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Remark on an elastic plate interacting with a gas in a semi-infinite tube: Periodic solutions
Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping
1. | Department of Mathematics, College of Charleston, 66 George Street, Charleston, SC, 29424, United States |
2. | University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152 |
References:
[1] |
H. Ashley and G. Zartarian, Piston theory: A new aerodynamic tool for the aeroelastician, Journal of the Aeronautical Sciences, 23 (1956), 1109-1118.
doi: 10.2514/8.3740. |
[2] |
M. Aouadi and A. Miranville, Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory, Evolutions Equations and Control Theory, 4 (2015), 241-263. |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
H. M. Berger, A new approach to the analysis of large deflections of plates, Journal of Applied Mechanics, 22 (1955), 465-472. |
[5] |
V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, The Macmillan Co., New York, 1963. |
[6] |
L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping, Journal of Differential Equations, 253 (2012), 3568-3609.
doi: 10.1016/j.jde.2012.08.004. |
[7] |
F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Communications on Pure and Applied Analysis, 6 (2007), 113-140. |
[8] |
F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Dynamical Systems, 22 (2008), 557-586.
doi: 10.3934/dcds.2008.22.557. |
[9] |
Z. Chbani and H. Riahi, Existence and asymptotic behavior for solutions of dynamical equilibrium systems, Evolution Equations and Control Theory, 3 (2014), 1-14. |
[10] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.
doi: 10.1007/978-3-319-22903-4. |
[11] |
I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, American Mathematical Soc., 2008. |
[12] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Science & Business Media, 2010.
doi: 10.1007/978-0-387-87712-9. |
[13] |
I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Mathematical aeroelasticity: A survey, Mathematics in Engineering, Science and Aerospace, 7 (2016), 5-29. |
[14] |
I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Applied Mathematics and Optimization, 73 (2016), 475-500.
doi: 10.1007/s00245-016-9349-1. |
[15] |
I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Communications in Partial Differential Equations, 39 (2014), 1965-1997.
doi: 10.1080/03605302.2014.930484. |
[16] |
I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions, Journal of Differential Equations, 254 (2013), 1741-1773.
doi: 10.1016/j.jde.2012.11.009. |
[17] |
P. G. Ciarlet, Mathematical Elasticity: Three-Dimensional Elasticity, Vol. 1, Elsevier, 1993. |
[18] |
E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. |
[19] |
E. H. Dowell, Nonlinear oscillations of a fluttering plate I. AIAA Journal, 4 (1967), 1267-1275. Nonlinear oscillations of a fluttering plate. II. AIAA Journal, 5 (1966), 1856-1862. |
[20] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete and Continuous Dynamical Systems, 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[21] |
P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, Journal of Differential Equations, 254 (2013), 1193-1229.
doi: 10.1016/j.jde.2012.10.016. |
[22] |
P. G. Geredeli and J. T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlinear Analysis B: Real World Applications, 31 (2016), 227-256.
doi: 10.1016/j.nonrwa.2016.02.002. |
[23] |
A. Haraux and M. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time decaying damping term, Evolution Equations and Control Theory, 2 (2013), 461-470. |
[24] |
A. A. Il'yushin, Law of plane sections in the aerodynamics of high supersonic velocities, Prikl. Mat. Mekh, 20 (1956), 733-755 (in Russian). |
[25] |
V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, Journal of Differential Equations, 247 (2009), 1120-1155.
doi: 10.1016/j.jde.2009.04.010. |
[26] |
A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, Journal of Mathematical Analysis and Applications, 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[27] |
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 1989.
doi: 10.1137/1.9781611970821. |
[28] |
J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, Journal of Differential Equations, 91 (1991), 355-388.
doi: 10.1016/0022-0396(91)90145-Y. |
[29] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories, Vol. 1, Cambridge University Press, 2000. |
[30] |
I. Lasiecka and J. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows, Communications on Pure & Applied Analysis, 13 (2014), 1935-1969. Updated version (May, 2015): http://arxiv.org/abs/1409.3308.
doi: 10.3934/cpaa.2014.13.1935. |
[31] |
I. Lasiecka and J. T. Webster, Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow, SIAM Journal on Mathematical Analysis, 48 (2016), 1848-1891.
doi: 10.1137/15M1040529. |
[32] |
M. J. Lighthill, Oscillating airfoils at high mach number, Journal of the Aeronautical Sciences, 20 (1953), 402-406.
doi: 10.2514/8.2657. |
[33] |
T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 3402-3412.
doi: 10.1016/j.na.2010.07.023. |
[34] |
J. Malek and D. Prazak, Large time behavior via the method of $l$-trajectories, Journal of Differential Equations, 181 (2002), 243-279.
doi: 10.1006/jdeq.2001.4087. |
[35] |
G. P. Menzala and E. Zuazua, Timoshenko's beam equation as limit of a nonlinear one-dimensional von Karman system, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 130 (2000), 855-875.
doi: 10.1017/S0308210500000470. |
[36] |
G. P. Menzala and E. Zuazua, Timoshenko's plate equation as a singular limit of the dynamical von Karman system, Journal de mathématiques pures et appliquees, 79 (2000), 73-94.
doi: 10.1016/S0021-7824(00)00149-5. |
[37] |
J. L. Nowinski and H. Ohnabe, On certain inconsistencies in Berger equations for large deflections of plastic plates, International Journal of Mechanical Sciences, 14 (1972), 165-170. |
[38] |
T. Saanouni, A note on global well-posedness and blow up of some semilinear evolution equations, Evolution Equations and Control Theory, 4 (2015), 355-372. |
[39] |
V. V. Vedeneev, Effect of damping on flutter of simply supported and clamped panels at low supersonic speeds, Journal of Fluids and Structures, 40 (2013), 366-372. |
[40] |
V. V. Vedeneev, Panel flutter at low supersonic speeds, Journal of Fluids and Structures, 29 (2012), 79-96. |
[41] |
C. P. Vendhan, A study of Berger equations applied to non-linear vibrations of elastic plates, International Journal of Mechanical Sciences, 17 (1975), 461-468. |
[42] |
J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: Semigroup approach, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3123-3136.
doi: 10.1016/j.na.2011.01.028. |
[43] |
Z. Yang, On an extensible beam equation with nonlinear damping and source terms, Journal of Differential Equations, 254 (2013), 3903-3927.
doi: 10.1016/j.jde.2013.02.008. |
show all references
References:
[1] |
H. Ashley and G. Zartarian, Piston theory: A new aerodynamic tool for the aeroelastician, Journal of the Aeronautical Sciences, 23 (1956), 1109-1118.
doi: 10.2514/8.3740. |
[2] |
M. Aouadi and A. Miranville, Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory, Evolutions Equations and Control Theory, 4 (2015), 241-263. |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
H. M. Berger, A new approach to the analysis of large deflections of plates, Journal of Applied Mechanics, 22 (1955), 465-472. |
[5] |
V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, The Macmillan Co., New York, 1963. |
[6] |
L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping, Journal of Differential Equations, 253 (2012), 3568-3609.
doi: 10.1016/j.jde.2012.08.004. |
[7] |
F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Communications on Pure and Applied Analysis, 6 (2007), 113-140. |
[8] |
F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Dynamical Systems, 22 (2008), 557-586.
doi: 10.3934/dcds.2008.22.557. |
[9] |
Z. Chbani and H. Riahi, Existence and asymptotic behavior for solutions of dynamical equilibrium systems, Evolution Equations and Control Theory, 3 (2014), 1-14. |
[10] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.
doi: 10.1007/978-3-319-22903-4. |
[11] |
I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, American Mathematical Soc., 2008. |
[12] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Science & Business Media, 2010.
doi: 10.1007/978-0-387-87712-9. |
[13] |
I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Mathematical aeroelasticity: A survey, Mathematics in Engineering, Science and Aerospace, 7 (2016), 5-29. |
[14] |
I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Applied Mathematics and Optimization, 73 (2016), 475-500.
doi: 10.1007/s00245-016-9349-1. |
[15] |
I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Communications in Partial Differential Equations, 39 (2014), 1965-1997.
doi: 10.1080/03605302.2014.930484. |
[16] |
I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions, Journal of Differential Equations, 254 (2013), 1741-1773.
doi: 10.1016/j.jde.2012.11.009. |
[17] |
P. G. Ciarlet, Mathematical Elasticity: Three-Dimensional Elasticity, Vol. 1, Elsevier, 1993. |
[18] |
E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. |
[19] |
E. H. Dowell, Nonlinear oscillations of a fluttering plate I. AIAA Journal, 4 (1967), 1267-1275. Nonlinear oscillations of a fluttering plate. II. AIAA Journal, 5 (1966), 1856-1862. |
[20] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete and Continuous Dynamical Systems, 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[21] |
P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, Journal of Differential Equations, 254 (2013), 1193-1229.
doi: 10.1016/j.jde.2012.10.016. |
[22] |
P. G. Geredeli and J. T. Webster, Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlinear Analysis B: Real World Applications, 31 (2016), 227-256.
doi: 10.1016/j.nonrwa.2016.02.002. |
[23] |
A. Haraux and M. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time decaying damping term, Evolution Equations and Control Theory, 2 (2013), 461-470. |
[24] |
A. A. Il'yushin, Law of plane sections in the aerodynamics of high supersonic velocities, Prikl. Mat. Mekh, 20 (1956), 733-755 (in Russian). |
[25] |
V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, Journal of Differential Equations, 247 (2009), 1120-1155.
doi: 10.1016/j.jde.2009.04.010. |
[26] |
A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, Journal of Mathematical Analysis and Applications, 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[27] |
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 1989.
doi: 10.1137/1.9781611970821. |
[28] |
J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, Journal of Differential Equations, 91 (1991), 355-388.
doi: 10.1016/0022-0396(91)90145-Y. |
[29] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories, Vol. 1, Cambridge University Press, 2000. |
[30] |
I. Lasiecka and J. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows, Communications on Pure & Applied Analysis, 13 (2014), 1935-1969. Updated version (May, 2015): http://arxiv.org/abs/1409.3308.
doi: 10.3934/cpaa.2014.13.1935. |
[31] |
I. Lasiecka and J. T. Webster, Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow, SIAM Journal on Mathematical Analysis, 48 (2016), 1848-1891.
doi: 10.1137/15M1040529. |
[32] |
M. J. Lighthill, Oscillating airfoils at high mach number, Journal of the Aeronautical Sciences, 20 (1953), 402-406.
doi: 10.2514/8.2657. |
[33] |
T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 3402-3412.
doi: 10.1016/j.na.2010.07.023. |
[34] |
J. Malek and D. Prazak, Large time behavior via the method of $l$-trajectories, Journal of Differential Equations, 181 (2002), 243-279.
doi: 10.1006/jdeq.2001.4087. |
[35] |
G. P. Menzala and E. Zuazua, Timoshenko's beam equation as limit of a nonlinear one-dimensional von Karman system, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 130 (2000), 855-875.
doi: 10.1017/S0308210500000470. |
[36] |
G. P. Menzala and E. Zuazua, Timoshenko's plate equation as a singular limit of the dynamical von Karman system, Journal de mathématiques pures et appliquees, 79 (2000), 73-94.
doi: 10.1016/S0021-7824(00)00149-5. |
[37] |
J. L. Nowinski and H. Ohnabe, On certain inconsistencies in Berger equations for large deflections of plastic plates, International Journal of Mechanical Sciences, 14 (1972), 165-170. |
[38] |
T. Saanouni, A note on global well-posedness and blow up of some semilinear evolution equations, Evolution Equations and Control Theory, 4 (2015), 355-372. |
[39] |
V. V. Vedeneev, Effect of damping on flutter of simply supported and clamped panels at low supersonic speeds, Journal of Fluids and Structures, 40 (2013), 366-372. |
[40] |
V. V. Vedeneev, Panel flutter at low supersonic speeds, Journal of Fluids and Structures, 29 (2012), 79-96. |
[41] |
C. P. Vendhan, A study of Berger equations applied to non-linear vibrations of elastic plates, International Journal of Mechanical Sciences, 17 (1975), 461-468. |
[42] |
J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: Semigroup approach, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3123-3136.
doi: 10.1016/j.na.2011.01.028. |
[43] |
Z. Yang, On an extensible beam equation with nonlinear damping and source terms, Journal of Differential Equations, 254 (2013), 3903-3927.
doi: 10.1016/j.jde.2013.02.008. |
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