# American Institute of Mathematical Sciences

May  2018, 23(3): 1037-1072. doi: 10.3934/dcdsb.2018141

## Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

 1 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA and IBS-Polish Academy of Sciences, Warsaw, Poland 2 Institute of Mathematical and Computer Sciences, University of São Paulo, 13566-590 São Carlos, SP, Brazil

* Corresponding author: Irena Lasiecka

Received  January 2017 Revised  June 2017 Published  May 2018 Early access  February 2018

This paper is concerned with long-time dynamics of a full von Karman system subject to nonlinear thermal coupling and free boundary conditions. In contrast with scalar von Karman system, vectorial full von Karman system accounts for both vertical and in plane displacements. The corresponding PDE is of critical interest in flow structure interactions where nonlinear plate/shell dynamics interacts with perturbed flows [vicid or invicid] [8,9,15]. In this paper it is shown that the system admits a global attractor which is also smooth and of finite fractal dimension. The above result is shown to hold for plates without regularizing effects of rotational inertia and without any mechanical dissipation imposed on vertical displacements. This is in contrast with the literature on this topic [15] and references therein. In order to handle highly supercritical nature of the von Karman nonlinearities, new results on "hidden" trace regularity generated by thermal effects are exploited. These lead to asymptotic compensated compactness of trajectories which then allows to use newly developed tools pertaining to quasi stable dynamical systems [8].

Citation: Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141
##### References:
 [1] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials With Memory. Theory and Applications, Springer, New York, 2012. [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.  doi: 10.1137/S0036141096300823. [3] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Dedicated to the memory of Pierre Grisvard, Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl., 1-28 (1997). [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992. [5] A. Benabdallah and I. Lasiecka, Exponential decay rates for a full von Karman system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93.  doi: 10.1006/jdeq.1999.3656. [6] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. Ⅱ. Systems & Control: Foundations & Applications, Birkhäuser Boston, Boston, 1993. [7] F. Bucci and I. Chueshov, Long time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discr. Cont. Dyn. Systems, 22 (2008), 557-586.  doi: 10.3934/dcds.2008.22.557. [8] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. [9] I. Chueshov, E. Dowell, I. Lasiecka and J. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500.  doi: 10.1007/s00245-016-9349-1. [10] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132. [11] I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Comm. Partial Differential Equations, 39 (2014), 1965-1997.  doi: 10.1080/03605302.2014.930484. [12] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolutions with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. [13] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010. [14] I. Chueshov and I. Lasiecka, Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241.  doi: 10.1007/s00245-007-9031-8. [15] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations, 254 (2013), 1833-1862.  doi: 10.1016/j.jde.2012.11.006. [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656. [17] P. G. Ciarlet and P. Rabier, Les Équations de von Kármán, Springer Verlag, 1980. [18] P. G. Ciarlet, Mathematical Elasticity, Vol. II, Theory of Plates. Studies in Mathematics and its Applications, 27. North-Holland, Amsterdam, 1997. [19] C. 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.  doi: 10.1007/BF00276727. [20] M. Eller, V. Isakov, G. Nakamura and D. Tataru, On the uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., North-Holland, Amsterdam, 31 (2002), 329-349. [21] P. G. Geredeli, I. Lasiecka and J. 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), 1193-1229.  doi: 10.1016/j.jde.2012.10.016. [22] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988. [23] S. Hansen, Boundary control of a one dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074.  doi: 10.1137/S0363012991222607. [24] M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11pp. [25] V. Isakov, A nonhyperbolic Cauchy problem for $\Box_a \Box_b$ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767.  doi: 10.1002/cpa.3160390603. [26] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047. [27] H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math., 58 (2000), 601-612.  doi: 10.1090/qam/1788420. [28] H. Koch and I. 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., Birkhäuser, Basel, 50 (2002), 197-216. [29] H. Koch and A. Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods Appl. Sci., 16 (1993), 581-586.  doi: 10.1002/mma.1670160806. [30] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. ⅷ+156 pp. [31] O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. [32] J. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal., 112 (1990), 223-267.  doi: 10.1007/BF00381235. [33] J. E. Lagnese, Uniform boundary stabilization of thermoelastic plates, Control of boundaries and stabilization, (Clermont-Ferrand, 1988), Lecture Notes in Control and Inform. Sci., 125 (1989), 154-167. [34] J. E Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. SIAM, Philadelphia, PA, 1989. [35] J. E. Lagnese and J. L. Lions, Modeling Analysis and Control of Thin Plates, Research in Applied Mathematics, 6. Masson, Paris, 1988. [36] I. Lasiecka, Weak, classical and intermediate solutions to full von Karman system of dynamic nonlinear elasticity, Appl. Anal., 68 (1998), 121-145.  doi: 10.1080/00036819808840625. [37] I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422.  doi: 10.1137/S0363012996301907. [38] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-553. [39] I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 1801-1847.  doi: 10.1080/03605309908821483. [40] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogenous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl.(9), 65 (1986), 149-192. [41] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4), 27 (1998), 457-482. [42] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. [43] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972. [44] Z. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q. [45] J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058. [46] 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), 229-271.  doi: 10.1007/s00245-015-9313-5. [47] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl.(9), 71 (1992), 455-467. [48] I. Ryzhkova, Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow, Z. Angew. Math. Phys., 58 (2007), 246-261.  doi: 10.1007/s00033-006-0080-7. [49] I. Ryzhkova, Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. Appl., 294 (2004), 462-481.  doi: 10.1016/j.jmaa.2004.02.021. [50] R. Sakamoto, Hyperbolic Boundary Value Problem, Cambridge University Press, 1982. [51] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988. [52] P.-F. Yao, Observability inequalities for shallow shells, SIAM J. Control Optim., 38 (2000), 1729-1756.  doi: 10.1137/S0363012999338692.

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##### References:
 [1] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials With Memory. Theory and Applications, Springer, New York, 2012. [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.  doi: 10.1137/S0036141096300823. [3] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Dedicated to the memory of Pierre Grisvard, Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl., 1-28 (1997). [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992. [5] A. Benabdallah and I. Lasiecka, Exponential decay rates for a full von Karman system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93.  doi: 10.1006/jdeq.1999.3656. [6] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. Ⅱ. Systems & Control: Foundations & Applications, Birkhäuser Boston, Boston, 1993. [7] F. Bucci and I. Chueshov, Long time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discr. Cont. Dyn. Systems, 22 (2008), 557-586.  doi: 10.3934/dcds.2008.22.557. [8] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. [9] I. Chueshov, E. Dowell, I. Lasiecka and J. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500.  doi: 10.1007/s00245-016-9349-1. [10] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132. [11] I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Comm. Partial Differential Equations, 39 (2014), 1965-1997.  doi: 10.1080/03605302.2014.930484. [12] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolutions with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. [13] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010. [14] I. Chueshov and I. Lasiecka, Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241.  doi: 10.1007/s00245-007-9031-8. [15] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations, 254 (2013), 1833-1862.  doi: 10.1016/j.jde.2012.11.006. [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656. [17] P. G. Ciarlet and P. Rabier, Les Équations de von Kármán, Springer Verlag, 1980. [18] P. G. Ciarlet, Mathematical Elasticity, Vol. II, Theory of Plates. Studies in Mathematics and its Applications, 27. North-Holland, Amsterdam, 1997. [19] C. 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.  doi: 10.1007/BF00276727. [20] M. Eller, V. Isakov, G. Nakamura and D. Tataru, On the uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., North-Holland, Amsterdam, 31 (2002), 329-349. [21] P. G. Geredeli, I. Lasiecka and J. 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), 1193-1229.  doi: 10.1016/j.jde.2012.10.016. [22] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988. [23] S. Hansen, Boundary control of a one dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074.  doi: 10.1137/S0363012991222607. [24] M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11pp. [25] V. Isakov, A nonhyperbolic Cauchy problem for $\Box_a \Box_b$ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767.  doi: 10.1002/cpa.3160390603. [26] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047. [27] H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math., 58 (2000), 601-612.  doi: 10.1090/qam/1788420. [28] H. Koch and I. 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., Birkhäuser, Basel, 50 (2002), 197-216. [29] H. Koch and A. Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods Appl. Sci., 16 (1993), 581-586.  doi: 10.1002/mma.1670160806. [30] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. ⅷ+156 pp. [31] O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. [32] J. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal., 112 (1990), 223-267.  doi: 10.1007/BF00381235. [33] J. E. Lagnese, Uniform boundary stabilization of thermoelastic plates, Control of boundaries and stabilization, (Clermont-Ferrand, 1988), Lecture Notes in Control and Inform. Sci., 125 (1989), 154-167. [34] J. E Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. SIAM, Philadelphia, PA, 1989. [35] J. E. Lagnese and J. L. Lions, Modeling Analysis and Control of Thin Plates, Research in Applied Mathematics, 6. Masson, Paris, 1988. [36] I. Lasiecka, Weak, classical and intermediate solutions to full von Karman system of dynamic nonlinear elasticity, Appl. Anal., 68 (1998), 121-145.  doi: 10.1080/00036819808840625. [37] I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422.  doi: 10.1137/S0363012996301907. [38] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-553. [39] I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 1801-1847.  doi: 10.1080/03605309908821483. [40] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogenous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl.(9), 65 (1986), 149-192. [41] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4), 27 (1998), 457-482. [42] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. [43] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972. [44] Z. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q. [45] J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058. [46] 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), 229-271.  doi: 10.1007/s00245-015-9313-5. [47] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl.(9), 71 (1992), 455-467. [48] I. Ryzhkova, Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow, Z. Angew. Math. Phys., 58 (2007), 246-261.  doi: 10.1007/s00033-006-0080-7. [49] I. Ryzhkova, Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. Appl., 294 (2004), 462-481.  doi: 10.1016/j.jmaa.2004.02.021. [50] R. Sakamoto, Hyperbolic Boundary Value Problem, Cambridge University Press, 1982. [51] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988. [52] P.-F. Yao, Observability inequalities for shallow shells, SIAM J. Control Optim., 38 (2000), 1729-1756.  doi: 10.1137/S0363012999338692.
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