# American Institute of Mathematical Sciences

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

 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.

Citation: Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure and 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). [2] D. Blanchard and O. Gulbé, Existence of a solution for a nonlinear system in thermoviscoelasticity, Adv. Differential Equations, 5 (2000), 1221-1252. [3] E. Bonetti and G. Bonfanti, Existence and uniqueness of the solution to a 3D thermoelastic system, Electron. J. Differential Equations, (2003), 1-15. [4] Y. S. Bugrov, Function spaces with mixed norm, Math. USSR. Izv., 5 (1971), 1145-1167. [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. [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. [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. [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. [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. [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. [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. [12] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [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. [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. [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. [16] K. K. Golovkin, On equivalent norms for fractional spaces, Amer. Math. Soc. Transl. Ser. 2, 81 (1969), 257-280. [17] N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations, Algebra i Analiz, 13 (2001), 1-25. [18] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian). [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. [20] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967. [21] I. Pawłlow, Three dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365. [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. [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. [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. [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. [26] T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys, Discrete Contin. Dyn. Syst. , Supplement (2007), 892-902. [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. [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. [29] T. Roubíček, Nonlinearly coupled thermo-visco-elasticity, Nonlinear Differ. Equ. Appl., 20 (2013), 1243-1275.  doi: 10.1007/s00030-012-0207-9. [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. [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. [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. [33] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type, Trudy MIAN, 83 (1965) (in Russian). [34] W. von Wahl, The Equations of the Navier-Stokes and Abstract Parabolic Equations, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9. [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.

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##### 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). [2] D. Blanchard and O. Gulbé, Existence of a solution for a nonlinear system in thermoviscoelasticity, Adv. Differential Equations, 5 (2000), 1221-1252. [3] E. Bonetti and G. Bonfanti, Existence and uniqueness of the solution to a 3D thermoelastic system, Electron. J. Differential Equations, (2003), 1-15. [4] Y. S. Bugrov, Function spaces with mixed norm, Math. USSR. Izv., 5 (1971), 1145-1167. [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. [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. [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. [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. [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. [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. [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. [12] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [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. [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. [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. [16] K. K. Golovkin, On equivalent norms for fractional spaces, Amer. Math. Soc. Transl. Ser. 2, 81 (1969), 257-280. [17] N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations, Algebra i Analiz, 13 (2001), 1-25. [18] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian). [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. [20] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967. [21] I. Pawłlow, Three dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365. [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. [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. [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. [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. [26] T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys, Discrete Contin. Dyn. Syst. , Supplement (2007), 892-902. [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. [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. [29] T. Roubíček, Nonlinearly coupled thermo-visco-elasticity, Nonlinear Differ. Equ. Appl., 20 (2013), 1243-1275.  doi: 10.1007/s00030-012-0207-9. [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. [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. [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. [33] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type, Trudy MIAN, 83 (1965) (in Russian). [34] W. von Wahl, The Equations of the Navier-Stokes and Abstract Parabolic Equations, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9. [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.
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