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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 |
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. 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. |
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). |
[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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