Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation

Lasiecka, I.; Ma, T.; Monteiro, R.

doi:10.1090/tran/7756

Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation
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by I. Lasiecka, T. F. Ma and R. N. Monteiro PDF

Trans. Amer. Math. Soc. 371 (2019), 8051-8096 Request permission

Abstract:

Nonlinear shallow shell models with thermal effects are considered. Such models provide basic prototypes for elastic bodies appearing in the flow/fluid structure interactions. It is assumed that shells are thin and do not account for the regularizing effects of rotary inertia. The nonlinear effects in the model become supercritical, and this raises a first fundamental question of Hadamard well-posedness in the class of weak solutions. The first main result of the present paper addresses the issue of generation of a nonlinear semigroup for such a model. The second result describes longtime behavior of the resulting dynamical system. It is shown that longtime dynamics admits finite-dimensional and smooth global attractors. This result is obtained without imposing any mechanical dissipation affecting the vertical displacements of the shell where the latter satisfy free boundary conditions. This particular feature, along with supercritical nonlinearity, leads to substantial challenges in the analysis. The resolution of the encountered difficulties rests on recently developed mathematical tools such as

[(1)] maximal regularity for thermal shells with free boundary conditions,
[(2)] “hidden” trace regularity propagated by thermal effects,
[(3)] compensated compactness and related theory of quasi-stable systems derived from books by Chueshov and by Chueshov and Lasiecka.

References

George Avalos and Irena Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29 (1998), no. 1, 155–182. MR 1617180, DOI 10.1137/S0036141096300823
George Avalos and Irena Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste 28 (1996), no. suppl., 1–28 (1997). Dedicated to the memory of Pierre Grisvard. MR 1602473
Moncef Aouadi and Kaouther Boulehmi, Partial exact controllability for inhomogeneous multidimensional thermoelastic diffusion problem, Evol. Equ. Control Theory 5 (2016), no. 2, 201–224. MR 3511695, DOI 10.3934/eect.2016001
Moncef Aouadi and Taoufik Moulahi, The controllability of a thermoelastic plate problem revisited, Evol. Equ. Control Theory 7 (2018), no. 1, 1–31. MR 3810184, DOI 10.3934/eect.2018001
A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492
Assia Benabdallah and Irena Lasiecka, Exponential decay rates for a full von Kármán system of dynamic thermoelasticity, J. Differential Equations 160 (2000), no. 1, 51–93. MR 1734529, DOI 10.1006/jdeq.1999.3656
Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter, Representation and control of infinite-dimensional systems. Vol. II, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1246331, DOI 10.1007/978-1-4612-2750-2
Michel Bernadou and J. Tinsley Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60 (1981), no. 3, 285–308. MR 633006
Tania Biswas and Sheetal Dharmatti, Control problems and invariant subspaces for sabra shell model of turbulence, Evol. Equ. Control Theory 7 (2018), no. 3, 417–445. MR 3825874, DOI 10.3934/eect.2018021
Anne Boutet de Monvel and Igor Chueshov, Uniqueness theorem for weak solutions of von Karman evolution equations, J. Math. Anal. Appl. 221 (1998), no. 2, 419–429. MR 1621722, DOI 10.1006/jmaa.1997.5681
Francesca Bucci and Igor Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Contin. Dyn. Syst. 22 (2008), no. 3, 557–586. MR 2429854, DOI 10.3934/dcds.2008.22.557
John Cagnol, Irena Lasiecka, Catherine Lebiedzik, and Richard Marchand, Hadamard well-posedness for a class of nonlinear shallow shell problems, Nonlinear Anal. 67 (2007), no. 8, 2452–2484. MR 2338113, DOI 10.1016/j.na.2006.09.004
Igor Chueshov, Remark on an elastic plate interacting with a gas in a semi-infinite tube: periodic solutions, Evol. Equ. Control Theory 5 (2016), no. 4, 561–566. MR 3603248, DOI 10.3934/eect.2016019
Igor Chueshov, Dynamics of quasi-stable dissipative systems, Universitext, Springer, Cham, 2015. MR 3408002, DOI 10.1007/978-3-319-22903-4
Igor Chueshov, Earl H. Dowell, Irena Lasiecka, and Justin T. Webster, Nonlinear elastic plate in a flow of gas: recent results and conjectures, Appl. Math. Optim. 73 (2016), no. 3, 475–500. MR 3498935, DOI 10.1007/s00245-016-9349-1
Igor Chueshov, Matthias Eller, and Irena Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations 27 (2002), no. 9-10, 1901–1951. MR 1941662, DOI 10.1081/PDE-120016132
Igor Chueshov, Irena Lasiecka, and Justin T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Comm. Partial Differential Equations 39 (2014), no. 11, 1965–1997. MR 3251861, DOI 10.1080/03605302.2014.930484
Igor Chueshov and Irena Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. 195 (2008), no. 912, viii+183. MR 2438025, DOI 10.1090/memo/0912
Igor Chueshov and Irena Lasiecka, Von Karman evolution equations, Springer Monographs in Mathematics, Springer, New York, 2010. Well-posedness and long-time dynamics. MR 2643040, DOI 10.1007/978-0-387-87712-9
Igor Chueshov and Irena Lasiecka, Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim. 58 (2008), no. 2, 195–241. MR 2439660, DOI 10.1007/s00245-007-9031-8
Igor Chueshov and Tamara Fastovska, On interaction of circular cylindrical shells with a Poiseuille type flow, Evol. Equ. Control Theory 5 (2016), no. 4, 605–629. MR 3603250, DOI 10.3934/eect.2016021
Igor Chueshov and Iryna Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations 254 (2013), no. 4, 1833–1862. MR 3003294, DOI 10.1016/j.jde.2012.11.006
Philippe G. Ciarlet, Mathematical elasticity. Vol. II, Studies in Mathematics and its Applications, vol. 27, North-Holland Publishing Co., Amsterdam, 1997. Theory of plates. MR 1477663
Philippe G. Ciarlet, Mathematical elasticity. Vol. III, Studies in Mathematics and its Applications, vol. 29, North-Holland Publishing Co., Amsterdam, 2000. Theory of shells. MR 1757535
Philippe G. Ciarlet and Patrick Rabier, Les équations de von Kármán, Lecture Notes in Mathematics, vol. 826, Springer, Berlin, 1980 (French). MR 595326
R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 233539, DOI 10.1007/BF00276727
Michel C. Delfour and Jean-Paul Zolésio, Differential equations for linear shells: comparison between intrinsic and classical models, Advances in mathematical sciences: CRM’s 25 years (Montreal, PQ, 1994) CRM Proc. Lecture Notes, vol. 11, Amer. Math. Soc., Providence, RI, 1997, pp. 41–124. MR 1479670, DOI 10.1090/crmp/011/04
Robert Denk and Yoshihiro Shibata, Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evol. Equ. 17 (2017), no. 1, 215–261. MR 3630321, DOI 10.1007/s00028-016-0367-x
M. Eller, V. Isakov, G. Nakamura, and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998) Stud. Math. Appl., vol. 31, North-Holland, Amsterdam, 2002, pp. 329–349. MR 1936000, DOI 10.1016/S0168-2024(02)80016-9
Pelin G. Geredeli, Irena Lasiecka, and Justin T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, J. Differential Equations 254 (2013), no. 3, 1193–1229. MR 2997367, DOI 10.1016/j.jde.2012.10.016
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
Scott W. Hansen, Boundary control of a one-dimensional linear thermoelastic rod, SIAM J. Control Optim. 32 (1994), no. 4, 1052–1074. MR 1280229, DOI 10.1137/S0363012991222607
Mary Ann Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control 8 (1998), no. 2, 11 pp.}, issn=1052-0600, review= MR 1651449,
Jason S. Howell, Irena Lasiecka, and Justin T. Webster, Quasi-stability and exponential attractors for a non-gradient system—applications to piston-theoretic plates with internal damping, Evol. Equ. Control Theory 5 (2016), no. 4, 567–603. MR 3603249, DOI 10.3934/eect.2016020
Victor Isakov, A nonhyperbolic Cauchy problem for $\square _b\square _c$ and its applications to elasticity theory, Comm. Pure Appl. Math. 39 (1986), no. 6, 747–767. MR 859272, DOI 10.1002/cpa.3160390603
Jong Uhn Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (1992), no. 4, 889–899. MR 1166563, DOI 10.1137/0523047
Herbert Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math. 58 (2000), no. 4, 601–612. MR 1788420, DOI 10.1090/qam/1788420
Herbert Koch and Irena Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Evolution equations, semigroups and functional analysis (Milano, 2000) Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 197–216. MR 1944164
Herbert Koch and Andreas Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods Appl. Sci. 16 (1993), no. 8, 581–586. MR 1233041, DOI 10.1002/mma.1670160806
V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627, DOI 10.1017/CBO9780511569418
John E. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal. 112 (1990), no. 3, 223–267. MR 1076073, DOI 10.1007/BF00381235
J. E. Lagnese, Uniform boundary stabilization of thermoelastic plates, Control of boundaries and stabilization (Clermont-Ferrand, 1988) Lect. Notes Control Inf. Sci., vol. 125, Springer, Berlin, 1989, pp. 154–167. MR 1015968, DOI 10.1007/BFb0043358
John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153, DOI 10.1137/1.9781611970821
J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 6, Masson, Paris, 1988. MR 953313
Irena Lasiecka, Weak, classical and intermediate solutions to full von Karman system of dynamic nonlinear elasticity, Appl. Anal. 68 (1998), no. 1-2, 121–145. MR 1623321, DOI 10.1080/00036819808840625
Irena Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim. 36 (1998), no. 4, 1376–1422. MR 1627581, DOI 10.1137/S0363012996301907
Irena Lasiecka, Uniform decay rates for full von Kármán system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Comm. Partial Differential Equations 24 (1999), no. 9-10, 1801–1847. MR 1708109, DOI 10.1080/03605309908821483
I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), no. 2, 149–192. MR 867669
Irena Lasiecka, To Fu Ma, and Rodrigo Nunes Monteiro, Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), no. 3, 1037–1072. MR 3810108, DOI 10.3934/dcdsb.2018141
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507–533. MR 1202555
Irena Lasiecka and Roberto Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 3-4, 457–482 (1999). MR 1678002
Irena Lasiecka and Roberto Triggiani, Control theory for partial differential equations: continuous and approximation theories. I, Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000. Abstract parabolic systems. MR 1745475
Irena Lasiecka and Roberto Triggiani, Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks, J. Math. Anal. Appl. 269 (2002), no. 2, 642–688. MR 1907136, DOI 10.1016/S0022-247X(02)00041-0
I. Lasiecka, R. Triggiani, and V. Valente, Uniform stabilization of spherical shells by boundary dissipation, Adv. Differential Equations 1 (1996), no. 4, 635–674. MR 1401407
I. Lasiecka, R. Triggiani, and Peng-Fei Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl. 235 (1999), no. 1, 13–57. MR 1758667, DOI 10.1006/jmaa.1999.6348
Irena Lasiecka and Vanda Valente, Uniform boundary stabilization of a nonlinear shallow and thin elastic spherical cap, J. Math. Anal. Appl. 202 (1996), no. 3, 951–994. MR 1408363, DOI 10.1006/jmaa.1996.0356
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
Z.-Y. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett. 8 (1995), no. 3, 1–6. MR 1356798, DOI 10.1016/0893-9659(95)00020-Q
Jaime E. Muñoz Rivera and Reinhard Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal. 26 (1995), no. 6, 1547–1563. MR 1356459, DOI 10.1137/S0036142993255058
Jaime E. Muñoz Rivera and Gustavo Perla Menzala, Uniform rates of decay for full von Kármán systems of dynamic viscoelasticity with memory, Asymptot. Anal. 27 (2001), no. 3-4, 335–357. MR 1858921
P. M. Naghdi, Foundations of elastic shell theory, Progress in Solid Mechanics, Vol. IV, North-Holland, Amsterdam, 1963, pp. 1–90. MR 0163488
G. Perla Menzala and F. Travessini De Cezaro, Global existence and uniqueness of weak and regular solutions of shallow shells with thermal effects, Appl. Math. Optim. 74 (2016), no. 2, 229–271. MR 3547006, DOI 10.1007/s00245-015-9313-5
A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl. (9) 71 (1992), no. 5, 455–467. MR 1191585
Iryna Ryzhkova, Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow, Z. Angew. Math. Phys. 58 (2007), no. 2, 246–261. MR 2305714, DOI 10.1007/s00033-006-0080-7
Reiko Sakamoto, Hyperbolic boundary value problems, Cambridge University Press, Cambridge-New York, 1982. Translated from the Japanese by Katsumi Miyahara. MR 666700
J. Lyell Sanders Jr., Nonlinear theories for thin shells, Quart. Appl. Math. 21 (1963), 21–36. MR 147023, DOI 10.1090/S0033-569X-1963-0147023-4
Giuseppe Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations 22 (1997), no. 5-6, 869–899. MR 1452171, DOI 10.1080/03605309708821287
V. I. Sedenko, On the uniqueness theorem for generalized solutions of initial-boundary problems for the Marguerre-Vlasov vibrations of shallow shells with clamped boundary conditions, Appl. Math. Optim. 39 (1999), no. 3, 309–326. MR 1675106, DOI 10.1007/s002459900108
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967, DOI 10.1007/978-1-4684-0313-8
Roberto Triggiani, Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space, Evol. Equ. Control Theory 5 (2016), no. 4, 489–514. MR 3603245, DOI 10.3934/eect.2016016
Roberto Triggiani and Jing Zhang, Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory 7 (2018), no. 1, 153–182. MR 3810191, DOI 10.3934/eect.2018008
Roberto Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim. 46 (2002), no. 2-3, 331–375. Special issue dedicated to the memory of Jacques-Louis Lions. MR 1944764, DOI 10.1007/s00245-002-0751-5
V. V. Vedeneev, Effect of damping on flutter of simply supported and clamped panels at low supersonic speeds, J. Fluids Struct. 40 (2013), 366–372.
I. I. Vorovich, Nonlinear theory of shallow shells, Applied Mathematical Sciences, vol. 133, Springer-Verlag, New York, 1999. Translated from the Russian by Michael Grinfeld; Translation edited and with a preface by L. P. Lebedev. MR 1712882
Peng-Fei Yao, Observability inequalities for shallow shells, SIAM J. Control Optim. 38 (2000), no. 6, 1729–1756. MR 1776654, DOI 10.1137/S0363012999338692