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May  2016, 15(3): 1009-1028. doi: 10.3934/cpaa.2016.15.1009

## Global regular solutions to two-dimensional thermoviscoelasticity

 1 Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland 2 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw

Received  March 2015 Revised  November 2015 Published  February 2016

A two-dimensional thermoviscoelastic system of Kelvin-Voigt type with strong dependence on temperature is considered. The existence and uniqueness of a global regular solution is proved without small data assumptions. The global existence is proved in two steps. First global a priori estimate is derived applying the theory of anisotropic Sobolev spaces with a mixed norm. Then local existence, proved by the method of successive approximations for a sufficiently small time interval, is extended step by step in time. By two-dimensional solution we mean that all its quantities depend on two space variables only.
Citation: Jerzy Gawinecki, Wojciech M. Zajączkowski. Global regular solutions to two-dimensional thermoviscoelasticity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1009-1028. doi: 10.3934/cpaa.2016.15.1009
##### References:
 [1] O. V. Besov, V. P. Il'in and S. M. Nikolskij, Integral Representation of Functions and Theorems of Imbeddings,, Nauka Moscow, (1975). Google Scholar [2] D. Blanchard and O. Guibé, Existence of a solution for nonlinear system in thermoviscoelasticity,, \emph{Adv. Diff. Equs.}, 5 (2000), 1221. Google Scholar [3] Y. S. Bugrov, Function spaces with mixed norm,, \emph{Math. USSR-Izv.}, 5 (1971), 1145. Google Scholar [4] C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity,, \emph{SIAM J. Math. Anal.}, 13 (1982), 397. doi: 10.1137/0513029. Google Scholar [5] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity,, \emph{Nonlin. Anal.}, 6 (1982), 435. doi: 10.1016/0362-546X(82)90058-X. Google Scholar [6] D. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems,, Pure and Applied Mathematics, (2005). doi: 10.1201/9781420027365. Google Scholar [7] J. A. Gawinecki, Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity,, \emph{Math. Meth. Appl. Sc.}, 26 (2003), 907. doi: 10.1002/mma.406. Google Scholar [8] J. A. Gawinecki and W. M. Zajączkowski, Global non-small data existence of spherically symmetric solutions to nonlinear viscoelasticity in a ball,, \emph{J. Anal. Appl.}, 30 (2011), 387. doi: 10.4171/ZAA/1441. Google Scholar [9] J. A. Gawinecki and W. M. Zajączkowski, On global existence of solutions of the Neumann problem for spherically symmetric nonlinear viscoelasticity in a ball,, \emph{Hindawi Publ. Corp. ISRN Math. Analysis}, (2013). Google Scholar [10] J. A. Gawinecki and W. M Zajączkowski,, Global existence of solutions to the nonlinear thermoviscoelasticity system with small data,, \emph{Top. Meth. Nonlin. Anal.}, 39 (2012), 263. Google Scholar [11] K. K. Golovkin, On equivalent norms for fractional spaces,, \emph{Amer. Math. Soc. Transl. Ser 2}, 81 (1969), 257. Google Scholar [12] N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations,, \emph{Algebra i analiz}, 13 (2001), 1. Google Scholar [13] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic type,, Nauka Moscow, (1967). Google Scholar [14] J. L. Lions and E. Magnes, Problémes aux limites non homogénes et applicationes,, Vol. 1, (1968). Google Scholar [15] I. Pawłow and W. M. Zajączkowski, Global regular solutions to a Kelvin-Voigt type thermoviscoelastic system,, \emph{SIAM J. Math. Anal.}, 45 (2013), 1997. doi: 10.1137/110859026. Google Scholar [16] I. Pawłow and W. M. Zajączkowski, Unique solvability of a nonlinear termoviscoelasticity system in Sobolev space with a mixed norm,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 4 (2011), 441. doi: 10.3934/dcdss.2011.4.441. Google Scholar [17] R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains,, \emph{Nonlin. Anal.}, 74 (2011), 3159. doi: 10.1016/j.na.2011.01.031. Google Scholar [18] T. Roubíček, Termoviscoelasticity at small strains with $L^1$-data,, \emph{Quart. Appl. Math.}, 67 (2009), 47. doi: 10.1090/S0033-569X-09-01094-3. Google Scholar [19] T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains,, \emph{SIAM J. Math. Anal.}, 42 (2010), 256. doi: 10.1137/080729992. Google Scholar [20] T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys,, \emph{Discrete and Continuous Dynamical Systems}, (2007), 892. Google Scholar [21] M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoviscoelasticity,, \emph{Arch. Ration. Mech. Anal.}, 76 (1981), 97. doi: 10.1007/BF00251248. Google Scholar [22] V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm,, \emph{Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov (POMI)}, 288 (2002), 204. doi: 10.1023/B:JOTH.0000041480.38912.3a. Google Scholar [23] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type,, \emph{Trudy MIAN}, 83 (1965). Google Scholar [24] B. D. Coleman, Thermodynamics of materials with memory,, \emph{Arch. Ration. Mech. Anal.}, 17 (1964), 1. Google Scholar

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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). Google Scholar [2] D. Blanchard and O. Guibé, Existence of a solution for nonlinear system in thermoviscoelasticity,, \emph{Adv. Diff. Equs.}, 5 (2000), 1221. Google Scholar [3] Y. S. Bugrov, Function spaces with mixed norm,, \emph{Math. USSR-Izv.}, 5 (1971), 1145. Google Scholar [4] C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity,, \emph{SIAM J. Math. Anal.}, 13 (1982), 397. doi: 10.1137/0513029. Google Scholar [5] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity,, \emph{Nonlin. Anal.}, 6 (1982), 435. doi: 10.1016/0362-546X(82)90058-X. Google Scholar [6] D. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems,, Pure and Applied Mathematics, (2005). doi: 10.1201/9781420027365. Google Scholar [7] J. A. Gawinecki, Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity,, \emph{Math. Meth. Appl. Sc.}, 26 (2003), 907. doi: 10.1002/mma.406. Google Scholar [8] J. A. Gawinecki and W. M. Zajączkowski, Global non-small data existence of spherically symmetric solutions to nonlinear viscoelasticity in a ball,, \emph{J. Anal. Appl.}, 30 (2011), 387. doi: 10.4171/ZAA/1441. Google Scholar [9] J. A. Gawinecki and W. M. Zajączkowski, On global existence of solutions of the Neumann problem for spherically symmetric nonlinear viscoelasticity in a ball,, \emph{Hindawi Publ. Corp. ISRN Math. Analysis}, (2013). Google Scholar [10] J. A. Gawinecki and W. M Zajączkowski,, Global existence of solutions to the nonlinear thermoviscoelasticity system with small data,, \emph{Top. Meth. Nonlin. Anal.}, 39 (2012), 263. Google Scholar [11] K. K. Golovkin, On equivalent norms for fractional spaces,, \emph{Amer. Math. Soc. Transl. Ser 2}, 81 (1969), 257. Google Scholar [12] N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations,, \emph{Algebra i analiz}, 13 (2001), 1. Google Scholar [13] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic type,, Nauka Moscow, (1967). Google Scholar [14] J. L. Lions and E. Magnes, Problémes aux limites non homogénes et applicationes,, Vol. 1, (1968). Google Scholar [15] I. Pawłow and W. M. Zajączkowski, Global regular solutions to a Kelvin-Voigt type thermoviscoelastic system,, \emph{SIAM J. Math. Anal.}, 45 (2013), 1997. doi: 10.1137/110859026. Google Scholar [16] I. Pawłow and W. M. Zajączkowski, Unique solvability of a nonlinear termoviscoelasticity system in Sobolev space with a mixed norm,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 4 (2011), 441. doi: 10.3934/dcdss.2011.4.441. Google Scholar [17] R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains,, \emph{Nonlin. Anal.}, 74 (2011), 3159. doi: 10.1016/j.na.2011.01.031. Google Scholar [18] T. Roubíček, Termoviscoelasticity at small strains with $L^1$-data,, \emph{Quart. Appl. Math.}, 67 (2009), 47. doi: 10.1090/S0033-569X-09-01094-3. Google Scholar [19] T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains,, \emph{SIAM J. Math. Anal.}, 42 (2010), 256. doi: 10.1137/080729992. Google Scholar [20] T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys,, \emph{Discrete and Continuous Dynamical Systems}, (2007), 892. Google Scholar [21] M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoviscoelasticity,, \emph{Arch. Ration. Mech. Anal.}, 76 (1981), 97. doi: 10.1007/BF00251248. Google Scholar [22] V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm,, \emph{Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov (POMI)}, 288 (2002), 204. doi: 10.1023/B:JOTH.0000041480.38912.3a. Google Scholar [23] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general type,, \emph{Trudy MIAN}, 83 (1965). Google Scholar [24] B. D. Coleman, Thermodynamics of materials with memory,, \emph{Arch. Ration. Mech. Anal.}, 17 (1964), 1. Google Scholar
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