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Global regular solutions to two-dimensional thermoviscoelasticity

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  • 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.
    Mathematics Subject Classification: Primary: 74B20, 35K50; Secondary: 35Q72, 74F05.

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  • [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. Guibé, Existence of a solution for nonlinear system in thermoviscoelasticity, Adv. Diff. Equs., 5 (2000), 1221-1252.

    [3]

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

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [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.

    [11]

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

    [12]

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

    [13]

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

    [14]

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

    [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.

    [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.

    [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.

    [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.

    [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.

    [20]

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

    [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.

    [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.

    [23]

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

    [24]

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

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