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, Amsterdam, 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, J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.  Google Scholar [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, 2002.  Google Scholar [4] 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/.  Google Scholar [5] 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.  Google Scholar [6] I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis, 95 (2014), 650-665. 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, Comm. Pure Appl. Anal., 11 (2012), 659-674. 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), 469-512. 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, J. Diff. Eqs., 233 (2007), 42-86. 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, vol.195, no. 912, AMS, Providence, RI, 2008. doi: 10.1090/memo/0912.  Google Scholar [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 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, Commun. Partial Dif. Eqs, 36 (2011), 67-99. 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, J. Abstr. Differ. Equ. Appl., 3 (2012), 1-27.  Google Scholar [14] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, in Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations (HCDTE Lecture Notes, Part I), AIMS on Applied Mathematics Vol.6, (eds G. Alberti et al.) AIMS, Springfield, 2013, 1-96. Google Scholar [15] I. Chueshov, I. Lasiecka and J. Webster, Evolution semigroups in supersonic flow-plate interactions, J. Diff. Eqs., 254 (2013), 1741-1773. 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. 12 (2013), 1635-1656. 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, J. Diff. Eqs., 254 (2013), 1833-1862. 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, Ukrainian Math. J., 65 (2013), 158-177 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 IFIP Advances in Information and Communication Technology, vol.391, (25th IFIP TC7 Conference, Berlin, Sept.2011), (eds D. Hömberg and F. Tröltzsch), Springer, Berlin, 2013, 328-337. Google Scholar [20] I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations, Evolution Equations and Control Theory, 1 (2012), 57-80. 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, New York, 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, Commun. Partial Dif. Eqs., 34 (2009), 137-170. doi: 10.1080/03605300802608247.  Google Scholar [23] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, 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, Birkhäuser, Basel, 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, SIAM Publications, 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, Paris, 1968. Google Scholar [27] J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair, Dunod, Paris, 1969.  Google Scholar [28] B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed., Taylor & Francis, New York, 2006. Google Scholar [29] 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.  Google Scholar [30] 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.  Google Scholar [31] R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer, New York, 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, Providence, RI, 2001.  Google Scholar [33] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.  Google Scholar [34] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach, Nonlinear Analysis, 74 (2011), 3123-3136. doi: 10.1016/j.na.2011.01.028.  Google Scholar

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References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 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, J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.  Google Scholar [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, 2002.  Google Scholar [4] 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/.  Google Scholar [5] 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.  Google Scholar [6] I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis, 95 (2014), 650-665. 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, Comm. Pure Appl. Anal., 11 (2012), 659-674. 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), 469-512. 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, J. Diff. Eqs., 233 (2007), 42-86. 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, vol.195, no. 912, AMS, Providence, RI, 2008. doi: 10.1090/memo/0912.  Google Scholar [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 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, Commun. Partial Dif. Eqs, 36 (2011), 67-99. 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, J. Abstr. Differ. Equ. Appl., 3 (2012), 1-27.  Google Scholar [14] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, in Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations (HCDTE Lecture Notes, Part I), AIMS on Applied Mathematics Vol.6, (eds G. Alberti et al.) AIMS, Springfield, 2013, 1-96. Google Scholar [15] I. Chueshov, I. Lasiecka and J. Webster, Evolution semigroups in supersonic flow-plate interactions, J. Diff. Eqs., 254 (2013), 1741-1773. 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. 12 (2013), 1635-1656. 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, J. Diff. Eqs., 254 (2013), 1833-1862. 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, Ukrainian Math. J., 65 (2013), 158-177 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 IFIP Advances in Information and Communication Technology, vol.391, (25th IFIP TC7 Conference, Berlin, Sept.2011), (eds D. Hömberg and F. Tröltzsch), Springer, Berlin, 2013, 328-337. Google Scholar [20] I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations, Evolution Equations and Control Theory, 1 (2012), 57-80. 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, New York, 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, Commun. Partial Dif. Eqs., 34 (2009), 137-170. doi: 10.1080/03605300802608247.  Google Scholar [23] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, 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, Birkhäuser, Basel, 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, SIAM Publications, 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, Paris, 1968. Google Scholar [27] J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair, Dunod, Paris, 1969.  Google Scholar [28] B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed., Taylor & Francis, New York, 2006. Google Scholar [29] 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.  Google Scholar [30] 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.  Google Scholar [31] R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer, New York, 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, Providence, RI, 2001.  Google Scholar [33] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.  Google Scholar [34] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach, Nonlinear Analysis, 74 (2011), 3123-3136. doi: 10.1016/j.na.2011.01.028.  Google Scholar
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