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December  2016, 5(4): 561-566. doi: 10.3934/eect.2016019

Remark on an elastic plate interacting with a gas in a semi-infinite tube: Periodic solutions

Igor Chueshov 1,

1. 

Kharkiv Karazin National University, 4 Svobody sq., 61077 Kharkiv, Ukraine

Received  January 2016 Revised  February 2016 Published  October 2016

We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.
Keywords: periodic solutions., fluid-structure interaction, Coupled wave and plate equations.
Mathematics Subject Classification: Primary: 74F10, 74K20; Secondary: 35B10, 35B3.
Citation: Igor Chueshov. Remark on an elastic plate interacting with a gas in a semi-infinite tube: Periodic solutions. Evolution Equations & Control Theory, 2016, 5 (4) : 561-566. doi: 10.3934/eect.2016019
References:
[1]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/ Google Scholar

[2]

I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: rigorous results and unsolved problems, Proceedings of the 16th IMACS World Congress, Lausanne (Switzerland), 2000, 1-6. Google Scholar

[3]

I. Chueshov, E. Dowell, I. Lasiecka and J. T. Webster, Von Karman plate in a gas flow: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500. doi: 10.1007/s00245-016-9349-1.  Google Scholar

[4]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discr. Cont. Dyn. Sys., 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777.  Google Scholar

[5]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Behavior, Monographs, {Springer-Verlag}, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions. Jour. Abstr. Differ. Equ. Appl., 3 (2012), 1-27.  Google Scholar

[7]

I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, Comm. in PDE, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484.  Google Scholar

[8]

I. Chueshov, I. Lasiecka and J. T. Webster, Flow-plate interactions: Well-posedness and long-time behavior, Discrete Contin. Dyn. Syst. Ser. S, Special Volume: New Developments in Mathematical Theory of Fluid Mechanics, 7 (2014), 925-965. doi: 10.3934/dcdss.2014.7.925.  Google Scholar

[9]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM Publications, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[10]

I. Lasiecka and J. T. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows, Comm. Pure Appl. Math., 13 (2014), 1935-1969, Updated version (May, 2015): Arxiv:1409.3308v5. doi: 10.3934/cpaa.2014.13.1935.  Google Scholar

[11]

I. Lasiecka and J. T. Webster, Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow, SIAM J. Math. Anal., 48 (2016), 1848-1891. doi: 10.1137/15M1040529.  Google Scholar

[12]

H. F. Walker, Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains, J. Diff. Eqs., 23 (1977), 459-471. doi: 10.1016/0022-0396(77)90123-1.  Google Scholar

show all references

References:
[1]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/ Google Scholar

[2]

I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: rigorous results and unsolved problems, Proceedings of the 16th IMACS World Congress, Lausanne (Switzerland), 2000, 1-6. Google Scholar

[3]

I. Chueshov, E. Dowell, I. Lasiecka and J. T. Webster, Von Karman plate in a gas flow: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500. doi: 10.1007/s00245-016-9349-1.  Google Scholar

[4]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discr. Cont. Dyn. Sys., 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777.  Google Scholar

[5]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Behavior, Monographs, {Springer-Verlag}, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions. Jour. Abstr. Differ. Equ. Appl., 3 (2012), 1-27.  Google Scholar

[7]

I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, Comm. in PDE, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484.  Google Scholar

[8]

I. Chueshov, I. Lasiecka and J. T. Webster, Flow-plate interactions: Well-posedness and long-time behavior, Discrete Contin. Dyn. Syst. Ser. S, Special Volume: New Developments in Mathematical Theory of Fluid Mechanics, 7 (2014), 925-965. doi: 10.3934/dcdss.2014.7.925.  Google Scholar

[9]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM Publications, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[10]

I. Lasiecka and J. T. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows, Comm. Pure Appl. Math., 13 (2014), 1935-1969, Updated version (May, 2015): Arxiv:1409.3308v5. doi: 10.3934/cpaa.2014.13.1935.  Google Scholar

[11]

I. Lasiecka and J. T. Webster, Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow, SIAM J. Math. Anal., 48 (2016), 1848-1891. doi: 10.1137/15M1040529.  Google Scholar

[12]

H. F. Walker, Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains, J. Diff. Eqs., 23 (1977), 459-471. doi: 10.1016/0022-0396(77)90123-1.  Google Scholar

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