American Institute of Mathematical Sciences

Advanced
July  2017, 16(4): 1331-1372. doi: 10.3934/cpaa.2017065

Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat

Irena PawŃow 1,2, and  Wojciech M. Zajączkowski 3,4,

1. 

Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
2. 

Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland
3. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
4. 

Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland

Received  April 2016 Revised  September 2016 Published  April 2017

A three-dimensional thermo-visco-elastic system for Kelvin-Voigt type material at small strains is considered. The system involves nonlinear temperature-dependent specific heat relevant in the limit of low temperature range. The existence of a unique global regular solution is proved without small data assumptions. The proof consists of two parts. First the existence of a local in time solution is proved by the Banach successive approximations method. Then a lower bound on temperature and global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time. The paper generalizes the results of the previous author's publication in SIAM J. Math. Anal. 45, No. 4 (2013), pp. 1997–2045.

Keywords: Thermo-visco-elastic system, Kelvin-Voigt materials, nonlinear specific heat, Sobolev spaces with a mixed norm, existence of global regular solutions.
Mathematics Subject Classification: Primary: 74B20, 35K50; Secondary: 35Q74, 74F05.
Citation: Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
References:
[1]

O. V. Besov, V. P. Il'in and S. M. Nikolskij, Integral Representation of Functions and Theorems of Imbeddings, Nauka, Moscow, 1975 (in Russian). Google Scholar

[2]

D. Blanchard and O. Gulbé, Existence of a solution for a nonlinear system in thermoviscoelasticity, Adv. Differential Equations, 5 (2000), 1221-1252.   Google Scholar

[3]

E. Bonetti and G. Bonfanti, Existence and uniqueness of the solution to a 3D thermoelastic system, Electron. J. Differential Equations, (2003), 1-15.  Google Scholar

[4]

Y. S. Bugrov, Function spaces with mixed norm, Math. USSR. Izv., 5 (1971), 1145-1167.   Google Scholar

[5]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.  Google Scholar

[6]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454.  doi: 10.1016/0362-546X(82)90058-X.  Google Scholar

[7]

R. Denk, M. Hieber and J. Prüss, Optimal Lp -Lq estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[8]

C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL 2005. doi: 10.1201/9781420027365.  Google Scholar

[9]

M. Fabrizio, D. Giorgi and A. Morro, A continuum theory for first-order phase transitions based on the balance of structure order, Math. Methods Appl. Sci., 31 (2008), 627-653.  doi: 10.1002/mma.930.  Google Scholar

[10]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.  doi: 10.1002/mma.1089.  Google Scholar

[11]

G. Francfort and P. M. Suquet, Homogenization and mechanical dissipation in thermoviscoelasticity, Arch. Ration. Mech. Anal., 96 (1986), 265-293.  doi: 10.1007/BF00251909.  Google Scholar

[12]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[13]

J. A. Gawinecki, Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity, Math. Methods Appl. Sci., 26 (2003), 907-936.  doi: 10.1002/mma.406.  Google Scholar

[14]

J. A. Gawinecki and W. M. Zajączkowski, Global existence of solutions to the nonlinear thermoviscoelasticity system with small data, Top. Meth. Nonlin. Anal., 39 (2012), 263-284.   Google Scholar

[15]

J. A. Gawinecki and W. M. Zajączkowski, Global regular solutions to two-dimensional thermoviscoelasticity, Commun. Pure Appl. Anal. , to appear. doi: 10.3934/cpaa.2016.15.1009.  Google Scholar

[16]

K. K. Golovkin, On equivalent norms for fractional spaces, Amer. Math. Soc. Transl. Ser. 2, 81 (1969), 257-280.   Google Scholar

[17]

N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations, Algebra i Analiz, 13 (2001), 1-25.   Google Scholar

[18]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian). Google Scholar

[19]

A. Miranville and G. Schimperna, Global solution to a phase transition model based on a microforce balance, J. Evol. Equ., 5 (2005), 253-276.  doi: 10.1007/s00028-005-0187-x.  Google Scholar

[20]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.  Google Scholar

[21]

I. Pawłlow, Three dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.   Google Scholar

[22]

I. Paw low and W. M. Zajączkowski, Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 441-466.  doi: 10.3934/dcdss.2011.4.441.  Google Scholar

[23]

I. Paw low and W. M. Zajączkowski, Global regular solutions to a Kelvin-Voigt type thermoviscoelastic system, SIAM J. Math. Anal., 45 (2013), 1997-2045.  doi: 10.1137/110859026.  Google Scholar

[24]

I. Paw low and A. Żochowski, Existence and uniqueness for a three-dimensional thermoelastic system, Dissertationes Math., 406 (2002), p.46. doi: 10.4064/dm406-0-1.  Google Scholar

[25]

R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., 74 (2011), 3159-3190.  doi: 10.1016/j.na.2011.01.031.  Google Scholar

[26]

T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys, Discrete Contin. Dyn. Syst. , Supplement (2007), 892-902.  Google Scholar

[27]

T. Roubíček, Thermo-viscoelasticity at small strains with L1-data, Quart. Appl. Math., 67 (2009), 47-71.  doi: 10.1090/S0033-569X-09-01094-3.  Google Scholar

[28]

T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.  doi: 10.1137/080729992.  Google Scholar

[29]

T. Roubíček, Nonlinearly coupled thermo-visco-elasticity, Nonlinear Differ. Equ. Appl., 20 (2013), 1243-1275.  doi: 10.1007/s00030-012-0207-9.  Google Scholar

[30]

Y. Shibata, Global in time existence of small solutions of non-linear thermoviscoelastic equations, Math. Methods Appl. Sci., 18 (1995), 871-895.  doi: 10.1002/mma.1670181104.  Google Scholar

[31]

M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity, Arch. Ration. Mech. Anal., 76 (1981), 97-133.  doi: 10.1007/BF00251248.  Google Scholar

[32]

V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm, Zap. Nauchn. Sem. S. Petersburg Otdel. Mat Inst. Steklov (POMI), 288 (2002), 204-231.  doi: 10.1023/B:JOTH.0000041480.38912.3a.  Google Scholar

[33]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type, Trudy MIAN, 83 (1965) (in Russian).  Google Scholar

[34]

W. von Wahl, The Equations of the Navier-Stokes and Abstract Parabolic Equations, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9.  Google Scholar

[35]

S. Y. Yoshikawa, I. Paw low and W. M. Zajączkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat, Commun. Pure Appl. Anal., 8 (2009), 1093-1115.  doi: 10.3934/cpaa.2009.8.1093.  Google Scholar

show all references

References:
[1]

O. V. Besov, V. P. Il'in and S. M. Nikolskij, Integral Representation of Functions and Theorems of Imbeddings, Nauka, Moscow, 1975 (in Russian). Google Scholar

[2]

D. Blanchard and O. Gulbé, Existence of a solution for a nonlinear system in thermoviscoelasticity, Adv. Differential Equations, 5 (2000), 1221-1252.   Google Scholar

[3]

E. Bonetti and G. Bonfanti, Existence and uniqueness of the solution to a 3D thermoelastic system, Electron. J. Differential Equations, (2003), 1-15.  Google Scholar

[4]

Y. S. Bugrov, Function spaces with mixed norm, Math. USSR. Izv., 5 (1971), 1145-1167.   Google Scholar

[5]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.  Google Scholar

[6]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454.  doi: 10.1016/0362-546X(82)90058-X.  Google Scholar

[7]

R. Denk, M. Hieber and J. Prüss, Optimal Lp -Lq estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[8]

C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL 2005. doi: 10.1201/9781420027365.  Google Scholar

[9]

M. Fabrizio, D. Giorgi and A. Morro, A continuum theory for first-order phase transitions based on the balance of structure order, Math. Methods Appl. Sci., 31 (2008), 627-653.  doi: 10.1002/mma.930.  Google Scholar

[10]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.  doi: 10.1002/mma.1089.  Google Scholar

[11]

G. Francfort and P. M. Suquet, Homogenization and mechanical dissipation in thermoviscoelasticity, Arch. Ration. Mech. Anal., 96 (1986), 265-293.  doi: 10.1007/BF00251909.  Google Scholar

[12]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[13]

J. A. Gawinecki, Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity, Math. Methods Appl. Sci., 26 (2003), 907-936.  doi: 10.1002/mma.406.  Google Scholar

[14]

J. A. Gawinecki and W. M. Zajączkowski, Global existence of solutions to the nonlinear thermoviscoelasticity system with small data, Top. Meth. Nonlin. Anal., 39 (2012), 263-284.   Google Scholar

[15]

J. A. Gawinecki and W. M. Zajączkowski, Global regular solutions to two-dimensional thermoviscoelasticity, Commun. Pure Appl. Anal. , to appear. doi: 10.3934/cpaa.2016.15.1009.  Google Scholar

[16]

K. K. Golovkin, On equivalent norms for fractional spaces, Amer. Math. Soc. Transl. Ser. 2, 81 (1969), 257-280.   Google Scholar

[17]

N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations, Algebra i Analiz, 13 (2001), 1-25.   Google Scholar

[18]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian). Google Scholar

[19]

A. Miranville and G. Schimperna, Global solution to a phase transition model based on a microforce balance, J. Evol. Equ., 5 (2005), 253-276.  doi: 10.1007/s00028-005-0187-x.  Google Scholar

[20]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.  Google Scholar

[21]

I. Pawłlow, Three dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.   Google Scholar

[22]

I. Paw low and W. M. Zajączkowski, Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 441-466.  doi: 10.3934/dcdss.2011.4.441.  Google Scholar

[23]

I. Paw low and W. M. Zajączkowski, Global regular solutions to a Kelvin-Voigt type thermoviscoelastic system, SIAM J. Math. Anal., 45 (2013), 1997-2045.  doi: 10.1137/110859026.  Google Scholar

[24]

I. Paw low and A. Żochowski, Existence and uniqueness for a three-dimensional thermoelastic system, Dissertationes Math., 406 (2002), p.46. doi: 10.4064/dm406-0-1.  Google Scholar

[25]

R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., 74 (2011), 3159-3190.  doi: 10.1016/j.na.2011.01.031.  Google Scholar

[26]

T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys, Discrete Contin. Dyn. Syst. , Supplement (2007), 892-902.  Google Scholar

[27]

T. Roubíček, Thermo-viscoelasticity at small strains with L1-data, Quart. Appl. Math., 67 (2009), 47-71.  doi: 10.1090/S0033-569X-09-01094-3.  Google Scholar

[28]

T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.  doi: 10.1137/080729992.  Google Scholar

[29]

T. Roubíček, Nonlinearly coupled thermo-visco-elasticity, Nonlinear Differ. Equ. Appl., 20 (2013), 1243-1275.  doi: 10.1007/s00030-012-0207-9.  Google Scholar

[30]

Y. Shibata, Global in time existence of small solutions of non-linear thermoviscoelastic equations, Math. Methods Appl. Sci., 18 (1995), 871-895.  doi: 10.1002/mma.1670181104.  Google Scholar

[31]

M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity, Arch. Ration. Mech. Anal., 76 (1981), 97-133.  doi: 10.1007/BF00251248.  Google Scholar

[32]

V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm, Zap. Nauchn. Sem. S. Petersburg Otdel. Mat Inst. Steklov (POMI), 288 (2002), 204-231.  doi: 10.1023/B:JOTH.0000041480.38912.3a.  Google Scholar

[33]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type, Trudy MIAN, 83 (1965) (in Russian).  Google Scholar

[34]

W. von Wahl, The Equations of the Navier-Stokes and Abstract Parabolic Equations, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9.  Google Scholar

[35]

S. Y. Yoshikawa, I. Paw low and W. M. Zajączkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat, Commun. Pure Appl. Anal., 8 (2009), 1093-1115.  doi: 10.3934/cpaa.2009.8.1093.  Google Scholar

[1]

Irena Pawłow, Wojciech M. Zajączkowski. Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 441-466. doi: 10.3934/dcdss.2011.4.441
[2]

Miroslav Bulíček, Josef Málek, K. R. Rajagopal. On Kelvin-Voigt model and its generalizations. Evolution Equations & Control Theory, 2012, 1 (1) : 17-42. doi: 10.3934/eect.2012.1.17
[3]

Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations & Control Theory, 2020, 9 (2) : 301-339. doi: 10.3934/eect.2020007
[4]

Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110
[5]

Maria-Magdalena Boureanu, Andaluzia Matei, Mircea Sofonea. Analysis of a contact problem for electro-elastic-visco-plastic materials. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1185-1203. doi: 10.3934/cpaa.2012.11.1185
[6]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[7]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029
[8]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[9]

Xiaobin Yao, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247
[10]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[11]

Khalid Addi, Oanh Chau, Daniel Goeleven. On some frictional contact problems with velocity condition for elastic and visco-elastic materials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1039-1051. doi: 10.3934/dcds.2011.31.1039
[12]

Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557
[13]

Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824
[14]

Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041
[15]

Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809
[16]

Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020067
[17]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[18]

Michela Eleuteri, Jana Kopfová, Pavel Krejčí. Fatigue accumulation in a thermo-visco-elastoplastic plate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2091-2109. doi: 10.3934/dcdsb.2014.19.2091
[19]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093
[20]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (23)
  • HTML views (49)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences