# 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 and 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), 163-184. 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 "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 15-54. 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., Ser.S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 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 "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 55-82. 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), 1173-207. 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), 21-52. doi: 10.1007/s00021-003-0082-5. [8] V. V. Bolotin, "Nonconservative Problems of Elastic Stability," Pergamon Press, Oxford, 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), 368-404. doi: 10.1007/s00021-004-0121-y. [10] I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/. [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), 1801-1812. 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), 659-674. 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), 469-512. 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, vol.195, no. 912, AMS, Providence, RI, 2008. [15] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations," Springer, New York, 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 "System Modeling and Optimization: 25th IFIP TC7 Conference, Berlin, Germany, Sept. 2011," Springer, in press. [19] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. 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), 41-74. 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), 633-650. 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), 716-737 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), 388-401. 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), 1053-1065. doi: 10.1080/00036810903114841. [25] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. 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), 215-269. 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), 459-472. [28] O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow," GIFML, Moscow, 1961 (1st Russian edition); Nauka, Moscow, 1970 (2nd Russian edition); Gordon and Breach, New York, 1963 and 1969 (English translations of the 1st Russian edition). [29] J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970821. [30] J. Lagnese, Modeling and stabilization of nonlinear plates, Int. Ser. Num. Math., 100 (1991), 247-264. [31] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates," Masson, Paris, 1988. [32] J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal. 43 (2011), 389-410. doi: 10.1137/10078983X. [33] J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1, (French), Dunod, Paris, 1968. [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod, Paris, 1969. [35] A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction, ESAIM Control, Optimisation and Calculus of Variations, 4 (1999), 497-513. doi: 10.1051/cocv:1999119. [36] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1986. [37] G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems," Elsevier Sciences, Amsterdam, 2 (2002), 885-992. 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), 5398-5443. doi: 10.1137/080744761. [39] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, Ser. 4, 148 (1987), 65-96. doi: 10.1007/BF01762360. [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [41] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [42] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978.

show all references

##### References:
 [1] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. 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 "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 15-54. 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., Ser.S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 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 "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 55-82. 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), 1173-207. 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), 21-52. doi: 10.1007/s00021-003-0082-5. [8] V. V. Bolotin, "Nonconservative Problems of Elastic Stability," Pergamon Press, Oxford, 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), 368-404. doi: 10.1007/s00021-004-0121-y. [10] I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/. [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), 1801-1812. 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), 659-674. 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), 469-512. 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, vol.195, no. 912, AMS, Providence, RI, 2008. [15] I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations," Springer, New York, 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 "System Modeling and Optimization: 25th IFIP TC7 Conference, Berlin, Germany, Sept. 2011," Springer, in press. [19] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. 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), 41-74. 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), 633-650. 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), 716-737 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), 388-401. 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), 1053-1065. doi: 10.1080/00036810903114841. [25] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. 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), 215-269. 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), 459-472. [28] O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow," GIFML, Moscow, 1961 (1st Russian edition); Nauka, Moscow, 1970 (2nd Russian edition); Gordon and Breach, New York, 1963 and 1969 (English translations of the 1st Russian edition). [29] J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970821. [30] J. Lagnese, Modeling and stabilization of nonlinear plates, Int. Ser. Num. Math., 100 (1991), 247-264. [31] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates," Masson, Paris, 1988. [32] J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal. 43 (2011), 389-410. doi: 10.1137/10078983X. [33] J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1, (French), Dunod, Paris, 1968. [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod, Paris, 1969. [35] A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction, ESAIM Control, Optimisation and Calculus of Variations, 4 (1999), 497-513. doi: 10.1051/cocv:1999119. [36] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1986. [37] G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems," Elsevier Sciences, Amsterdam, 2 (2002), 885-992. 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), 5398-5443. doi: 10.1137/080744761. [39] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, Ser. 4, 148 (1987), 65-96. doi: 10.1007/BF01762360. [40] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [41] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [42] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978.
 [1] Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151 [2] Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59 [3] Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7001-7020. doi: 10.3934/dcds.2016104 [4] Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 [5] Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189 [6] Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012 [7] Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022062 [8] Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045 [9] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [10] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [11] Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 [12] Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081 [13] Yutaka Tsuzuki. Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains. Conference Publications, 2015, 2015 (special) : 1079-1088. doi: 10.3934/proc.2015.1079 [14] Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481 [15] Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093 [16] C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253 [17] I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759 [18] M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503 [19] A. Jiménez-Casas, Mario Castro, Justine Yassapan. Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid. Conference Publications, 2013, 2013 (special) : 375-384. doi: 10.3934/proc.2013.2013.375 [20] Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151

2021 Impact Factor: 1.273