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

Interaction of an elastic plate with a linearized inviscid incompressible fluid

I. D. Chueshov 1,

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.
Keywords: nonlinear plate, linearized 3D Euler equations, global attractor., well-posedness, Fluid--structure interaction.
Mathematics Subject Classification: Primary: 74F10; Secondary: 35B41, 35Q30, 74K2.
Citation: 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
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[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.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, 2002.

[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/.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511662898.

[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.

[25]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM Publications, 2001. doi: 10.1137/1.9780898717099.

[26]

J.-L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Vol. 1, Dunod, Paris, 1968.

[27]

J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair, Dunod, Paris, 1969.

[28]

B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed., Taylor & Francis, New York, 2006.

[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.

[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.

[31]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[32]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

[33]

H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.

[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.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[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.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, 2002.

[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/.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511662898.

[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.

[25]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM Publications, 2001. doi: 10.1137/1.9780898717099.

[26]

J.-L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Vol. 1, Dunod, Paris, 1968.

[27]

J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair, Dunod, Paris, 1969.

[28]

B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed., Taylor & Francis, New York, 2006.

[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.

[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.

[31]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[32]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

[33]

H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.

[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.

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