# American Institue of Mathematical Sciences

2013, 12(4): 1635-1656. doi: 10.3934/cpaa.2013.12.1635

## A global attractor for a fluid--plate interaction model

 1 Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sq., 61077, Kharkov, Ukraine 2 Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61022, Ukraine

Received  February 2011 Revised  June 2012 Published  November 2012

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system. To achieve the result we first study the corresponding linearized model and show that this linear model generates strongly continuous exponentially stable semigroup.
Citation: I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635
##### References:
 [1] G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163. doi: 10.1007/s00245-006-0884-z. [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction I. Explicit semigroup generator and its spectral properties,, in, 440 (2007), 15. doi: 10.1090/conm/440/08475. [3] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolc-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discr. Contin. Dyn. Sys., 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992). [5] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in, 440 (2007), 55. doi: 10.1090/conm/440/08476. [6] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model,, Indiana Univ. Math. J., 57 (2008), 1173. doi: 10.1512/iumj.2008.57.3284. [7] H. Beirão da Veiga, On the existence of strong solution to a coupled fluid-structure evolution problem,, J. Math. Fluid Mech., 6 (2004), 21. doi: 10.1007/s00021-003-0082-5. [8] V. V. Bolotin, "Nonconservative Problems of Elastic Stability,", Pergamon Press, (1963). [9] A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, J. Math. Fluid Mech., 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. [10] I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (1999). [11] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Math. Meth. Appl. Sci., 34 (2011), 1801. doi: 10.1002/mma.1496. [12] I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, Comm. Pure Appl. Anal., 11 (2012), 659. doi: 10.3934/cpaa.2012.11.659. [13] I. Chueshov and I. Lasiecka, Attractors for second order evolution equations,, J. Dynam. Diff. Eqs., 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. [14] I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,", Memoirs of AMS, (2008). [15] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations,", Sprin\-ger, (2010). doi: 10.1007/978-0-387-87712-9. [16] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, Preprint \arXiv{1204.5864v1}., (). [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, Preprint \arXiv{1112.6094v1}., (). [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in, (2011). [19] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Ration. Mech. Anal., 176 (2005), 25. doi: 10.1007/s00205-004-0340-7. [20] G. Galdi, C. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$,, Math. Annalen, 331 (2005), 41. doi: 10.1007/s00208-004-0573-7. [21] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete Contin. Dyn. Syst., 9 (2003), 633. doi: 10.3934/dcds.2003.9.633. [22] C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, SIAM J. Math. Anal., 40 (2008), 716. doi: 10.1137/070699196. [23] M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, J. Math. Fluid Mech., 10 (2008), 388. doi: 10.1007/s00021-006-0236-4. [24] M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model,, Applicable Analysis, 88 (2009), 1053. doi: 10.1080/00036810903114841. [25] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model,, Math. Methods Appl. Sci., 32 (2009), 1452. doi: 10.1002/mma.1104. [26] M. Guidorzi, M. Padula and P. I. Plotnikov, Hopf solutions to a fluid-elastic interaction model,, Math. Models Methods Appl. Sci., 18 (2008), 215. doi: 10.1142/S0218202508002668. [27] N. Kopachevskii and Yu. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,, Russian J. Math. Phys., 5 (1998), 459. [28] O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow,", GIFML, (1961). [29] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989). doi: 10.1137/1.9781611970821. [30] J. Lagnese, Modeling and stabilization of nonlinear plates,, Int. Ser. Num. Math., 100 (1991), 247. [31] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates,", Masson, (1988). [32] J. Lequeurre, Existence of strong solutions to a fluid-structure system,, SIAM J. Math. Anal. \textbf{43} (2011), 43 (2011), 389. doi: 10.1137/10078983X. [33] J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1,, (French), (1968). [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", (French), (1969). [35] A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction,, ESAIM Control, 4 (1999), 497. doi: 10.1051/cocv:1999119. [36] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1986). [37] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. [38] J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM Journal on Control and Optimization, 48 (2010), 5398. doi: 10.1137/080744761. [39] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura ed Applicata, 148 (1987), 65. doi: 10.1007/BF01762360. [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988). doi: 10.1007/978-1-4684-0313-8. [41] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis,", Reprint of the 1984 edition, (1984). [42] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North Holland, (1978).

show all references

##### References:
 [1] G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163. doi: 10.1007/s00245-006-0884-z. [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction I. Explicit semigroup generator and its spectral properties,, in, 440 (2007), 15. doi: 10.1090/conm/440/08475. [3] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolc-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discr. Contin. Dyn. Sys., 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992). [5] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in, 440 (2007), 55. doi: 10.1090/conm/440/08476. [6] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model,, Indiana Univ. Math. J., 57 (2008), 1173. doi: 10.1512/iumj.2008.57.3284. [7] H. Beirão da Veiga, On the existence of strong solution to a coupled fluid-structure evolution problem,, J. Math. Fluid Mech., 6 (2004), 21. doi: 10.1007/s00021-003-0082-5. [8] V. V. Bolotin, "Nonconservative Problems of Elastic Stability,", Pergamon Press, (1963). [9] A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, J. Math. Fluid Mech., 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. [10] I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (1999). [11] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Math. Meth. Appl. Sci., 34 (2011), 1801. doi: 10.1002/mma.1496. [12] I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, Comm. Pure Appl. Anal., 11 (2012), 659. doi: 10.3934/cpaa.2012.11.659. [13] I. Chueshov and I. Lasiecka, Attractors for second order evolution equations,, J. Dynam. Diff. Eqs., 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. [14] I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,", Memoirs of AMS, (2008). [15] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations,", Sprin\-ger, (2010). doi: 10.1007/978-0-387-87712-9. [16] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, Preprint \arXiv{1204.5864v1}., (). [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, Preprint \arXiv{1112.6094v1}., (). [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in, (2011). [19] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Ration. Mech. Anal., 176 (2005), 25. doi: 10.1007/s00205-004-0340-7. [20] G. Galdi, C. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$,, Math. Annalen, 331 (2005), 41. doi: 10.1007/s00208-004-0573-7. [21] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete Contin. Dyn. Syst., 9 (2003), 633. doi: 10.3934/dcds.2003.9.633. [22] C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, SIAM J. Math. Anal., 40 (2008), 716. doi: 10.1137/070699196. [23] M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, J. Math. Fluid Mech., 10 (2008), 388. doi: 10.1007/s00021-006-0236-4. [24] M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model,, Applicable Analysis, 88 (2009), 1053. doi: 10.1080/00036810903114841. [25] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model,, Math. Methods Appl. Sci., 32 (2009), 1452. doi: 10.1002/mma.1104. [26] M. Guidorzi, M. Padula and P. I. Plotnikov, Hopf solutions to a fluid-elastic interaction model,, Math. Models Methods Appl. Sci., 18 (2008), 215. doi: 10.1142/S0218202508002668. [27] N. Kopachevskii and Yu. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,, Russian J. Math. Phys., 5 (1998), 459. [28] O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow,", GIFML, (1961). [29] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989). doi: 10.1137/1.9781611970821. [30] J. Lagnese, Modeling and stabilization of nonlinear plates,, Int. Ser. Num. Math., 100 (1991), 247. [31] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates,", Masson, (1988). [32] J. Lequeurre, Existence of strong solutions to a fluid-structure system,, SIAM J. Math. Anal. \textbf{43} (2011), 43 (2011), 389. doi: 10.1137/10078983X. [33] J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1,, (French), (1968). [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", (French), (1969). [35] A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction,, ESAIM Control, 4 (1999), 497. doi: 10.1051/cocv:1999119. [36] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1986). [37] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. [38] J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM Journal on Control and Optimization, 48 (2010), 5398. doi: 10.1137/080744761. [39] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura ed Applicata, 148 (1987), 65. doi: 10.1007/BF01762360. [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988). doi: 10.1007/978-1-4684-0313-8. [41] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis,", Reprint of the 1984 edition, (1984). [42] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North Holland, (1978).
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