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October  2014, 7(5): 925-965. doi: 10.3934/dcdss.2014.7.925

Flow-plate interactions: Well-posedness and long-time behavior

Igor Chueshov 1, , Irena Lasiecka 2, and  Justin Webster 3,

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.
Keywords: global attractors, nonlinear semigroups, well-posedness, Flow-structure interaction, nonlinear plate, PDE with delay..
Mathematics Subject Classification: Primary: 35M33, 74F10; Secondary: 35B41, 35Q7.
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

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I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2002). doi: 10.1137/1.9780898717099. Google Scholar

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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

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I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. I, II,, Cambridge University Press, (2000). Google Scholar

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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

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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]

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