American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data
  • EECT Home
  • This Issue
  • Next Article
    Visualization of the convex integration solutions to the Monge-Ampère equation
June  2019, 8(2): 301-313. doi: 10.3934/eect.2019016

Generation of semigroups for the thermoelastic plate equation with free boundary conditions

Robert Denk 1, and  Yoshihiro Shibata 2,

1. 

University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany
2. 

Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Ohkobu 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Received  June 2018 Revised  October 2018 Published  March 2019

We consider the linear thermoelastic plate equations with free boundary conditions in uniform $ C^4 $-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in $ L^p $-spaces for all $ p\in(1, \infty) $ and has maximal $ L^q $-$ L^p $-regularity on finite time intervals. On bounded $ C^4 $-domains, we obtain exponential stability.

Keywords: Thermoelastic plate equations, operator-valued Fourier multipliers, generation of analytic semigroups.
Mathematics Subject Classification: Primary: 35K35, 35J40; Secondary: 42B15.
Citation: Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
References:
[1]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010, Well-posedness and long-time dynamics. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[2]

R. Denk, M. Hieber and J. Prüss, $ \mathcal R $-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114pp. doi: 10.1090/memo/0788.  Google Scholar

[3]

R. Denk and R. Racke, $ L^p $-resolvent estimates and time decay for generalized thermoelastic plate equations, Electron. J. Differential Equations, No. 48, 16 pp(electronic).  Google Scholar

[4]

R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet–Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.  doi: 10.1016/j.jde.2015.02.043.  Google Scholar

[5]

R. Denk and Y. Shibata, Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evol. Equ., 17 (2017), 215-261.  doi: 10.1007/s00028-016-0367-x.  Google Scholar

[6]

Y. Enomoto and Y. Shibata, On the $ \mathcal R $-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcial. Ekvac., 56 (2013), 441-505.  doi: 10.1619/fesi.56.441.  Google Scholar

[7]

M. Girardi and L. Weis, Criteria for R-boundedness of operator families, in Evolution Equations, vol. 234 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2003,203–221.  Google Scholar

[8]

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

[9]

P. C. Kunstmann and L. Weis, Maximal $ L_p $-regularity for parabolic equations, Fourier multiplier theorems and $ H^\infty $-functional calculus, in Functional Analytic Methods for Evolution Equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, 65–311. doi: 10.1007/978-3-540-44653-8_2.  Google Scholar

[10]

J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[11]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, in Control and Partial Differential Equations (Marseille-Luminy, 1997), vol. 4 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, 1998,199–222(electronic). doi: 10.1051/proc:1998029.  Google Scholar

[12]

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(1999), URL http://www.numdam.org/item?id=ASNSP_1998_4_27_3-4_457_0.  Google Scholar

[13]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189.  Google Scholar

[14]

K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.  doi: 10.1007/s000330050071.  Google Scholar

[15]

Z.-Y. 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

[16]

Z. Liu and J. Yong, Qualitative properties of certain $ C_0 $ semigroups arising in elastic systems with various dampings, Adv. Differential Equations, 3 (1998), 643-686.   Google Scholar

[17]

Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564.  doi: 10.1090/qam/1466148.  Google Scholar

[18]

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.  Google Scholar

[19]

Y. Naito, On the $ L_p $-$ L_q $ maximal regularity for the linear thermoelastic plate equation in a bounded domain, Math. Methods Appl. Sci., 32 (2009), 1609-1637.  doi: 10.1002/mma.1100.  Google Scholar

[20]

Y. Naito and Y. Shibata, On the $ L_p $ analytic semigroup associated with the linear thermoelastic plate equations in the half-space, J. Math. Soc. Japan, 61 (2009), 971–1011, URL http://projecteuclid.org/euclid.jmsj/1257520498. doi: 10.2969/jmsj/06140971.  Google Scholar

[21]

K. Schade and Y. Shibata, On strong dynamics of compressible nematic liquid crystals, SIAM J. Math. Anal., 47 (2015), 3963-3992.  doi: 10.1137/140970628.  Google Scholar

[22]

Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102.   Google Scholar

[23]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.  Google Scholar

show all references

References:
[1]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010, Well-posedness and long-time dynamics. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[2]

R. Denk, M. Hieber and J. Prüss, $ \mathcal R $-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114pp. doi: 10.1090/memo/0788.  Google Scholar

[3]

R. Denk and R. Racke, $ L^p $-resolvent estimates and time decay for generalized thermoelastic plate equations, Electron. J. Differential Equations, No. 48, 16 pp(electronic).  Google Scholar

[4]

R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet–Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.  doi: 10.1016/j.jde.2015.02.043.  Google Scholar

[5]

R. Denk and Y. Shibata, Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evol. Equ., 17 (2017), 215-261.  doi: 10.1007/s00028-016-0367-x.  Google Scholar

[6]

Y. Enomoto and Y. Shibata, On the $ \mathcal R $-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcial. Ekvac., 56 (2013), 441-505.  doi: 10.1619/fesi.56.441.  Google Scholar

[7]

M. Girardi and L. Weis, Criteria for R-boundedness of operator families, in Evolution Equations, vol. 234 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2003,203–221.  Google Scholar

[8]

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

[9]

P. C. Kunstmann and L. Weis, Maximal $ L_p $-regularity for parabolic equations, Fourier multiplier theorems and $ H^\infty $-functional calculus, in Functional Analytic Methods for Evolution Equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, 65–311. doi: 10.1007/978-3-540-44653-8_2.  Google Scholar

[10]

J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[11]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, in Control and Partial Differential Equations (Marseille-Luminy, 1997), vol. 4 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, 1998,199–222(electronic). doi: 10.1051/proc:1998029.  Google Scholar

[12]

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(1999), URL http://www.numdam.org/item?id=ASNSP_1998_4_27_3-4_457_0.  Google Scholar

[13]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189.  Google Scholar

[14]

K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.  doi: 10.1007/s000330050071.  Google Scholar

[15]

Z.-Y. 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

[16]

Z. Liu and J. Yong, Qualitative properties of certain $ C_0 $ semigroups arising in elastic systems with various dampings, Adv. Differential Equations, 3 (1998), 643-686.   Google Scholar

[17]

Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564.  doi: 10.1090/qam/1466148.  Google Scholar

[18]

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.  Google Scholar

[19]

Y. Naito, On the $ L_p $-$ L_q $ maximal regularity for the linear thermoelastic plate equation in a bounded domain, Math. Methods Appl. Sci., 32 (2009), 1609-1637.  doi: 10.1002/mma.1100.  Google Scholar

[20]

Y. Naito and Y. Shibata, On the $ L_p $ analytic semigroup associated with the linear thermoelastic plate equations in the half-space, J. Math. Soc. Japan, 61 (2009), 971–1011, URL http://projecteuclid.org/euclid.jmsj/1257520498. doi: 10.2969/jmsj/06140971.  Google Scholar

[21]

K. Schade and Y. Shibata, On strong dynamics of compressible nematic liquid crystals, SIAM J. Math. Anal., 47 (2015), 3963-3992.  doi: 10.1137/140970628.  Google Scholar

[22]

Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102.   Google Scholar

[23]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.  Google Scholar

[1]

Qi Lü, Xu Zhang. Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application. Mathematical Control & Related Fields, 2018, 8 (1) : 337-381. doi: 10.3934/mcrf.2018014
[2]

Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[3]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
[4]

Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595
[5]

Moncef Aouadi, Taoufik Moulahi. The controllability of a thermoelastic plate problem revisited. Evolution Equations & Control Theory, 2018, 7 (1) : 1-31. doi: 10.3934/eect.2018001
[6]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[7]

Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557
[8]

Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467
[9]

Lassi Roininen, Markku S. Lehtinen, Petteri Piiroinen, Ilkka I. Virtanen. Perfect radar pulse compression via unimodular fourier multipliers. Inverse Problems & Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831
[10]

Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139
[11]

Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189
[12]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241
[13]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214
[14]

Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249
[15]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007
[16]

Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441
[17]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[18]

Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1409-1415. doi: 10.3934/dcdss.2013.6.1409
[19]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[20]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (73)
  • HTML views (362)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences