Long-time dynamics for a coupled system modeling the oscillations of suspension bridges

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Abstract

This paper is concerned with a coupled system modeling the oscillations of suspension bridges, which consists of a beam equation and a viscoelastic string equation. We first transformed the original initial-boundary value problem into an equivalent one in the history space framework. Then we obtained the global well-posedness and regularity of mild solutions by using the semigroup theory. In addition, we employed the perturbed energy method to establish a stabilizability estimate. By verifying the gradient property and quasi-stability of the corresponding dynamical system, we derived the existence of a global attractor with finite fractal dimension.

Keywords

coupled beam-string system / global well-posedness / global attractors / gradient dynamical system / quasi-stability

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References

[1] A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst., 35 (2015), 5879-5908. https://doi.org/10.3934/dcds.2015.35.5879

[2] F. Gazzola, Mathematical Models for Suspension Bridges: Nonlinear Structural Instability , Cham: Springer, 2015. https://doi.org/10.1007/978-3-319-15434-3

[3] R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability , Reston, Virginia: ASCE Press, 2001.

[4] J. H. G. Macdonald, Lateral excitation of bridges by balancing pedestrians, Proc. R. Soc. A Math. Phys. Eng. Sci., 465 (2009), 1055-1073. https://doi.org/10.1098/rspa.2008.0367

[5] S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, E. Ott, Crowd synchrony on the Millennium Bridge, Nature, 438 (2005), 43-44. https://doi.org/10.1038/438043a

[6] N. U. Ahmed, H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853-874. https://doi.org/10.1137/S0036139996308698

[7] A. C. Lazer, P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. https://doi.org/10.1137/1032120

[8] M. Aouadi, Robustness of global attractors for extensible coupled suspension bridge equations with fractional damping, Appl. Math. Optim., 84 (2021), 403-435. https://doi.org/10.1007/s00245-021-09774-8

[9] G. Arioli, F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the tacoma narrows bridge collapse, Appl. Math. Model., 39 (2015), 901-912. https://doi.org/10.1016/j.apm.2014.06.022

[10] U. Battisti, E. Berchio, A. Ferrero, F. Gazzola, Energy transfer between modes in a nonlinear beam equation, J. Math. Pures Appl., 108 (2017), 885-917. https://doi.org/10.1016/j.matpur.2017.05.010

[11] V. Benci, D. Fortunato, F. Gazzola, Existence of torsional solitons in a beam model of suspension bridge, Arch. Ration. Mech. Anal., 226 (2017), 559-585. https://doi.org/10.1007/s00205-017-1138-8

[12] I. Bochicchio, C. Giorgi, E. Vuk, Long-term dynamics of the coupled suspension bridge system, Math. Models Methods Appl. Sci., 22 (2012), 1250021. https://doi.org/10.1142/S0218202512500212

[13] F. Dell’Oro, C. Giorgi, V. Pata, Asymptotic behavior of coupled linear systems modeling suspension bridges, Z. Angew. Math. Phys., 66 (2015), 1095-1108. https://doi.org/10.1007/s00033-014-0414-9

[14] F. Gazzola, A. Soufyane, Long-time behavior of partially damped systems modeling degenerate plates with piers, Nonlinearity, 34 (2021), 7705-7727. https://doi.org/10.1088/1361-6544/ac24e2

[15] A. C. Lazer, P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 243-274. https://doi.org/10.1016/S0294-1449(16)30368-7

[16] Q. Z. Ma, C. K. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775. https://doi.org/10.1016/j.jde.2009.02.022

[17] P. J. McKenna, Large torsional oscillations in suspension bridges revisited: fixing an old approximation, Amer. Math. Monthly, 106 (1999), 1-18. https://doi.org/10.1080/00029890.1999.12005001

[18] P. J. McKenna, C. ó. Tuama, Large torsional oscillations in suspension bridges visited again: vertical forcing creates torsional response, Amer. Math. Monthly, 108 (2001), 738-745. https://doi.org/10.1080/00029890.2001.11919805

[19] P. J. McKenna, W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Ration. Mech. Anal., 98 (1987), 167-177. https://doi.org/10.1007/BF00251232

[20] P. J. McKenna, W. Walter, Travelling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715. https://doi.org/10.1137/0150041

[21] P. R. S. Antunes, F. Gazzola, Some solutions of minimaxmax problems for the torsional displacements of rectangular plates, ZAMM Z. Angew. Math. Mech., 98 (2018), 1974-1991. https://doi.org/10.1002/zamm.201800065

[22] E. Berchio, D. Buoso, F. Gazzola, D. Zucco, A minimaxmax problem for improving the torsional stability of rectangular plates, J. Optim. Theory Appl., 177 (2018), 64-92. https://doi.org/10.1007/s10957-018-1261-1

[23] E. Berchio, A. Falocchi, A. Ferrero, D. Ganguly, On the first frequency of reinforced partially hinged plates, Commun. Contemp. Math., 23 (2021), 1950074. https://doi.org/10.1142/S0219199719500743

[24] E. Berchio, A. Ferrero, F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions, Nonlinear Anal. Real World Appl., 28 (2016), 91-125. https://doi.org/10.1016/j.nonrwa.2015.09.005

[25] D. Bonheure, F. Gazzola, E. M. Dos Santos, Periodic solutions and torsional instability in a nonlinear nonlocal plate equation, SIAM J. Math. Anal., 51 (2019), 3052-3091. https://doi.org/10.1137/18M1221242

[26] V. Ferreira Jr, F. Gazzola, E. M. dos Santos, Instability of modes in a partially hinged rectangular plate, J. Differential Equations, 261 (2016), 6302-6340. https://doi.org/10.1016/j.jde.2016.08.037

[27] Y. Liu, Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges, Commun. Anal. Mech., 15 (2023), 436-456. https://doi.org/10.3934/cam.2023021

[28] Y. Liu, J. Mu, Y. J. Jiao, A class of fourth order damped wave equations with arbitrary positive initial energy, Proc. Edinburgh Math. Soc., 62 (2019), 165-178. https://doi.org/10.1017/S0013091518000330

[29] R. Z. Xu, X. C. Wang, Y. B. Yang, S. H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503. https://doi.org/10.1063/1.5006728

[30] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics , 2 Eds., New York: Springer-Verlag, 1997. https://doi.org/10.1007/978-1-4684-0313-8

[31] A. V. Babin, M. I. Vishik, Attractors of Evolution Equations , In: Studies in Mathematics and its Applications, Amsterdam: North-Holland, 1992.

[32] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations , Cambridge: Cambridge University Press, 1991. https://doi.org/10.1017/cbo9780511569418

[33] J. K. Hale, Asymptotic Behavior of Dissipative Systems , In: Mathematical Surveys and Monographs, American Mathematical Society, 1988. http://doi.org/10.1090/surv/025

[34] Q. F. Ma, S. H. Wang, C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. https://doi.org/10.1512/iumj.2002.51.2255

[35] M. Conti, V. Danese, C. Giorgi, V. Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math., 140 (2018), 349-389. https://doi.org/10.1353/ajm.2018.0008

[36] M. Conti, F. Dell’Oro, V. Pata, Some unexplored questions arising in linear viscoelasticity, J. Funct. Anal., 282 (2022), 109422. https://doi.org/10.1016/j.jfa.2022.109422

[37] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. https://doi.org/10.1007/BF00251609

[38] L. Deseri, M. Fabrizio, M. Golden, The concept of a minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96. https://doi.org/10.1007/s00205-005-0406-1

[39] F. Di Plinio, V. Pata, S. Zelik, On the strongly damped wave equation with memory, Indiana Univ. Math. J., 57 (2008), 757-780. https://doi.org/10.1512/iumj.2008.57.3266

[40] M. Fabrizio, C. Giorgi, V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232. https://doi.org/10.1007/s00205-010-0300-3

[41] M. Fabrizio, B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. https://doi.org/10.1007/BF00375589

[42] J. C. O. Faria, A. Y. Souza Franco, Well-posedness and exponential stability for a Klein-Gordon system with locally distributed viscoelastic dampings in a past-history framework, J. Differential Equations, 346 (2023), 108-144. https://doi.org/10.1016/j.jde.2022.11.022

[43] V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[44] I. Chueshov, I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics , New York: Springer-Verlag, 2010. https://doi.org/10.1007/978-0-387-87712-9

[45] J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. https://doi.org/10.1090/S0002-9939-1977-0442748-6

[46] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , New York: Springer-Verlag, 1983. https://doi.org/10.1007/978-1-4612-5561-1

[47] M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510. https://doi.org/10.57262/die/1356060344

[48] M. M. Cavalcanti, V. N. Domingos Cavalcanti, P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. https://doi.org/10.1016/j.jde.2004.04.011

[49] Y. Liu, B. Moon, V. D. Rădulescu, R. Z. Xu, C. Yang, Qualitative properties of solution to a viscoelastic Kirchhoff-like plate equation, J. Math. Phys., 64 (2023), 051511. https://doi.org/10.1063/5.0149240