# American Institute of Mathematical Sciences

July  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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] V. V. Bolotin, "Nonconservative Problems of Elastic Stability,", Pergamon Press, (1963).   Google Scholar [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.  Google Scholar [10] I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (1999).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,", Memoirs of AMS, (2008).   Google Scholar [15] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations,", Sprin\-ger, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar [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}., ().   Google Scholar [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}., ().   Google Scholar [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in, (2011).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model,, Math. Methods Appl. Sci., 32 (2009), 1452.  doi: 10.1002/mma.1104.  Google Scholar [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.  Google Scholar [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.   Google Scholar [28] O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow,", GIFML, (1961).   Google Scholar [29] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [30] J. Lagnese, Modeling and stabilization of nonlinear plates,, Int. Ser. Num. Math., 100 (1991), 247.   Google Scholar [31] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates,", Masson, (1988).   Google Scholar [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.  Google Scholar [33] J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1,, (French), (1968).   Google Scholar [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", (French), (1969).   Google Scholar [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.  Google Scholar [36] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1986).   Google Scholar [37] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar [38] J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM Journal on Control and Optimization, 48 (2010), 5398.  doi: 10.1137/080744761.  Google Scholar [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.  Google Scholar [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [41] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis,", Reprint of the 1984 edition, (1984).   Google Scholar [42] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North Holland, (1978).   Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] V. V. Bolotin, "Nonconservative Problems of Elastic Stability,", Pergamon Press, (1963).   Google Scholar [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.  Google Scholar [10] I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", Acta, (1999).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,", Memoirs of AMS, (2008).   Google Scholar [15] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations,", Sprin\-ger, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar [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}., ().   Google Scholar [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}., ().   Google Scholar [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in, (2011).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model,, Math. Methods Appl. Sci., 32 (2009), 1452.  doi: 10.1002/mma.1104.  Google Scholar [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.  Google Scholar [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.   Google Scholar [28] O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow,", GIFML, (1961).   Google Scholar [29] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [30] J. Lagnese, Modeling and stabilization of nonlinear plates,, Int. Ser. Num. Math., 100 (1991), 247.   Google Scholar [31] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates,", Masson, (1988).   Google Scholar [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.  Google Scholar [33] J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1,, (French), (1968).   Google Scholar [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", (French), (1969).   Google Scholar [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.  Google Scholar [36] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1986).   Google Scholar [37] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar [38] J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM Journal on Control and Optimization, 48 (2010), 5398.  doi: 10.1137/080744761.  Google Scholar [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.  Google Scholar [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [41] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis,", Reprint of the 1984 edition, (1984).   Google Scholar [42] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North Holland, (1978).   Google Scholar
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