# American Institute of Mathematical Sciences

September  2014, 13(5): 1759-1778. doi: 10.3934/cpaa.2014.13.1759

## Interaction of an elastic plate with a linearized inviscid incompressible fluid

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

Received  June 2013 Revised  September 2013 Published  June 2014

We prove well-posedness of energy type solutions to an interacting system consisting of the 3D linearized Euler equations and a (possibly nonlinear) elastic plate equation describing large deflections of a flexible part of the boundary. In the damped case under some conditions concerning the plate nonlinearity we prove the existence of a compact global attractor for the corresponding dynamical system and describe the situations when this attractor has a finite fractal dimension.
Citation: I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759
##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar [2] 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,, \emph{J. Math. Fluid Mech.}, 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS, (2002). Google Scholar [4] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999). Google Scholar [5] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, \emph{Math. Meth. Appl. Sci.}, 34 (2011), 1801. doi: 10.1002/mma.1496. Google Scholar [6] I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid,, \emph{Nonlinear Analysis}, 95 (2014), 650. doi: 10.1016/j.na.2013.10.018. Google Scholar [7] I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \emph{Comm. Pure Appl. Anal.}, 11 (2012), 659. doi: 10.3934/cpaa.2012.11.659. Google Scholar [8] 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 [9] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping,, \emph{J. Diff. Eqs.}, 233 (2007), 42. doi: 10.1016/j.jde.2006.09.019. Google Scholar [10] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Memoirs of AMS, (2008). doi: 10.1090/memo/0912. Google Scholar [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [12] I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, \emph{Commun. Partial Dif. Eqs}, 36 (2011), 67. doi: 10.1080/03605302.2010.484472. Google Scholar [13] I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions,, \emph{J. Abstr. Differ. Equ. Appl.}, 3 (2012), 1. Google Scholar [14] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, in \emph{Nonlinear Hyperbolic PDEs, (2013), 1. Google Scholar [15] I. Chueshov, I. Lasiecka and J. Webster, Evolution semigroups in supersonic flow-plate interactions,, \emph{J. Diff. Eqs.}, 254 (2013), 1741. doi: 10.1016/j.jde.2012.11.009. Google Scholar [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Comm. Pure Appl. Anal. \textbf{12} (2013), 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, \emph{J. Diff. Eqs.}, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. Google Scholar [18] I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille type flow,, \emph{Ukrainian Math. J.}, 65 (2013), 158. doi: 10.1007/s11253-013-0771-0. Google Scholar [19] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in \emph{IFIP Advances in Information and Communication Technology}, (2013), 328. Google Scholar [20] I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations,, \emph{Evolution Equations and Control Theory}, 1 (2012), 57. doi: 10.3934/eect.2012.1.57. Google Scholar [21] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd ed.,, Springer, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar [22] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Commun. Partial Dif. Eqs.}, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar [23] M. S. Howe, Acoustics of Fluid-Structure Interactions,, Cambridge University Press, (1998). doi: 10.1017/CBO9780511662898. Google Scholar [24] N. Kopachevskii and S. Krein, Operator Approach to Linear Problems of Hydrodynamics: Volume 1: Self-adjoint Problems for an Ideal Fluid,, Birkh\, (2001). doi: 10.1007/978-3-0348-8342-9. Google Scholar [25] I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2001). doi: 10.1137/1.9780898717099. Google Scholar [26] J.-L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Vol. 1,, Dunod, (1968). Google Scholar [27] J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair,, Dunod, (1969). Google Scholar [28] B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed.,, Taylor & Francis, (2006). Google Scholar [29] A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction,, \emph{ESAIM: Control, 4 (1999), 497. doi: 10.1051/cocv:1999119. Google Scholar [30] J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Matematica Pura ed Applicata}, 148 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [31] R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [32] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition,, AMS Chelsea Publishing, (2001). Google Scholar [33] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators,, North Holland, (1978). Google Scholar [34] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach,, \emph{Nonlinear Analysis}, 74 (2011), 3123. doi: 10.1016/j.na.2011.01.028. Google Scholar

show all references

##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar [2] 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,, \emph{J. Math. Fluid Mech.}, 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS, (2002). Google Scholar [4] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999). Google Scholar [5] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, \emph{Math. Meth. Appl. Sci.}, 34 (2011), 1801. doi: 10.1002/mma.1496. Google Scholar [6] I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid,, \emph{Nonlinear Analysis}, 95 (2014), 650. doi: 10.1016/j.na.2013.10.018. Google Scholar [7] I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \emph{Comm. Pure Appl. Anal.}, 11 (2012), 659. doi: 10.3934/cpaa.2012.11.659. Google Scholar [8] 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 [9] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping,, \emph{J. Diff. Eqs.}, 233 (2007), 42. doi: 10.1016/j.jde.2006.09.019. Google Scholar [10] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Memoirs of AMS, (2008). doi: 10.1090/memo/0912. Google Scholar [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [12] I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, \emph{Commun. Partial Dif. Eqs}, 36 (2011), 67. doi: 10.1080/03605302.2010.484472. Google Scholar [13] I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions,, \emph{J. Abstr. Differ. Equ. Appl.}, 3 (2012), 1. Google Scholar [14] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, in \emph{Nonlinear Hyperbolic PDEs, (2013), 1. Google Scholar [15] I. Chueshov, I. Lasiecka and J. Webster, Evolution semigroups in supersonic flow-plate interactions,, \emph{J. Diff. Eqs.}, 254 (2013), 1741. doi: 10.1016/j.jde.2012.11.009. Google Scholar [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Comm. Pure Appl. Anal. \textbf{12} (2013), 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, \emph{J. Diff. Eqs.}, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. Google Scholar [18] I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille type flow,, \emph{Ukrainian Math. J.}, 65 (2013), 158. doi: 10.1007/s11253-013-0771-0. Google Scholar [19] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in \emph{IFIP Advances in Information and Communication Technology}, (2013), 328. Google Scholar [20] I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations,, \emph{Evolution Equations and Control Theory}, 1 (2012), 57. doi: 10.3934/eect.2012.1.57. Google Scholar [21] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd ed.,, Springer, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar [22] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Commun. Partial Dif. Eqs.}, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar [23] M. S. Howe, Acoustics of Fluid-Structure Interactions,, Cambridge University Press, (1998). doi: 10.1017/CBO9780511662898. Google Scholar [24] N. Kopachevskii and S. Krein, Operator Approach to Linear Problems of Hydrodynamics: Volume 1: Self-adjoint Problems for an Ideal Fluid,, Birkh\, (2001). doi: 10.1007/978-3-0348-8342-9. Google Scholar [25] I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2001). doi: 10.1137/1.9780898717099. Google Scholar [26] J.-L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Vol. 1,, Dunod, (1968). Google Scholar [27] J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair,, Dunod, (1969). Google Scholar [28] B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed.,, Taylor & Francis, (2006). Google Scholar [29] A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction,, \emph{ESAIM: Control, 4 (1999), 497. doi: 10.1051/cocv:1999119. Google Scholar [30] J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Matematica Pura ed Applicata}, 148 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [31] R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [32] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition,, AMS Chelsea Publishing, (2001). Google Scholar [33] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators,, North Holland, (1978). Google Scholar [34] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach,, \emph{Nonlinear Analysis}, 74 (2011), 3123. doi: 10.1016/j.na.2011.01.028. Google Scholar
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