# American Institute of Mathematical Sciences

September  2014, 13(5): 1935-1969. doi: 10.3934/cpaa.2014.13.1935

## Eliminating flutter for clamped von Karman plates immersed in subsonic flows

 1 University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152 2 Oregon State University, Department of Mathematics, 368 Kidder Hall, Corvallis, OR 97330

Received  January 2014 Revised  April 2014 Published  June 2014

We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system, full trajectories---both plate and flow---converge strongly to (the set of) stationary states. Weak convergence results require minimal" interior damping, and strong convergence of the dynamics are shown with sufficiently large damping. We require the existence of a good" energy balance equation, which is only available when the flows are subsonic. Our proof is based on first showing the convergence properties for regular solutions, which in turn requires propagation of initial regularity on the infinite horizon. Then, we utilize the exponential decay of the difference of two plate trajectories to show that full flow-plate trajectories are uniform-in-time Hadamard continuous. This allows us to pass convergence properties of smooth initial data to finite energy type initial data. Physically, our results imply that flutter (a non-static end behavior) does not occur in subsonic dynamics. While such results were known for rotational (compact/regular) plate dynamics [14] (and references therein), the result presented herein is the first such result obtained for non-regularized---the most physically relevant---models.
Citation: Irena Lasiecka, Justin Webster. Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1935-1969. doi: 10.3934/cpaa.2014.13.1935
##### References:
 [1] A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [2] A. V. Balakrishnan, Aeroelasticity-Continuum Theory, Springer-Verlag, 2012. doi: 10.1007/978-1-4614-3609-6. [3] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [4] R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, Wiley, 1962; also Dover, New York, 1975. [5] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976. [6] V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963. [7] A. Boutet de Monvel and I. Chueshov, The problem of interaction of von Karman plate with subsonic flow gas, Math. Methods in Appl. Sc., 22 (1999), 801-810. doi: 10.1002/(SICI)1099-1476(19990710)22:10<801::AID-MMA61>3.0.CO;2-T. [8] L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas, C.R. Acad. Sci. Paris, Ser.I, 322 (1996), 1001-1006. [9] L. Boutet de Monvel and I. Chueshov, Oscillation of von Karman's plate in a potential flow of gas, Izvestiya RAN: Ser. Mat., 63 (1999), 219-244. doi: 10.1070/im1999v063n02ABEH000237. [10] L. Boutet de Monvel, I. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Comm. PDEs, 22 (1997), 1453-1474. doi: 10.1080/03605309708821307. [11] I. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity, Teor. Funktsii Funktsional. Anal. i Prilozhen, 54 (1990), 123-130 (in Russian); translation in J. Soviet Math., 58 (1992), 385-390. doi: 10.1007/BF01097291. [12] 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. [13] I. Chueshov and I. Lasiecka, Long-time Behavior of Second-order Evolutions with Nonlinear Damping, Memoires of AMS, v. 195, 2008. doi: 10.1090/memo/0912. [14] 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. [15] 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. [16] 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, arXiv:1208.5245, submitted June 2012. [17] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups for supersonic flow-plate interactions, J. of Diff. Eqs., 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009. [18] 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), 5. [19] P. Ciarlet and P. Rabier, Les Equations de Von Karman, Springer, 1980. [20] C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Func. Anal, 13 (1973), 97-106. [21] E. Dowell, Nonlinear Oscillations of a Fluttering Plate, I and II, AIAA J., 4 (1966), 1267-1275; and 5 (1967), 1857-1862. [22] E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells, AIAA Journal, 8 (1970), 385-399. [23] E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. [24] E. H. Dowell, Some recent advances in nonlinear aeroelasticity: fluid-structure interaction in the 21st century, Proceedings of the 51st AIAA/ASME/ASCE/AHS/ASC Structures, AIAA, 3137 (2010). [25] D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge Univ. Press, 2002. [26] E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow, Nauka, Moscow, 1978, in Russian. [27] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Prog. Nonlin. Differential Eqs. and Their App., 50 (2002), 197-216. [28] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. [29] P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, New York, 1967. [30] I. Lasiecka and J. T. Webster, Generation of bounded semigroups in nonlinear flow-structure interactions with boundary damping, Math. Methods in App. Sc., DOI: 10.1002/mma.1518, published online December, 2011. [31] J.-L. Lions and E. Magenes, Problmes aux limites non homognes et applications, vol. 1, Dunod, Paris, 1968. [32] E. Livne, Future of Airplane Aeroelasticity, J. of Aircraft, 40 (2003), 1066-1092. [33] Y. Lu, Uniform decay rates for the energy in nonlinear fluid structure interactions with monotone viscous damping, Palestine J. Mathematics, 2 (2013), 215-232. [34] S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487. [35] A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handboook of Differential Equations, Vol 4, Elsevier 2008. doi: 10.1016/S1874-5717(08)00003-0. [36] I. Ryzhkova, Stabilization of a von Karman plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. and Appl., 294 (2004), 462-481. doi: 10.1016/j.jmaa.2004.02.021. [37] I. Ryzhkova, Dynamics of a thermoelastic von Karman plate in a subsonic gas flow, Zeitschrift Ang. Math. Phys., 58 (2007), 246-261. doi: 10.1007/s00033-006-0080-7. [38] M. Slemrod, Weak asymptotic decay via a "relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping, Proc. Royal Soc., Edinburgh Sect. A., 113 (1989), 87-97. doi: 10.1017/S0308210500023970. [39] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Normale. Sup. di Pisa., 26 (1998), 185-206. [40] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8. [41] 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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##### References:
 [1] A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [2] A. V. Balakrishnan, Aeroelasticity-Continuum Theory, Springer-Verlag, 2012. doi: 10.1007/978-1-4614-3609-6. [3] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [4] R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, Wiley, 1962; also Dover, New York, 1975. [5] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976. [6] V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963. [7] A. Boutet de Monvel and I. Chueshov, The problem of interaction of von Karman plate with subsonic flow gas, Math. Methods in Appl. Sc., 22 (1999), 801-810. doi: 10.1002/(SICI)1099-1476(19990710)22:10<801::AID-MMA61>3.0.CO;2-T. [8] L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas, C.R. Acad. Sci. Paris, Ser.I, 322 (1996), 1001-1006. [9] L. Boutet de Monvel and I. Chueshov, Oscillation of von Karman's plate in a potential flow of gas, Izvestiya RAN: Ser. Mat., 63 (1999), 219-244. doi: 10.1070/im1999v063n02ABEH000237. [10] L. Boutet de Monvel, I. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Comm. PDEs, 22 (1997), 1453-1474. doi: 10.1080/03605309708821307. [11] I. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity, Teor. Funktsii Funktsional. Anal. i Prilozhen, 54 (1990), 123-130 (in Russian); translation in J. Soviet Math., 58 (1992), 385-390. doi: 10.1007/BF01097291. [12] 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. [13] I. Chueshov and I. Lasiecka, Long-time Behavior of Second-order Evolutions with Nonlinear Damping, Memoires of AMS, v. 195, 2008. doi: 10.1090/memo/0912. [14] 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. [15] 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. [16] 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, arXiv:1208.5245, submitted June 2012. [17] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups for supersonic flow-plate interactions, J. of Diff. Eqs., 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009. [18] 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), 5. [19] P. Ciarlet and P. Rabier, Les Equations de Von Karman, Springer, 1980. [20] C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Func. Anal, 13 (1973), 97-106. [21] E. Dowell, Nonlinear Oscillations of a Fluttering Plate, I and II, AIAA J., 4 (1966), 1267-1275; and 5 (1967), 1857-1862. [22] E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells, AIAA Journal, 8 (1970), 385-399. [23] E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. [24] E. H. Dowell, Some recent advances in nonlinear aeroelasticity: fluid-structure interaction in the 21st century, Proceedings of the 51st AIAA/ASME/ASCE/AHS/ASC Structures, AIAA, 3137 (2010). [25] D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge Univ. Press, 2002. [26] E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow, Nauka, Moscow, 1978, in Russian. [27] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Prog. Nonlin. Differential Eqs. and Their App., 50 (2002), 197-216. [28] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. [29] P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, New York, 1967. [30] I. Lasiecka and J. T. Webster, Generation of bounded semigroups in nonlinear flow-structure interactions with boundary damping, Math. Methods in App. Sc., DOI: 10.1002/mma.1518, published online December, 2011. [31] J.-L. Lions and E. Magenes, Problmes aux limites non homognes et applications, vol. 1, Dunod, Paris, 1968. [32] E. Livne, Future of Airplane Aeroelasticity, J. of Aircraft, 40 (2003), 1066-1092. [33] Y. Lu, Uniform decay rates for the energy in nonlinear fluid structure interactions with monotone viscous damping, Palestine J. Mathematics, 2 (2013), 215-232. [34] S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487. [35] A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handboook of Differential Equations, Vol 4, Elsevier 2008. doi: 10.1016/S1874-5717(08)00003-0. [36] I. Ryzhkova, Stabilization of a von Karman plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. and Appl., 294 (2004), 462-481. doi: 10.1016/j.jmaa.2004.02.021. [37] I. Ryzhkova, Dynamics of a thermoelastic von Karman plate in a subsonic gas flow, Zeitschrift Ang. Math. Phys., 58 (2007), 246-261. doi: 10.1007/s00033-006-0080-7. [38] M. Slemrod, Weak asymptotic decay via a "relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping, Proc. Royal Soc., Edinburgh Sect. A., 113 (1989), 87-97. doi: 10.1017/S0308210500023970. [39] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Normale. Sup. di Pisa., 26 (1998), 185-206. [40] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8. [41] 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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