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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 and 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. 

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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. 

[13]

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

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[23]

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

[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. 

[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.

[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.

[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.

[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. 

[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. 
[30]

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

[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.

[32]

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

[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).

[34]

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

[35]

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

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. 

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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. 

[13]

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

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[23]

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

[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. 

[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.

[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.

[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.

[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. 

[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. 
[30]

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

[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.

[32]

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

[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).

[34]

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

[35]

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

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