# American Institute of Mathematical Sciences

• Previous Article
Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions
• DCDS-S Home
• This Issue
• Next Article
Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model
October  2014, 7(5): 925-965. doi: 10.3934/dcdss.2014.7.925

## 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

Received  March 2013 Revised  June 2013 Published  May 2014

We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent results on existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting sets corresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to a PDE system with delay.
Citation: Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925
##### References:
 [1] A. Babin and M. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (1992). Google Scholar [2] A. V. Balakrishnan, Aeroelasticity. Continuum Theory,, Springer, (2012). doi: 10.1007/978-1-4614-3609-6. Google Scholar [3] A. V. Balakrishnan, Nonlinear aeroelastic theory: Continuum models,, in Control Methods in PDE-Dynamical Systems, (2007), 79. doi: 10.1090/conm/426/08185. Google Scholar [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. doi: 10.1098/rspa.2003.1217. Google Scholar [5] H. M. Berger, A new approach to the analysis of large deflections of plates,, J. Appl. Mech., 22 (1955), 465. Google Scholar [6] R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity,, John Wiley and Sons, (1962). Google Scholar [7] L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping,, J. Diff. Eqs., 253 (2012), 3568. doi: 10.1016/j.jde.2012.08.004. Google Scholar [8] V. V. Bolotin, Nonconservative Problems of Elastic Stability,, Pergamon Press, (1963). Google Scholar [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. doi: 10.1002/(SICI)1099-1476(19990710)22:10<801::AID-MMA61>3.0.CO;2-T. Google Scholar [10] L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas,, C. R. Acad. Sci. Paris, 322 (1996), 1001. Google Scholar [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. doi: 10.1070/im1999v063n02ABEH000237. Google Scholar [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. doi: 10.1080/03605309708821307. Google Scholar [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. doi: 10.1007/s00021-004-0121-y. Google Scholar [14] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics,, 3rd edition, (1993). Google Scholar [15] I. Chueshov, On a system of equations with delay that arises in aero-elasticity,, (in Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen, 58 (1990), 123. doi: 10.1007/BF01097291. Google Scholar [16] I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems,, (in Russian) Acta, (1999). Google Scholar [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, (2000), 1. Google Scholar [18] I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid,, Nonlinear Analysis: Theory, 95 (2014), 650. doi: 10.1016/j.na.2013.10.018. Google Scholar [19] I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid,, Commun. Pure Appl. Anal., (2014). doi: 10.1016/j.na.2013.10.018. Google Scholar [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. doi: 10.1007/s10884-004-4289-x. Google Scholar [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. Google Scholar [22] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics,, Springer Monographs in Mathematics, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [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. Google Scholar [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, (2012). Google Scholar [25] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups for supersonic flow-plate interactions,, J. of Diff. Eqs., 254 (2013), 1741. doi: 10.1016/j.jde.2012.11.009. Google Scholar [26] I. Chueshov and A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations,, C. R. Acad. Sci. Paris, 321 (1995), 607. Google Scholar [27] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Comm. Pure Appl. Anal., 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar [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. doi: 10.1016/j.jde.2012.11.006. Google Scholar [29] I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille-type flow,, Ukrainian Mathematical J., 65 (2013), 158. doi: 10.1007/s11253-013-0771-0. Google Scholar [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, (2011), 12. doi: 10.1007/978-3-642-36062-6_33. Google Scholar [31] P. Ciarlet and P. Rabier, Les Équations de Von Kármán,, Lecture Notes in Mathematics, (1980). Google Scholar [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. Google Scholar [33] K. F. Clancey, On finite Hilbert transforms,, Transactions AMS, 212 (1975), 347. doi: 10.1090/S0002-9947-1975-0377598-5. Google Scholar [34] D. G. Crighton, The Kutta condition in unsteady flow,, Ann. Rev. Fluid Mech., 17 (1985), 411. doi: 10.1146/annurev.fluid.17.1.411. Google Scholar [35] O. Diekmann, S. van Gils, S. Lunel and H. O. Walther, Delay Equations,, Springer, (1995). Google Scholar [36] E. Dowell, Nonlinear oscillations of a fluttering plate, I and II,, AIAA J., 4 (1966), 1267. Google Scholar [37] E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells,, AIAA Journal, 8 (1970), 385. Google Scholar [38] E. Dowell, O. Bendiksen, J. Edwards and T. Strganac, Transonic Nonlinear Aeroelasticity,, Encyclopedia of Aerospace Engineering, (2003). doi: 10.1002/9780470686652.eae151. Google Scholar [39] , E. Dowell,, Private Communication., (). Google Scholar [40] E. Dowell, A Modern Course in Aeroelasticity,, Kluwer Academic Publishers, (2004). Google Scholar [41] C. Eloy, C. Souilliez and L. Schouveiler, Flutter of a rectangular plate,, J. Fluids and Structures, 23 (2007), 904. doi: 10.1016/j.jfluidstructs.2007.02.002. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF00039321. Google Scholar [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. doi: 10.1016/j.jde.2012.10.016. Google Scholar [45] D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity,, Cambridge Univ. Press, (2002). doi: 10.1115/1.1566393. Google Scholar [46] P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis,, Automatica, 14 (1978), 367. doi: 10.1016/0005-1098(78)90036-5. Google Scholar [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. doi: 10.1063/1.4766724. Google Scholar [48] T. Von Kármán, Festigkeitsprobleme in Maschinenbau,, Encyklopedie der Mathematischen Wissenschaften, (1910), 348. Google Scholar [49] A. K. Khanmmamedov, Global attractors for von Karman equations with non-linear dissipation,, J. Math. Anal. Appl., 318 (2006), 92. doi: 10.1016/j.jmaa.2005.05.031. Google Scholar [50] A. Kornecki, E. H. Dowell and J. O'Brien, On the aeroelastic instability of two-dimensional panes in uniform incompressible flow,, 47 (1976), 47 (1976), 163. Google Scholar [51] E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow,, (in Russian) {Nauka}, (1978). Google Scholar [52] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar [53] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM, (1989). doi: 10.1137/1.9781611970821. Google Scholar [54] L. Landau and E. Lifshitz, Course of Theoretical Physics. Vol. 6. Fluid Mechanics,, Pergamon Press, (1963). Google Scholar [55] I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2002). doi: 10.1137/1.9780898717099. Google Scholar [56] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogenuous boundary value problems for second order hyperbolic operators,, J. Math. Pure et Appliques, 65 (1986), 149. Google Scholar [57] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. I, II,, Cambridge University Press, (2000). Google Scholar [58] I. Lasiecka and J. T. Webster, Generation of bounded semigroups in nonlinear flow-structure interactions with boundary damping,, Math. Methods in App. Sc., 36 (2013), 1995. doi: 10.1002/mma.1518. Google Scholar [59] I. Lasiecka and J. T. Webster, Long-time dynamics and control of subsonic flow-structure interactions,, Proceedings of the 2012 American Control Conference, (2012). Google Scholar [60] I. Laseicka and J. T. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows,, Discrete Contin. Dyn. Syst. Ser. S, (2014). Google Scholar [61] E. Livne, Future of Airplane Aeroelasticity,, J. of Aircraft, 40 (2003), 1066. doi: 10.2514/2.7218. Google Scholar [62] S. Miyatake, Mixed problem for hyperbolic equation of second order,, J. Math. Kyoto Univ., 13 (1973), 435. Google Scholar [63] S. Okada and D. Elliott, The finite Hilbert transform in $L_2$,, Math. Nachr., 153 (1991), 43. doi: 10.1002/mana.19911530105. Google Scholar [64] 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. doi: 10.1016/j.jmaa.2004.02.021. Google Scholar [65] I. Ryzhkova, Dynamics of a thermoelastic von Karman plate in a subsonic gas flow,, Zeitschrift Ang. Math. Phys., 58 (2007), 246. doi: 10.1007/s00033-006-0080-7. Google Scholar [66] M. Shubov, Riesz basis property of mode shapes for aircraft wing model (subsonic case),, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 607. doi: 10.1098/rspa.2005.1579. Google Scholar [67] M. Shubov, Solvability of reduced Possio integral equation in theoretical aeroelasticity,, Adv. Diff. Eqs., 15 (2010), 801. Google Scholar [68] D. Tataru, On the regularity of boundary traces for the wave equation., Ann. Scuola Normale. Sup. di Pisa., 26 (1998), 185. Google Scholar [69] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [70] F. G. Tricomi, Integral Equations,, Interscience Publishers Inc., (1957). Google Scholar [71] 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. doi: 10.1016/j.jfluidstructs.2013.04.004. Google Scholar [72] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar [73] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach,, Nonlinear Analysis, 74 (2011), 3123. doi: 10.1016/j.na.2011.01.028. Google Scholar [74] H. Widom, Integral Equations in $L_p$,, Transactions AMS, 97 (1960), 131. Google Scholar

show all references

##### References:
 [1] A. Babin and M. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (1992). Google Scholar [2] A. V. Balakrishnan, Aeroelasticity. Continuum Theory,, Springer, (2012). doi: 10.1007/978-1-4614-3609-6. Google Scholar [3] A. V. Balakrishnan, Nonlinear aeroelastic theory: Continuum models,, in Control Methods in PDE-Dynamical Systems, (2007), 79. doi: 10.1090/conm/426/08185. Google Scholar [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. doi: 10.1098/rspa.2003.1217. Google Scholar [5] H. M. Berger, A new approach to the analysis of large deflections of plates,, J. Appl. Mech., 22 (1955), 465. Google Scholar [6] R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity,, John Wiley and Sons, (1962). Google Scholar [7] L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping,, J. Diff. Eqs., 253 (2012), 3568. doi: 10.1016/j.jde.2012.08.004. Google Scholar [8] V. V. Bolotin, Nonconservative Problems of Elastic Stability,, Pergamon Press, (1963). Google Scholar [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. doi: 10.1002/(SICI)1099-1476(19990710)22:10<801::AID-MMA61>3.0.CO;2-T. Google Scholar [10] L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas,, C. R. Acad. Sci. Paris, 322 (1996), 1001. Google Scholar [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. doi: 10.1070/im1999v063n02ABEH000237. Google Scholar [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. doi: 10.1080/03605309708821307. Google Scholar [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. doi: 10.1007/s00021-004-0121-y. Google Scholar [14] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics,, 3rd edition, (1993). Google Scholar [15] I. Chueshov, On a system of equations with delay that arises in aero-elasticity,, (in Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen, 58 (1990), 123. doi: 10.1007/BF01097291. Google Scholar [16] I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems,, (in Russian) Acta, (1999). Google Scholar [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, (2000), 1. Google Scholar [18] I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid,, Nonlinear Analysis: Theory, 95 (2014), 650. doi: 10.1016/j.na.2013.10.018. Google Scholar [19] I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid,, Commun. Pure Appl. Anal., (2014). doi: 10.1016/j.na.2013.10.018. Google Scholar [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. doi: 10.1007/s10884-004-4289-x. Google Scholar [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. Google Scholar [22] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics,, Springer Monographs in Mathematics, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [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. Google Scholar [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, (2012). Google Scholar [25] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups for supersonic flow-plate interactions,, J. of Diff. Eqs., 254 (2013), 1741. doi: 10.1016/j.jde.2012.11.009. Google Scholar [26] I. Chueshov and A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations,, C. R. Acad. Sci. Paris, 321 (1995), 607. Google Scholar [27] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Comm. Pure Appl. Anal., 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar [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. doi: 10.1016/j.jde.2012.11.006. Google Scholar [29] I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille-type flow,, Ukrainian Mathematical J., 65 (2013), 158. doi: 10.1007/s11253-013-0771-0. Google Scholar [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, (2011), 12. doi: 10.1007/978-3-642-36062-6_33. Google Scholar [31] P. Ciarlet and P. Rabier, Les Équations de Von Kármán,, Lecture Notes in Mathematics, (1980). Google Scholar [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. Google Scholar [33] K. F. Clancey, On finite Hilbert transforms,, Transactions AMS, 212 (1975), 347. doi: 10.1090/S0002-9947-1975-0377598-5. Google Scholar [34] D. G. Crighton, The Kutta condition in unsteady flow,, Ann. Rev. Fluid Mech., 17 (1985), 411. doi: 10.1146/annurev.fluid.17.1.411. Google Scholar [35] O. Diekmann, S. van Gils, S. Lunel and H. O. Walther, Delay Equations,, Springer, (1995). Google Scholar [36] E. Dowell, Nonlinear oscillations of a fluttering plate, I and II,, AIAA J., 4 (1966), 1267. Google Scholar [37] E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells,, AIAA Journal, 8 (1970), 385. Google Scholar [38] E. Dowell, O. Bendiksen, J. Edwards and T. Strganac, Transonic Nonlinear Aeroelasticity,, Encyclopedia of Aerospace Engineering, (2003). doi: 10.1002/9780470686652.eae151. Google Scholar [39] , E. Dowell,, Private Communication., (). Google Scholar [40] E. Dowell, A Modern Course in Aeroelasticity,, Kluwer Academic Publishers, (2004). Google Scholar [41] C. Eloy, C. Souilliez and L. Schouveiler, Flutter of a rectangular plate,, J. Fluids and Structures, 23 (2007), 904. doi: 10.1016/j.jfluidstructs.2007.02.002. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF00039321. Google Scholar [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. doi: 10.1016/j.jde.2012.10.016. Google Scholar [45] D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity,, Cambridge Univ. Press, (2002). doi: 10.1115/1.1566393. Google Scholar [46] P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis,, Automatica, 14 (1978), 367. doi: 10.1016/0005-1098(78)90036-5. Google Scholar [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. doi: 10.1063/1.4766724. Google Scholar [48] T. Von Kármán, Festigkeitsprobleme in Maschinenbau,, Encyklopedie der Mathematischen Wissenschaften, (1910), 348. Google Scholar [49] A. K. Khanmmamedov, Global attractors for von Karman equations with non-linear dissipation,, J. Math. Anal. Appl., 318 (2006), 92. doi: 10.1016/j.jmaa.2005.05.031. Google Scholar [50] A. Kornecki, E. H. Dowell and J. O'Brien, On the aeroelastic instability of two-dimensional panes in uniform incompressible flow,, 47 (1976), 47 (1976), 163. Google Scholar [51] E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow,, (in Russian) {Nauka}, (1978). Google Scholar [52] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar [53] J. Lagnese, Boundary Stabilization of Thin Plates,, SIAM, (1989). doi: 10.1137/1.9781611970821. Google Scholar [54] L. Landau and E. Lifshitz, Course of Theoretical Physics. Vol. 6. Fluid Mechanics,, Pergamon Press, (1963). Google Scholar [55] I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2002). doi: 10.1137/1.9780898717099. Google Scholar [56] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogenuous boundary value problems for second order hyperbolic operators,, J. Math. Pure et Appliques, 65 (1986), 149. Google Scholar [57] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. I, II,, Cambridge University Press, (2000). Google Scholar [58] I. Lasiecka and J. T. Webster, Generation of bounded semigroups in nonlinear flow-structure interactions with boundary damping,, Math. Methods in App. Sc., 36 (2013), 1995. doi: 10.1002/mma.1518. Google Scholar [59] I. Lasiecka and J. T. Webster, Long-time dynamics and control of subsonic flow-structure interactions,, Proceedings of the 2012 American Control Conference, (2012). Google Scholar [60] I. Laseicka and J. T. Webster, Eliminating flutter for clamped von Karman plates immersed in subsonic flows,, Discrete Contin. Dyn. Syst. Ser. S, (2014). Google Scholar [61] E. Livne, Future of Airplane Aeroelasticity,, J. of Aircraft, 40 (2003), 1066. doi: 10.2514/2.7218. Google Scholar [62] S. Miyatake, Mixed problem for hyperbolic equation of second order,, J. Math. Kyoto Univ., 13 (1973), 435. Google Scholar [63] S. Okada and D. Elliott, The finite Hilbert transform in $L_2$,, Math. Nachr., 153 (1991), 43. doi: 10.1002/mana.19911530105. Google Scholar [64] 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. doi: 10.1016/j.jmaa.2004.02.021. Google Scholar [65] I. Ryzhkova, Dynamics of a thermoelastic von Karman plate in a subsonic gas flow,, Zeitschrift Ang. Math. Phys., 58 (2007), 246. doi: 10.1007/s00033-006-0080-7. Google Scholar [66] M. Shubov, Riesz basis property of mode shapes for aircraft wing model (subsonic case),, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 607. doi: 10.1098/rspa.2005.1579. Google Scholar [67] M. Shubov, Solvability of reduced Possio integral equation in theoretical aeroelasticity,, Adv. Diff. Eqs., 15 (2010), 801. Google Scholar [68] D. Tataru, On the regularity of boundary traces for the wave equation., Ann. Scuola Normale. Sup. di Pisa., 26 (1998), 185. Google Scholar [69] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [70] F. G. Tricomi, Integral Equations,, Interscience Publishers Inc., (1957). Google Scholar [71] 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. doi: 10.1016/j.jfluidstructs.2013.04.004. Google Scholar [72] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar [73] J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach,, Nonlinear Analysis, 74 (2011), 3123. doi: 10.1016/j.na.2011.01.028. Google Scholar [74] H. Widom, Integral Equations in $L_p$,, Transactions AMS, 97 (1960), 131. Google Scholar
 [1] George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417 [2] Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181 [3] Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261 [4] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37 [5] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [6] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023 [7] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389 [8] Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 [9] John R. Tucker. Attractors and kernels: Linking nonlinear PDE semigroups to harmonic analysis state-space decomposition. Conference Publications, 2001, 2001 (Special) : 366-370. doi: 10.3934/proc.2001.2001.366 [10] Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253 [11] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [12] Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455 [13] Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387 [14] Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 [15] Tristan Roy. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1307-1323. doi: 10.3934/dcds.2009.24.1307 [16] Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1 [17] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 [18] Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019 [19] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [20] Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605

2018 Impact Factor: 0.545