Strong stability for a fluid-structure interaction model

Mathematical Methods in the Applied Sciences

Volumn 32, Issue 11, 2009, Pages 1452-1466

  Grobbelaar Van Dalsen, Marié ^a  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

Author keywords

Fluid solid; Fluid structure model; Interaction; Resolvent criterion; Strong stability

Indexed keywords

DISSIPATIVE MECHANISMS; FLEXIBLE PARTS; FLUID-STRUCTURE MODEL; INCOMPRESSIBLE FLUID; INTERACTION; RESOLVENT CRITERION; STOKES EQUATIONS; STRONG STABILITY;

FLEXIBLE STRUCTURES; FLUID STRUCTURE INTERACTION; MATHEMATICAL MODELS; PLATES (STRUCTURAL COMPONENTS); THREE DIMENSIONAL; TWO DIMENSIONAL;

FLUIDS;

EID: 67650838205     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.1104     Document Type: Article

References (36)

1. You Y. Pointwise boundary stabilizability of hyperbolic evolution equations: two-dimensional hybrid elastic structures. Journal of Mathematical Analysis and Applications 1992; 165:239-265.
2. Lagnese JE, Leugering G, Schmidt EJPG. Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures. Birkhauser: Boston, 1994.
3. Grobbelaar-Van Dalsen M. Dynamic boundary stabilization of a Reissner-Mindlin plate with Timoshenko beam. Mathematical Methods in the Applied Sciences 2004; 27:1301-1315.
4. Avalos G. The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract and Applied Analysis 1996; 1:203-217.
5. Lasiecka I. Mathematical control theory in structural acoustic problems. Mathematical Models and Methods in Applied Sciences 1998; 8:1119-1153.
6. Lasiecka I. Boundary stabilization of a three-dimensional structural acoustics model. Journal de Mathematiques Pures et Appliquees 1999; 78:203-232.
7. Lasiecka I, Lebiedzik C. Decay rates on interactive hyperbolic-parabolic interactive PDE models with thermal effects on the interface. Applied Mathematics and Optimization 2000; 42:127-167.
8. Lasiecka I, Lebiedzik C. Asymptotic behaviour of nonlinear structural acoustic interactions with thermal effects. Nonlinear Analysis 2002; 49:703-735.
9. Grobbelaar-Van Dalsen M. On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam. Journal of Mathematical Analysis and Applications 2005; 320:121-144.
10. Grobbelaar-Van Dalsen M. Uniform stabilization of a nonlinear structural acoustic model with Timoshenko beam interface. Mathematical Methods in the Applied Sciences 2006; 29:1749-1766.
11. Howe MS. Acoustics of Fluid-Structure Interaction. University Press: Cambridge, MA, 1998.
12. Grobbelaar-Van Dalsen M. On a structural acoustic model which incorporates shear and thermal effects in the structural component. Journal of Mathematical Analysis and Applications 2008; 341:1253-1270.
13. Dautray R, Lions J. Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer: Berlin, 1985.
14. Sun X, Yu C, Wang Y, Liu Y. Numerical simulation of soft palate movement and airflow in human upper airway by fluid-structure interaction method. Acta Mechanica Sinica 2007; 23:359-367.
15. Conca C, San Martin J, Tucsnak M. Existence of solutions for the equations modelling the motions of a rigid body in a viscous fluid. Communications in Partial Differential Equations 2000; 25:1019-1042.
16. Beirao da Veiga H. On the existence of strong solutions to a coupled fluid-structure problem. Journal of Mathematical Fluid Mechanics 2004; 6:21-52.
17. Chambolle A, Desjardins B, Esteban MJ, Grandmont C. Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. Journal of Mathematical Fluid Mechanics 2005; 7:368-404.
18. Du Q, Gunzburger MD, Hou LS, Lee J. Analysis of a linear fluid-structure interaction problem. Discrete and Continuous Dynamical Systems 2003; 9:633-650.
19. Barbu V, Grujic G, Lasiecka I, Tuffaha A. Existence of the Energy-level Weak Solutions for a Nonlinear Fluid-Structure Interaction Model. Contemporary Mathematics, vol. 440. American Mathematical Society: Providence, RI, 2007; 55-82.
20. Avalos G, Triggiani R. The Coupled PDE System Arising in Fluid-Structure Interaction. Part I: Explicit Semigroup Generator and its Spectral Properties. Contemporary Mathematics, vol. 440. American Mathematical Society: Providence, RI, 2007; 163-184.
21. Avalos G. The strong stability of a fluid-structure semigroup. Applied Mathematics and Optimization 2007; 55:163-184.
22. Tomilov Y. A resolvent approach to stability of operator semigroups. Journal of Operator Theory 2001; 46:63-98.
23. Arendt W, Batty CJK. Tauberian theorems and stability of one-parameter semigroups. Transactions of the American Mathematical Society 1988; 306:837-852.
24. Arendt W, Batty CJK, Hieber M, Neubrander F. Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhauser: Basel, 2001.
25. Zhang X, Zuazua E. Long-time behaviour of a coupled heat-wave system arising in fluid-structure interaction. Archives for Rational Mechanics and Analysis 2007; 184:49-120.
26. Slemrod M. Weak asymptotic decay via a 'relaxed invariance principle' for a wave equation with nonlinear, monotone damping. Proceedings of the Royal Society of Edinburgh Section A 1989; 113:87-97.
27. Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. I. Cambridge University Press: Cambridge, 2000.
28. Grobbelaar-Van Dalsen M. On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid. Journal of Mathematical Fluid Mechanics 2008; 10:388-401.
29. Pedley TJ. The Fluid Mechanics of Large Blood Vessels. Cambridge University Press: Cambridge, 1980.
30. Sz-Nagy B, Foias C. Harmonic Analysis of Operators on Hilbert Space. North-Holland: Amsterdam, 1970.
31. Cremer L, Heckl M, Ungar EE. Structure-Borne Sound (2nd edn). Springer: Berlin, 1988.
32. Lagnese J. Boundary Stabilization of Thin Plates. SIAM: Philadelphia, PA, 1989.
33. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer: New York, 1983.
34. Sohr H. The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhauser: Boston, 2001.
35. Chill R, Tomilov Y. Stability of Operator Semigroups: Ideas and Results. Perspectives in Operator Theory, vol. 75. Banach Center Publications, Polish Academy of Science: Warsaw, 2007; 71-109.
36. Boyadzhiev KN, Levan N. Strong stability of Hilbert space contraction semigroups. Studia Scientiarum Mathematicarum Hungariae 1995; 30:165-182.