American Institute of Mathematical Sciences

April  2011, 4(2): 441-466. doi: 10.3934/dcdss.2011.4.441

Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm

 1 System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw 2 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw

Received  February 2009 Published  November 2010

In this paper we study a nonlinear thermoviscoelasticity system within the framework of parabolic theory in anisotropic Sobolev spaces with a mixed norm. The application of such a framework allows to generalize the previous results by admitting stronger thermomechanical nonlinearity and a broader class of solution spaces.
Citation: Irena Pawłow, Wojciech M. Zajączkowski. Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 441-466. doi: 10.3934/dcdss.2011.4.441
References:
 [1] O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings,", Nauka, (1975). Google Scholar [2] Ya. S. Bugrov, Function spaces with mixed norm,, Izv. AN SSSR, 35 (1971), 1137. doi: 10.1070/IM1971v005n05ABEH001213. Google Scholar [3] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity,, Nonlinear Anal., 6 (1982), 435. doi: 10.1016/0362-546X(82)90058-X. Google Scholar [4] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$ estimates for parabolic boundary value problems with inhomogeneous data,, Math. Z., 257 (2007), 193. doi: 10.1007/s00209-007-0120-9. Google Scholar [5] F. Falk, Elastic phase transitions and nonconvex energy functions,, in, (1990), 45. Google Scholar [6] F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys,, Journal of Physics: Condensed Matter, 2 (1990), 61. doi: 10.1088/0953-8984/2/1/005. Google Scholar [7] K. K. Golovkin, On equivalent norms for fractional spaces,, Trudy Mat. Inst. Steklov, 66 (1962), 364. Google Scholar [8] M. Hieber and J. Prüss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations,, Commun. in PDEs, 22 (1997), 1647. doi: 10.1080/03605309708821314. Google Scholar [9] N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations,, Algebra i Analiz., 13 (2001), 1. Google Scholar [10] P. Maremonti and V. A. Solonnikov, On the estimates of solutions of evolution Stokes problem in anisotropic Sobolev spaces with mixed norm,, Zap. Nauchn. Semin. POMI, 222 (1995), 124. Google Scholar [11] S. M. Nikolskij, "Approximation of Functions of Several Variables and Imbedding Theorems,", Nauka, (1977). Google Scholar [12] I. Pawłow and W. M. Zajączkowski, Unique global solvability in two-dimensional non-linear thermoelasticity,, Math. Meth. Appl. Sci., 28 (2005), 551. doi: 10.1002/mma.582. Google Scholar [13] I. Pawłow and W. M. Zajączkowski, Global existence to a three-dimensional non-linear thermoelasticity system arising in shape memory materials,, Math. Meth. Appl. Sci., 28 (2005), 407. doi: 10.1002/mma.574. Google Scholar [14] I. Pawłow and W. M. Zajączkowski, New existence result for a 3-D shape memory model,, in, 71 (2006), 201. Google Scholar [15] I. Pawłow and A. Żochowski, Existence and uniqueness for a three-dimensional thermoelastic system,, Dissertationes Mathematicae, 406 (2002). doi: 10.4064/dm406-0-1. Google Scholar [16] V. A. Solonnikov, Boundary value problems for linear parabolic systems of general type,, Trudy Mat. Inst. Steklov, 83 (1965), 1. Google Scholar [17] V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm,, Zapiski Naucz. Sem. LOMI, 288 (2002), 204. Google Scholar [18] P. Weidemeier, Existence results in $L_p-L_q$ spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions,, in, 384 (1998), 189. Google Scholar [19] P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm,, Electr. Res. Announc. Am. Math. Soc., 8 (2002), 47. doi: 10.1090/S1079-6762-02-00104-X. Google Scholar [20] S. Yoshikawa, Unique global existence for a three-dimensional thermoelastic system of shape memory alloys,, Adv. Math. Sci Appl., 15 (2005), 603. Google Scholar [21] S. Yoshikawa, Small energy global existence for a two-dimensional thermoelastic system of shape memory materials,, in, 23 (2005), 297. Google Scholar [22] S. Yoshikawa, Global solutions for shape memory alloy systems,, Tohoku Math. Publ., 32 (2007). doi: 10.2748/tmpub.32.1. Google Scholar [23] S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials,, SIAM J. Math. Anal., 38 (2007), 1733. doi: 10.1137/060653159. Google Scholar [24] S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat,, Commun. Pure Appl. Anal., 8 (2009), 1093. doi: 10.3934/cpaa.2009.8.1093. 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, (1975). Google Scholar [2] Ya. S. Bugrov, Function spaces with mixed norm,, Izv. AN SSSR, 35 (1971), 1137. doi: 10.1070/IM1971v005n05ABEH001213. Google Scholar [3] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity,, Nonlinear Anal., 6 (1982), 435. doi: 10.1016/0362-546X(82)90058-X. Google Scholar [4] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$ estimates for parabolic boundary value problems with inhomogeneous data,, Math. Z., 257 (2007), 193. doi: 10.1007/s00209-007-0120-9. Google Scholar [5] F. Falk, Elastic phase transitions and nonconvex energy functions,, in, (1990), 45. Google Scholar [6] F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys,, Journal of Physics: Condensed Matter, 2 (1990), 61. doi: 10.1088/0953-8984/2/1/005. Google Scholar [7] K. K. Golovkin, On equivalent norms for fractional spaces,, Trudy Mat. Inst. Steklov, 66 (1962), 364. Google Scholar [8] M. Hieber and J. Prüss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations,, Commun. in PDEs, 22 (1997), 1647. doi: 10.1080/03605309708821314. Google Scholar [9] N. V. Krylov, The Calderon-Zygmund theorem and its application for parabolic equations,, Algebra i Analiz., 13 (2001), 1. Google Scholar [10] P. Maremonti and V. A. Solonnikov, On the estimates of solutions of evolution Stokes problem in anisotropic Sobolev spaces with mixed norm,, Zap. Nauchn. Semin. POMI, 222 (1995), 124. Google Scholar [11] S. M. Nikolskij, "Approximation of Functions of Several Variables and Imbedding Theorems,", Nauka, (1977). Google Scholar [12] I. Pawłow and W. M. Zajączkowski, Unique global solvability in two-dimensional non-linear thermoelasticity,, Math. Meth. Appl. Sci., 28 (2005), 551. doi: 10.1002/mma.582. Google Scholar [13] I. Pawłow and W. M. Zajączkowski, Global existence to a three-dimensional non-linear thermoelasticity system arising in shape memory materials,, Math. Meth. Appl. Sci., 28 (2005), 407. doi: 10.1002/mma.574. Google Scholar [14] I. Pawłow and W. M. Zajączkowski, New existence result for a 3-D shape memory model,, in, 71 (2006), 201. Google Scholar [15] I. Pawłow and A. Żochowski, Existence and uniqueness for a three-dimensional thermoelastic system,, Dissertationes Mathematicae, 406 (2002). doi: 10.4064/dm406-0-1. Google Scholar [16] V. A. Solonnikov, Boundary value problems for linear parabolic systems of general type,, Trudy Mat. Inst. Steklov, 83 (1965), 1. Google Scholar [17] V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm,, Zapiski Naucz. Sem. LOMI, 288 (2002), 204. Google Scholar [18] P. Weidemeier, Existence results in $L_p-L_q$ spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions,, in, 384 (1998), 189. Google Scholar [19] P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm,, Electr. Res. Announc. Am. Math. Soc., 8 (2002), 47. doi: 10.1090/S1079-6762-02-00104-X. Google Scholar [20] S. Yoshikawa, Unique global existence for a three-dimensional thermoelastic system of shape memory alloys,, Adv. Math. Sci Appl., 15 (2005), 603. Google Scholar [21] S. Yoshikawa, Small energy global existence for a two-dimensional thermoelastic system of shape memory materials,, in, 23 (2005), 297. Google Scholar [22] S. Yoshikawa, Global solutions for shape memory alloy systems,, Tohoku Math. Publ., 32 (2007). doi: 10.2748/tmpub.32.1. Google Scholar [23] S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials,, SIAM J. Math. Anal., 38 (2007), 1733. doi: 10.1137/060653159. Google Scholar [24] S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat,, Commun. Pure Appl. Anal., 8 (2009), 1093. doi: 10.3934/cpaa.2009.8.1093. Google Scholar
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