B. Kellogg, "Properties of solutions of elliptic boundary value problems", in The Mathe- matical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Edited by A. K. Aziz, Academic Press, New York (1972), pp. 47-81.
26. S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, New York (1989).
27. Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, "Analysis of a linear fluid-structure inter- action problem", Discrete and Continuous Dynamical Systems, Volume 9, No. 3 (May 2003), pp. 633-650.
28. J. Lagnese, "Decay of solutions of wave equations in a bounded region with boundary dissi- pation", J. Differential Equations 50 (1983), pp. 163-182.
29. I. Lasiecka, R. Triggiani, and M. Renardy, "Backwards uniqueness for thermoelastic plates with rotational forces", Semigroup Forum, Vol. 62 (2001), pp. 217-242.
30. I. Lasiecka and R. Triggiani, "Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions", Appl. Math. Optim. 25 (1992), pp. 189-224.
31. J.L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris (1969).
32. J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Springer-Verlag (1972).
33. W. Littman and B. Liu, "On the spectral properties and stabilization of acoustic flow", IMA, University of Minnesota, IMA Preprint Series #1436, (November, 1996).
34. Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Matematica, LXXXVII (1988), pp. 37-42.
35. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York (1983).
36. J. Quinn and D. L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), pp. 97-127.
37. R. Temam, Navier-Stokes Equations, North Holland, Amsterdam (1978).
38. Y. Tomilov, A resolvent approach to stability of operator semigroups, J. Operator Theory 46 (2001), pp. 63-98.
39. R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, Journal of Mathematical Analysis and Applications 137 (1989), pp. 438-461.
40. J. Zabczyk, Stabilization of boundary control systems, in "International Symposium on Sys- tems Optimization and Analysis, December 1978", Lecture Notes in Control and Information Sciences (A. Bensoussan and J. L. Lions, Eds.), Vol. 14, Springer-Verlag, New York/Berlin.
41. J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser, Boston (1992).
42. I. Lasiecka and R. Triggiani, Uniform stabilization of a shollow shell model with nonlinear boundary feedback. J. Math Analysis and applications, vol. 269 (2002), pp. 642-688.