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December  2021, 10(4): 767-796. doi: 10.3934/eect.2020092

Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions

Catherine Lebiedzik

Wayne State University, Department of Mathematics, Detroit, MI 48201 USA

Received  February 2020 Revised  August 2020 Published  December 2021 Early access  September 2020

We will consider the full von Kármán thermoelastic system with free boundary conditions and dissipation imposed only on the in-plane displacement. It will be shown that the corresponding solutions are exponentially stable, though there is no mechanical dissipation on the vertical displacements. The main tools used are: (i) partial analyticity of the linearized semigroup and (ii) trace estimates which exploit the hidden regularity harvested from partial analyticity.

Keywords: Uniform stability, nonlinear thermoelasticity, Von Kármán plates, free boundary conditions.
Mathematics Subject Classification: Primary:35B35, 35M30, 74B20, 74F05, 74K20;Secondary:35A01, 35A02.
Citation: Catherine Lebiedzik. Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions. Evolution Equations & Control Theory, 2021, 10 (4) : 767-796. doi: 10.3934/eect.2020092
References:
[1]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.   Google Scholar

[2]

M. Belishev and I. Lasiecka, The dynamical Lamé system: Regularity of solutions, boundary controllability and boundary data continuation, ESAIM, Control Optim. Calc. Var., 8 (2002), 143-167.  doi: 10.1051/cocv:2002058.  Google Scholar

[3]

A. Benabdallah, Modeling of Von Kármán system with thermal effects, Prepublications de L'equipe de Mathematiques de Besancon, 99 (1999). Google Scholar

[4]

A. Benabdallah and I. Lasiecka, Exponential decay rates for a full Von Kármán system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93.  doi: 10.1006/jdeq.1999.3656.  Google Scholar

[5]

A. Benabdallah and I. Lasiecka, Exponential decay rates for a full Von Kármán thermoelastic system with nonlinear thermal coupling, ESIAM: Proceedings. Contrôle des systèmes gouvernés par des équations aux dérivées partielles, 8 (2000), 13-38.  doi: 10.1051/proc:2000002.  Google Scholar

[6]

A. Benabdallah and D. Teniou, Exponential stability of a Von Kármán model with thermal effects, Electronic Journal of Differential Equations, (1998), 1–13.  Google Scholar

[7]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter and K. Sanjoy, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[8]

P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅲ. Theory of Shells, in Studies in Mathematics and its Applications, Vol. 29, North-Holland Publishing Co., Amsterdam, 2000.  Google Scholar

[9]

P. G. Ciarlet and P.Rabier, Les Equations de Von Karman, Springer, Berlin, 1980.  Google Scholar

[10]

C. M. Dafermos, On the existence and asymptotic stability of solutions to the equations of linear thermoelasticity, Archive for Rational Mechanics and Analysis, 29 (1968), 241-271.  doi: 10.1007/BF00276727.  Google Scholar

[11]

E. H. Dowell, A Modern Course in Aeroelasticity, 5th edition, Springer, Cham, 2015. doi: 10.1007/978-3-319-09453-3.  Google Scholar

[12]

A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equation, 9 (1996), 267-294.   Google Scholar

[13]

I. Chueshov and I. Lasiecka, Von Kármán Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[14]

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.  Google Scholar

[15]

H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math, 58 (2000), 601-612.  doi: 10.1090/qam/1788420.  Google Scholar

[16]

H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity – full von Kármán systems, in Progress in Nonlinear Differential Equations and their Appl., Vol. 50, Birkhäuser, Basel, 2002.  Google Scholar

[17]

W. T. Koiter, On the nonlinear theory of thin elastic shells, Nederl. Akad. Wetensch. Proc. Ser. B, 69 (1966), 1-54.   Google Scholar

[18]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[19]

J. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations, 91 (1991), 355-388.  doi: 10.1016/0022-0396(91)90145-Y.  Google Scholar

[20]

J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, in Research in Applied Mathematics, Vol. 6, Masson, Paris, 1988.  Google Scholar

[21]

I. Lasiecka, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422.  doi: 10.1137/S0363012996301907.  Google Scholar

[22]

I. 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), 1801-1847.  doi: 10.1080/03605309908821483.  Google Scholar

[23]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[24]

I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.   Google Scholar

[25]

I. Lasiecka, T. Ma and R. Montiero, Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation, Trans. Amer. Math. Soc., 371 (2019), 8051-8096.  doi: 10.1090/tran/7756.  Google Scholar

[26]

I. Lasiecka, T. Ma and R. Montiero, Long-time dynamics of vectorial von Kármán system with nonlinear thermal effects and free boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1037-1072.  doi: 10.3934/dcdsb.2018141.  Google Scholar

[27]

I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Part Ⅰ. $L_2$ nonhomogenous data, Ann. Mat. Pura Appl., 157 (1990), 285-367.  doi: 10.1007/BF01765322.  Google Scholar

[28]

I. Lasiecka and R. Triggiani, Analyticity of thermoelastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1999), 457-487.   Google Scholar

[29] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, in Encyclopedia of Mathematics and its Applications, Volume Ⅰ and Ⅱ, Cambridge University Press, Cambridge, 2000.   Google Scholar
[30]

J. L. Lions, Quelques Methodes de Résolution des Problèmes aux Limits Nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[31]

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.  Google Scholar

[32]

S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487.  doi: 10.1215/kjm/1250523319.  Google Scholar

[33]

V. I. Sedenko, On uniqueness of the generalized solutions of initial boundary value problem for Marguerre-Vlasov nonlinear oscillations of the shallow shells, Russian Izvestiya, North-Caucasus Region, Ser. Natural Sciences, 1-2 (1994). Google Scholar

[34]

D. Tataru, On the regularity of boundary traces for the wave equation, Annali di Scuola Normale Superiore, 26 (1998), 185-206.   Google Scholar

[35]

T. von Kármán, Festigkeitprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4 (1910), 314-385.   Google Scholar

show all references

References:
[1]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.   Google Scholar

[2]

M. Belishev and I. Lasiecka, The dynamical Lamé system: Regularity of solutions, boundary controllability and boundary data continuation, ESAIM, Control Optim. Calc. Var., 8 (2002), 143-167.  doi: 10.1051/cocv:2002058.  Google Scholar

[3]

A. Benabdallah, Modeling of Von Kármán system with thermal effects, Prepublications de L'equipe de Mathematiques de Besancon, 99 (1999). Google Scholar

[4]

A. Benabdallah and I. Lasiecka, Exponential decay rates for a full Von Kármán system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93.  doi: 10.1006/jdeq.1999.3656.  Google Scholar

[5]

A. Benabdallah and I. Lasiecka, Exponential decay rates for a full Von Kármán thermoelastic system with nonlinear thermal coupling, ESIAM: Proceedings. Contrôle des systèmes gouvernés par des équations aux dérivées partielles, 8 (2000), 13-38.  doi: 10.1051/proc:2000002.  Google Scholar

[6]

A. Benabdallah and D. Teniou, Exponential stability of a Von Kármán model with thermal effects, Electronic Journal of Differential Equations, (1998), 1–13.  Google Scholar

[7]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter and K. Sanjoy, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[8]

P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅲ. Theory of Shells, in Studies in Mathematics and its Applications, Vol. 29, North-Holland Publishing Co., Amsterdam, 2000.  Google Scholar

[9]

P. G. Ciarlet and P.Rabier, Les Equations de Von Karman, Springer, Berlin, 1980.  Google Scholar

[10]

C. M. Dafermos, On the existence and asymptotic stability of solutions to the equations of linear thermoelasticity, Archive for Rational Mechanics and Analysis, 29 (1968), 241-271.  doi: 10.1007/BF00276727.  Google Scholar

[11]

E. H. Dowell, A Modern Course in Aeroelasticity, 5th edition, Springer, Cham, 2015. doi: 10.1007/978-3-319-09453-3.  Google Scholar

[12]

A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equation, 9 (1996), 267-294.   Google Scholar

[13]

I. Chueshov and I. Lasiecka, Von Kármán Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[14]

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.  Google Scholar

[15]

H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math, 58 (2000), 601-612.  doi: 10.1090/qam/1788420.  Google Scholar

[16]

H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity – full von Kármán systems, in Progress in Nonlinear Differential Equations and their Appl., Vol. 50, Birkhäuser, Basel, 2002.  Google Scholar

[17]

W. T. Koiter, On the nonlinear theory of thin elastic shells, Nederl. Akad. Wetensch. Proc. Ser. B, 69 (1966), 1-54.   Google Scholar

[18]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[19]

J. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations, 91 (1991), 355-388.  doi: 10.1016/0022-0396(91)90145-Y.  Google Scholar

[20]

J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, in Research in Applied Mathematics, Vol. 6, Masson, Paris, 1988.  Google Scholar

[21]

I. Lasiecka, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422.  doi: 10.1137/S0363012996301907.  Google Scholar

[22]

I. 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), 1801-1847.  doi: 10.1080/03605309908821483.  Google Scholar

[23]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[24]

I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.   Google Scholar

[25]

I. Lasiecka, T. Ma and R. Montiero, Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation, Trans. Amer. Math. Soc., 371 (2019), 8051-8096.  doi: 10.1090/tran/7756.  Google Scholar

[26]

I. Lasiecka, T. Ma and R. Montiero, Long-time dynamics of vectorial von Kármán system with nonlinear thermal effects and free boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1037-1072.  doi: 10.3934/dcdsb.2018141.  Google Scholar

[27]

I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Part Ⅰ. $L_2$ nonhomogenous data, Ann. Mat. Pura Appl., 157 (1990), 285-367.  doi: 10.1007/BF01765322.  Google Scholar

[28]

I. Lasiecka and R. Triggiani, Analyticity of thermoelastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1999), 457-487.   Google Scholar

[29] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, in Encyclopedia of Mathematics and its Applications, Volume Ⅰ and Ⅱ, Cambridge University Press, Cambridge, 2000.   Google Scholar
[30]

J. L. Lions, Quelques Methodes de Résolution des Problèmes aux Limits Nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[31]

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.  Google Scholar

[32]

S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487.  doi: 10.1215/kjm/1250523319.  Google Scholar

[33]

V. I. Sedenko, On uniqueness of the generalized solutions of initial boundary value problem for Marguerre-Vlasov nonlinear oscillations of the shallow shells, Russian Izvestiya, North-Caucasus Region, Ser. Natural Sciences, 1-2 (1994). Google Scholar

[34]

D. Tataru, On the regularity of boundary traces for the wave equation, Annali di Scuola Normale Superiore, 26 (1998), 185-206.   Google Scholar

[35]

T. von Kármán, Festigkeitprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4 (1910), 314-385.   Google Scholar

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