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Flow-plate interactions: Well-posedness and long-time behavior
1. | Kharkov University, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov |
2. | University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152, United States |
3. | Oregon State University, Department of Mathematics, 368 Kidder Hall, Corvallis, OR 97330, United States |
References:
[1] |
A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
A. V. Balakrishnan, Aeroelasticity. Continuum Theory, Springer, New York, 2012.
doi: 10.1007/978-1-4614-3609-6. |
[3] |
A. V. Balakrishnan, Nonlinear aeroelastic theory: Continuum models, in Control Methods in PDE-Dynamical Systems, Contemp. Math., 426, Amer. Math. Soc., Providence, RI, 2007, 79-101.
doi: 10.1090/conm/426/08185. |
[4] |
A. V. Balakrishnan and M. A. Shubov, Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1057-1091.
doi: 10.1098/rspa.2003.1217. |
[5] |
H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. |
[6] |
R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London 1962; also Dover, New York, 1975. |
[7] |
L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping, J. Diff. Eqs., 253 (2012), 3568-3609.
doi: 10.1016/j.jde.2012.08.004. |
[8] |
V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer, New York, 1993. |
[15] |
I. Chueshov, On a system of equations with delay that arises in aero-elasticity, (in Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen, No. 54, (1990), 123-130; translation in J. Soviet Math., 58 (1992), 385-390.
doi: 10.1007/BF01097291. |
[16] |
I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, (in Russian) Acta, Kharkov, 1999; English translation in Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/ |
[17] |
I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: Rigorous results and unsolved problems, in Proceedings of the 16th IMACS World Congress, Lausanne, Switzerland, 2000, 1-6. |
[18] |
I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 650-665.
doi: 10.1016/j.na.2013.10.018. |
[19] |
I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid, Commun. Pure Appl. Anal., to appear, (2014).
doi: 10.1016/j.na.2013.10.018. |
[20] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equations, 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[21] |
I. Chueshov and I. Lasiecka, Long-time behavior of second-order evolutions with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008).
doi: 10.1090/memo/0912. |
[22] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[23] |
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. |
[24] |
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, submitted, arXiv:1208.5245, 2012. |
[25] |
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. |
[26] |
I. Chueshov and A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. 1, 321 (1995), 607-612. |
[27] |
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. |
[28] |
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. |
[29] |
I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical J., 65 (2013), 158-177.
doi: 10.1007/s11253-013-0771-0. |
[30] |
I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in System Modeling and Optimization. 25th IFIP TC7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers (eds. D. Hömberg and F. Tröltzsch), IFIP Advances in Information and Communication Technology, 391, Springer, Berlin, 2013, 328-337.
doi: 10.1007/978-3-642-36062-6_33. |
[31] |
P. Ciarlet and P. Rabier, Les Équations de Von Kármán, Lecture Notes in Mathematics, 826, Springer, Berlin, 1980. |
[32] |
C. Mei, K. Abdel-Motagaly and R. Chen, Review of nonlinear panel flutter at supersonic and hypersonic speeds, Appl. Mech. Rev., 52 (1999), 321-332. |
[33] |
K. F. Clancey, On finite Hilbert transforms, Transactions AMS, 212 (1975), 347-354.
doi: 10.1090/S0002-9947-1975-0377598-5. |
[34] |
D. G. Crighton, The Kutta condition in unsteady flow, Ann. Rev. Fluid Mech., 17 (1985), 411-445.
doi: 10.1146/annurev.fluid.17.1.411. |
[35] |
O. Diekmann, S. van Gils, S. Lunel and H. O. Walther, Delay Equations, Springer, 1995. |
[36] |
E. Dowell, Nonlinear oscillations of a fluttering plate, I and II, AIAA J., 4 (1966), 1267-1275; and 5 (1967), 1857-1862. |
[37] |
E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells, AIAA Journal, 8 (1970), 385-399. |
[38] |
E. Dowell, O. Bendiksen, J. Edwards and T. Strganac, Transonic Nonlinear Aeroelasticity, Encyclopedia of Aerospace Engineering, 2003.
doi: 10.1002/9780470686652.eae151. |
[39] | |
[40] |
E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. |
[41] |
C. Eloy, C. Souilliez and L. Schouveiler, Flutter of a rectangular plate, J. Fluids and Structures, 23 (2007), 904-919.
doi: 10.1016/j.jfluidstructs.2007.02.002. |
[42] |
A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Diff. Int. Eqs, 9 (1966), 267-294.; Addendum, Diff. Int. Eqs., 10 (1997), 197-220. |
[43] |
W. Frederiks, H. C. J. Hilbering and J. A. Sparenberg, On the Kutta condition for the flow along a semi-infinite elastic plate, J. Engin. Math., 20 (1986), 27-50.
doi: 10.1007/BF00039321. |
[44] |
P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear damping localized in a boundary layer, J. of Diff. Eqs., 254 (2013), 1193-1229.
doi: 10.1016/j.jde.2012.10.016. |
[45] |
D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge Univ. Press, 2002.
doi: 10.1115/1.1566393. |
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P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, 14 (1978), 367-384.
doi: 10.1016/0005-1098(78)90036-5. |
[47] |
M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a free boundary fluid-structure model, J. of Math. Phys., 53 (2012), 115624-115624.
doi: 10.1063/1.4766724. |
[48] |
T. Von Kármán, Festigkeitsprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4, Leipzig, 1910, 348-352. |
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A. K. Khanmmamedov, Global attractors for von Karman equations with non-linear dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[50] |
A. Kornecki, E. H. Dowell and J. O'Brien, On the aeroelastic instability of two-dimensional panes in uniform incompressible flow, 47 (1976), 163-178. |
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E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow, (in Russian) {Nauka}, Moscow, 1978. |
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O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
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show all references
References:
[1] |
A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
A. V. Balakrishnan, Aeroelasticity. Continuum Theory, Springer, New York, 2012.
doi: 10.1007/978-1-4614-3609-6. |
[3] |
A. V. Balakrishnan, Nonlinear aeroelastic theory: Continuum models, in Control Methods in PDE-Dynamical Systems, Contemp. Math., 426, Amer. Math. Soc., Providence, RI, 2007, 79-101.
doi: 10.1090/conm/426/08185. |
[4] |
A. V. Balakrishnan and M. A. Shubov, Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1057-1091.
doi: 10.1098/rspa.2003.1217. |
[5] |
H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. |
[6] |
R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London 1962; also Dover, New York, 1975. |
[7] |
L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping, J. Diff. Eqs., 253 (2012), 3568-3609.
doi: 10.1016/j.jde.2012.08.004. |
[8] |
V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer, New York, 1993. |
[15] |
I. Chueshov, On a system of equations with delay that arises in aero-elasticity, (in Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen, No. 54, (1990), 123-130; translation in J. Soviet Math., 58 (1992), 385-390.
doi: 10.1007/BF01097291. |
[16] |
I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, (in Russian) Acta, Kharkov, 1999; English translation in Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/ |
[17] |
I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: Rigorous results and unsolved problems, in Proceedings of the 16th IMACS World Congress, Lausanne, Switzerland, 2000, 1-6. |
[18] |
I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 650-665.
doi: 10.1016/j.na.2013.10.018. |
[19] |
I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid, Commun. Pure Appl. Anal., to appear, (2014).
doi: 10.1016/j.na.2013.10.018. |
[20] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equations, 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[21] |
I. Chueshov and I. Lasiecka, Long-time behavior of second-order evolutions with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008).
doi: 10.1090/memo/0912. |
[22] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[23] |
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. |
[24] |
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, submitted, arXiv:1208.5245, 2012. |
[25] |
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. |
[26] |
I. Chueshov and A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. 1, 321 (1995), 607-612. |
[27] |
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. |
[28] |
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. |
[29] |
I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical J., 65 (2013), 158-177.
doi: 10.1007/s11253-013-0771-0. |
[30] |
I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in System Modeling and Optimization. 25th IFIP TC7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers (eds. D. Hömberg and F. Tröltzsch), IFIP Advances in Information and Communication Technology, 391, Springer, Berlin, 2013, 328-337.
doi: 10.1007/978-3-642-36062-6_33. |
[31] |
P. Ciarlet and P. Rabier, Les Équations de Von Kármán, Lecture Notes in Mathematics, 826, Springer, Berlin, 1980. |
[32] |
C. Mei, K. Abdel-Motagaly and R. Chen, Review of nonlinear panel flutter at supersonic and hypersonic speeds, Appl. Mech. Rev., 52 (1999), 321-332. |
[33] |
K. F. Clancey, On finite Hilbert transforms, Transactions AMS, 212 (1975), 347-354.
doi: 10.1090/S0002-9947-1975-0377598-5. |
[34] |
D. G. Crighton, The Kutta condition in unsteady flow, Ann. Rev. Fluid Mech., 17 (1985), 411-445.
doi: 10.1146/annurev.fluid.17.1.411. |
[35] |
O. Diekmann, S. van Gils, S. Lunel and H. O. Walther, Delay Equations, Springer, 1995. |
[36] |
E. Dowell, Nonlinear oscillations of a fluttering plate, I and II, AIAA J., 4 (1966), 1267-1275; and 5 (1967), 1857-1862. |
[37] |
E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells, AIAA Journal, 8 (1970), 385-399. |
[38] |
E. Dowell, O. Bendiksen, J. Edwards and T. Strganac, Transonic Nonlinear Aeroelasticity, Encyclopedia of Aerospace Engineering, 2003.
doi: 10.1002/9780470686652.eae151. |
[39] | |
[40] |
E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. |
[41] |
C. Eloy, C. Souilliez and L. Schouveiler, Flutter of a rectangular plate, J. Fluids and Structures, 23 (2007), 904-919.
doi: 10.1016/j.jfluidstructs.2007.02.002. |
[42] |
A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Diff. Int. Eqs, 9 (1966), 267-294.; Addendum, Diff. Int. Eqs., 10 (1997), 197-220. |
[43] |
W. Frederiks, H. C. J. Hilbering and J. A. Sparenberg, On the Kutta condition for the flow along a semi-infinite elastic plate, J. Engin. Math., 20 (1986), 27-50.
doi: 10.1007/BF00039321. |
[44] |
P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear damping localized in a boundary layer, J. of Diff. Eqs., 254 (2013), 1193-1229.
doi: 10.1016/j.jde.2012.10.016. |
[45] |
D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge Univ. Press, 2002.
doi: 10.1115/1.1566393. |
[46] |
P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, 14 (1978), 367-384.
doi: 10.1016/0005-1098(78)90036-5. |
[47] |
M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a free boundary fluid-structure model, J. of Math. Phys., 53 (2012), 115624-115624.
doi: 10.1063/1.4766724. |
[48] |
T. Von Kármán, Festigkeitsprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4, Leipzig, 1910, 348-352. |
[49] |
A. K. Khanmmamedov, Global attractors for von Karman equations with non-linear dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[50] |
A. Kornecki, E. H. Dowell and J. O'Brien, On the aeroelastic instability of two-dimensional panes in uniform incompressible flow, 47 (1976), 163-178. |
[51] |
E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow, (in Russian) {Nauka}, Moscow, 1978. |
[52] |
O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[53] |
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