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

Global regular solutions to two-dimensional thermoviscoelasticity

Jerzy Gawinecki 1, and  Wojciech M. Zajączkowski 2,

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.
Keywords: a priori estimates, Thermoviscoelastic system, Kelvin-Voigt type materials, anisotropic Sobolev spaces with a mixed norm, global existence..
Mathematics Subject Classification: Primary: 74B20, 35K50; Secondary: 35Q72, 74F0.
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 (in Russian). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlin. Anal., 6 (1982), 435-454. 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, Chapman & Hall/CRC, Boca Raton, FL, 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, Math. Meth. Appl. Sc., 26 (2003), 907-936. 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, J. Anal. Appl., 30 (2011), 387-419. 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, Hindawi Publ. Corp. ISRN Math. Analysis, Vol. 2013, article ID268505.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic type, Nauka Moscow, 1967 (in Russian). Google Scholar

[14]

J. L. Lions and E. Magnes, Problémes aux limites non homogénes et applicationes, Vol. 1, Dunod, Paris, 1968.  Google Scholar

[15]

I. Pawłow 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

[16]

I. Pawłow and W. M. Zajączkowski, Unique solvability of a nonlinear termoviscoelasticity 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

[17]

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

[18]

T. Roubíček, Termoviscoelasticity at small strains with $L^1$-data, Quart. Appl. Math., 67 (2009), 47-71. doi: 10.1090/S0033-569X-09-01094-3.  Google Scholar

[19]

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

[20]

T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys, Discrete and Continuous Dynamical Systems, Supplement (2007), 892-902.  Google Scholar

[21]

M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoviscoelasticity, Arch. Ration. Mech. Anal., 76 (1981), 97-133. 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, Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov (POMI), 288 (2002), 204-231. 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, Trudy MIAN, 83 (1965), (in Russian).  Google Scholar

[24]

B. D. Coleman, Thermodynamics of materials with memory, Arch. Ration. Mech. Anal., 17 (1964), 1-46.  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. Guibé, Existence of a solution for nonlinear system in thermoviscoelasticity, Adv. Diff. Equs., 5 (2000), 1221-1252.  Google Scholar

[3]

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

[4]

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

[5]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlin. Anal., 6 (1982), 435-454. 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, Chapman & Hall/CRC, Boca Raton, FL, 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, Math. Meth. Appl. Sc., 26 (2003), 907-936. 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, J. Anal. Appl., 30 (2011), 387-419. 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, Hindawi Publ. Corp. ISRN Math. Analysis, Vol. 2013, article ID268505.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic type, Nauka Moscow, 1967 (in Russian). Google Scholar

[14]

J. L. Lions and E. Magnes, Problémes aux limites non homogénes et applicationes, Vol. 1, Dunod, Paris, 1968.  Google Scholar

[15]

I. Pawłow 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

[16]

I. Pawłow and W. M. Zajączkowski, Unique solvability of a nonlinear termoviscoelasticity 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

[17]

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

[18]

T. Roubíček, Termoviscoelasticity at small strains with $L^1$-data, Quart. Appl. Math., 67 (2009), 47-71. doi: 10.1090/S0033-569X-09-01094-3.  Google Scholar

[19]

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

[20]

T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape memory alloys, Discrete and Continuous Dynamical Systems, Supplement (2007), 892-902.  Google Scholar

[21]

M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoviscoelasticity, Arch. Ration. Mech. Anal., 76 (1981), 97-133. 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, Zap. Nauchn. Sem. S. Petersburg. Otdel. Mat. Inst. Steklov (POMI), 288 (2002), 204-231. 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, Trudy MIAN, 83 (1965), (in Russian).  Google Scholar

[24]

B. D. Coleman, Thermodynamics of materials with memory, Arch. Ration. Mech. Anal., 17 (1964), 1-46.  Google Scholar

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