# American Institute of Mathematical Sciences

December  2016, 5(4): 567-603. doi: 10.3934/eect.2016020

## 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

Received  July 2016 Revised  September 2016 Published  October 2016

We consider a nonlinear (Berger or Von Karman) clamped plate model with a piston-theoretic right hand side---which includes non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of strong tools such as backward-in-time smallness of velocities and finiteness of the dissipation integral. Modern quasi-stability techniques are utilized to show the existence of compact global attractors and generalized fractal exponential attractors. Specific results here depend on the size of the damping parameter and the nonlinearity in force. For the Berger plate, in the presence of large damping, the existence of a proper global attractor (whose fractal dimension is finite in the state space) is shown via a decomposition of the nonlinear dynamics. This leads to the construction of a compact set upon which quasi-stability theory can be implemented. Numerical investigations for appropriate 1-D models are presented which explore and support the abstract results presented herein.
Citation: Jason S. Howell, Irena Lasiecka, Justin T. Webster. Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping. Evolution Equations & Control Theory, 2016, 5 (4) : 567-603. doi: 10.3934/eect.2016020
##### References:
 [1] H. Ashley and G. Zartarian, Piston theory: A new aerodynamic tool for the aeroelastician,, Journal of the Aeronautical Sciences, 23 (1956), 1109.  doi: 10.2514/8.3740.  Google Scholar [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.   Google Scholar [3] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing Co., (1992).   Google Scholar [4] H. M. Berger, A new approach to the analysis of large deflections of plates,, Journal of Applied Mechanics, 22 (1955), 465.   Google Scholar [5] V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability,, The Macmillan Co., (1963).   Google Scholar [6] L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping,, Journal of Differential Equations, 253 (2012), 3568.  doi: 10.1016/j.jde.2012.08.004.  Google Scholar [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.   Google Scholar [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.  doi: 10.3934/dcds.2008.22.557.  Google Scholar [9] Z. Chbani and H. Riahi, Existence and asymptotic behavior for solutions of dynamical equilibrium systems,, Evolution Equations and Control Theory, 3 (2014), 1.   Google Scholar [10] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015).  doi: 10.1007/978-3-319-22903-4.  Google Scholar [11] I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping,, American Mathematical Soc., (2008).   Google Scholar [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.  Google Scholar [13] I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Mathematical aeroelasticity: A survey,, Mathematics in Engineering, 7 (2016), 5.   Google Scholar [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.  doi: 10.1007/s00245-016-9349-1.  Google Scholar [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.  doi: 10.1080/03605302.2014.930484.  Google Scholar [16] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions,, Journal of Differential Equations, 254 (2013), 1741.  doi: 10.1016/j.jde.2012.11.009.  Google Scholar [17] P. G. Ciarlet, Mathematical Elasticity: Three-Dimensional Elasticity,, Vol. 1, (1993).   Google Scholar [18] E. Dowell, A Modern Course in Aeroelasticity,, Kluwer Academic Publishers, (2004).   Google Scholar [19] E. H. Dowell, Nonlinear oscillations of a fluttering plate I., AIAA Journal, 4 (1967), 1267.   Google Scholar [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.  doi: 10.3934/dcds.2004.10.211.  Google Scholar [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.  doi: 10.1016/j.jde.2012.10.016.  Google Scholar [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.  doi: 10.1016/j.nonrwa.2016.02.002.  Google Scholar [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.   Google Scholar [24] A. A. Il'yushin, Law of plane sections in the aerodynamics of high supersonic velocities,, Prikl. Mat. Mekh, 20 (1956), 733.   Google Scholar [25] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, Journal of Differential Equations, 247 (2009), 1120.  doi: 10.1016/j.jde.2009.04.010.  Google Scholar [26] A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation,, Journal of Mathematical Analysis and Applications, 318 (2006), 92.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar [27] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM Studies in Applied Mathematics, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [28] J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback,, Journal of Differential Equations, 91 (1991), 355.  doi: 10.1016/0022-0396(91)90145-Y.  Google Scholar [29] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories,, Vol. 1, (2000).   Google Scholar [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.  doi: 10.3934/cpaa.2014.13.1935.  Google Scholar [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.  doi: 10.1137/15M1040529.  Google Scholar [32] M. J. Lighthill, Oscillating airfoils at high mach number,, Journal of the Aeronautical Sciences, 20 (1953), 402.  doi: 10.2514/8.2657.  Google Scholar [33] T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms,, Nonlinear Analysis: Theory, 73 (2010), 3402.  doi: 10.1016/j.na.2010.07.023.  Google Scholar [34] J. Malek and D. Prazak, Large time behavior via the method of $l$-trajectories,, Journal of Differential Equations, 181 (2002), 243.  doi: 10.1006/jdeq.2001.4087.  Google Scholar [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.  doi: 10.1017/S0308210500000470.  Google Scholar [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.  doi: 10.1016/S0021-7824(00)00149-5.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [40] V. V. Vedeneev, Panel flutter at low supersonic speeds,, Journal of Fluids and Structures, 29 (2012), 79.   Google Scholar [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.   Google Scholar [42] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: Semigroup approach,, Nonlinear Analysis: Theory, 74 (2011), 3123.  doi: 10.1016/j.na.2011.01.028.  Google Scholar [43] Z. Yang, On an extensible beam equation with nonlinear damping and source terms,, Journal of Differential Equations, 254 (2013), 3903.  doi: 10.1016/j.jde.2013.02.008.  Google Scholar

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##### References:
 [1] H. Ashley and G. Zartarian, Piston theory: A new aerodynamic tool for the aeroelastician,, Journal of the Aeronautical Sciences, 23 (1956), 1109.  doi: 10.2514/8.3740.  Google Scholar [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.   Google Scholar [3] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing Co., (1992).   Google Scholar [4] H. M. Berger, A new approach to the analysis of large deflections of plates,, Journal of Applied Mechanics, 22 (1955), 465.   Google Scholar [5] V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability,, The Macmillan Co., (1963).   Google Scholar [6] L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping,, Journal of Differential Equations, 253 (2012), 3568.  doi: 10.1016/j.jde.2012.08.004.  Google Scholar [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.   Google Scholar [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.  doi: 10.3934/dcds.2008.22.557.  Google Scholar [9] Z. Chbani and H. Riahi, Existence and asymptotic behavior for solutions of dynamical equilibrium systems,, Evolution Equations and Control Theory, 3 (2014), 1.   Google Scholar [10] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015).  doi: 10.1007/978-3-319-22903-4.  Google Scholar [11] I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping,, American Mathematical Soc., (2008).   Google Scholar [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.  Google Scholar [13] I. Chueshov, E. H. Dowell, I. Lasiecka and J. T. Webster, Mathematical aeroelasticity: A survey,, Mathematics in Engineering, 7 (2016), 5.   Google Scholar [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.  doi: 10.1007/s00245-016-9349-1.  Google Scholar [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.  doi: 10.1080/03605302.2014.930484.  Google Scholar [16] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions,, Journal of Differential Equations, 254 (2013), 1741.  doi: 10.1016/j.jde.2012.11.009.  Google Scholar [17] P. G. Ciarlet, Mathematical Elasticity: Three-Dimensional Elasticity,, Vol. 1, (1993).   Google Scholar [18] E. Dowell, A Modern Course in Aeroelasticity,, Kluwer Academic Publishers, (2004).   Google Scholar [19] E. H. Dowell, Nonlinear oscillations of a fluttering plate I., AIAA Journal, 4 (1967), 1267.   Google Scholar [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.  doi: 10.3934/dcds.2004.10.211.  Google Scholar [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.  doi: 10.1016/j.jde.2012.10.016.  Google Scholar [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.  doi: 10.1016/j.nonrwa.2016.02.002.  Google Scholar [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.   Google Scholar [24] A. A. Il'yushin, Law of plane sections in the aerodynamics of high supersonic velocities,, Prikl. Mat. Mekh, 20 (1956), 733.   Google Scholar [25] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, Journal of Differential Equations, 247 (2009), 1120.  doi: 10.1016/j.jde.2009.04.010.  Google Scholar [26] A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation,, Journal of Mathematical Analysis and Applications, 318 (2006), 92.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar [27] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM Studies in Applied Mathematics, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [28] J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback,, Journal of Differential Equations, 91 (1991), 355.  doi: 10.1016/0022-0396(91)90145-Y.  Google Scholar [29] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories,, Vol. 1, (2000).   Google Scholar [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.  doi: 10.3934/cpaa.2014.13.1935.  Google Scholar [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.  doi: 10.1137/15M1040529.  Google Scholar [32] M. J. Lighthill, Oscillating airfoils at high mach number,, Journal of the Aeronautical Sciences, 20 (1953), 402.  doi: 10.2514/8.2657.  Google Scholar [33] T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms,, Nonlinear Analysis: Theory, 73 (2010), 3402.  doi: 10.1016/j.na.2010.07.023.  Google Scholar [34] J. Malek and D. Prazak, Large time behavior via the method of $l$-trajectories,, Journal of Differential Equations, 181 (2002), 243.  doi: 10.1006/jdeq.2001.4087.  Google Scholar [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.  doi: 10.1017/S0308210500000470.  Google Scholar [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.  doi: 10.1016/S0021-7824(00)00149-5.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [40] V. V. Vedeneev, Panel flutter at low supersonic speeds,, Journal of Fluids and Structures, 29 (2012), 79.   Google Scholar [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.   Google Scholar [42] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: Semigroup approach,, Nonlinear Analysis: Theory, 74 (2011), 3123.  doi: 10.1016/j.na.2011.01.028.  Google Scholar [43] Z. Yang, On an extensible beam equation with nonlinear damping and source terms,, Journal of Differential Equations, 254 (2013), 3903.  doi: 10.1016/j.jde.2013.02.008.  Google Scholar
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