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 and 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, Moscow, 1975, (in Russian).

[2]

Ya. S. Bugrov, Function spaces with mixed norm, Izv. AN SSSR, Ser. Mat., 35 (1971), 1137-1158; Eng. transl.: Math. USSR-Izv., 5 (1971), 1145-1167. doi: 10.1070/IM1971v005n05ABEH001213.

[3]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454. doi: 10.1016/0362-546X(82)90058-X.

[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), 193-224. doi: 10.1007/s00209-007-0120-9.

[5]

F. Falk, Elastic phase transitions and nonconvex energy functions, in "Free Boundary Problems: Theory and Applications" (I. K.-H. Hoffmann and J. Sprekels, eds.), Longman, London, (1990), 45-59.

[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), 61-77. doi: 10.1088/0953-8984/2/1/005.

[7]

K. K. Golovkin, On equivalent norms for fractional spaces, Trudy Mat. Inst. Steklov, 66 (1962), 364-383 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Ser 2, 81 (1969), 257-280.

[8]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations, Commun. in PDEs, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314.

[9]

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

[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), 124-150, (in Russian).

[11]

S. M. Nikolskij, "Approximation of Functions of Several Variables and Imbedding Theorems," Nauka, Moscow, 1977.

[12]

I. Pawłow and W. M. Zajączkowski, Unique global solvability in two-dimensional non-linear thermoelasticity, Math. Meth. Appl. Sci., 28 (2005), 551-592. doi: 10.1002/mma.582.

[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), 407-442. doi: 10.1002/mma.574.

[14]

I. Pawłow and W. M. Zajączkowski, New existence result for a 3-D shape memory model, in "Dissipative Phase Transitions" (P. Colli, N. Kenmochi and J. Sprekels, eds.), Ser. Adv., Math. Appl. Sci., 71, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2006), 201-224.

[15]

I. Pawłow and A. Żochowski, Existence and uniqueness for a three-dimensional thermoelastic system, Dissertationes Mathematicae, 406 (2002), 46. doi: 10.4064/dm406-0-1.

[16]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of general type, Trudy Mat. Inst. Steklov, 83 (1965), 1-162, (in Russian).

[17]

V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm, Zapiski Naucz. Sem. LOMI, T., 288 (2002), 204-231.

[18]

P. Weidemeier, Existence results in $L_p-L_q$ spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions, in "Progress in PDEs" (H. Amann and et al., eds.), Pitman Research Notes in Math., 384, Longman, Harlow, (1998), 189-200.

[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), 47-52. doi: 10.1090/S1079-6762-02-00104-X.

[20]

S. Yoshikawa, Unique global existence for a three-dimensional thermoelastic system of shape memory alloys, Adv. Math. Sci Appl., 15 (2005), 603-627.

[21]

S. Yoshikawa, Small energy global existence for a two-dimensional thermoelastic system of shape memory materials, in "Mathematical Approach to Nonlinear Phenomena," GAKUTO Internat. Ser. Math. Sci. Appl., 23, Gakkotosho, Tokyo, (2005), 297-306.

[22]

S. Yoshikawa, Global solutions for shape memory alloy systems, Tohoku Math. Publ., 32 (2007), 105. doi: 10.2748/tmpub.32.1.

[23]

S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials, SIAM J. Math. Anal., 38 (2007), 1733-1759. doi: 10.1137/060653159.

[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), 1093-1115. doi: 10.3934/cpaa.2009.8.1093.

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]

Ya. S. Bugrov, Function spaces with mixed norm, Izv. AN SSSR, Ser. Mat., 35 (1971), 1137-1158; Eng. transl.: Math. USSR-Izv., 5 (1971), 1145-1167. doi: 10.1070/IM1971v005n05ABEH001213.

[3]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454. doi: 10.1016/0362-546X(82)90058-X.

[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), 193-224. doi: 10.1007/s00209-007-0120-9.

[5]

F. Falk, Elastic phase transitions and nonconvex energy functions, in "Free Boundary Problems: Theory and Applications" (I. K.-H. Hoffmann and J. Sprekels, eds.), Longman, London, (1990), 45-59.

[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), 61-77. doi: 10.1088/0953-8984/2/1/005.

[7]

K. K. Golovkin, On equivalent norms for fractional spaces, Trudy Mat. Inst. Steklov, 66 (1962), 364-383 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Ser 2, 81 (1969), 257-280.

[8]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations, Commun. in PDEs, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314.

[9]

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

[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), 124-150, (in Russian).

[11]

S. M. Nikolskij, "Approximation of Functions of Several Variables and Imbedding Theorems," Nauka, Moscow, 1977.

[12]

I. Pawłow and W. M. Zajączkowski, Unique global solvability in two-dimensional non-linear thermoelasticity, Math. Meth. Appl. Sci., 28 (2005), 551-592. doi: 10.1002/mma.582.

[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), 407-442. doi: 10.1002/mma.574.

[14]

I. Pawłow and W. M. Zajączkowski, New existence result for a 3-D shape memory model, in "Dissipative Phase Transitions" (P. Colli, N. Kenmochi and J. Sprekels, eds.), Ser. Adv., Math. Appl. Sci., 71, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2006), 201-224.

[15]

I. Pawłow and A. Żochowski, Existence and uniqueness for a three-dimensional thermoelastic system, Dissertationes Mathematicae, 406 (2002), 46. doi: 10.4064/dm406-0-1.

[16]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of general type, Trudy Mat. Inst. Steklov, 83 (1965), 1-162, (in Russian).

[17]

V. A. Solonnikov, Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm, Zapiski Naucz. Sem. LOMI, T., 288 (2002), 204-231.

[18]

P. Weidemeier, Existence results in $L_p-L_q$ spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions, in "Progress in PDEs" (H. Amann and et al., eds.), Pitman Research Notes in Math., 384, Longman, Harlow, (1998), 189-200.

[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), 47-52. doi: 10.1090/S1079-6762-02-00104-X.

[20]

S. Yoshikawa, Unique global existence for a three-dimensional thermoelastic system of shape memory alloys, Adv. Math. Sci Appl., 15 (2005), 603-627.

[21]

S. Yoshikawa, Small energy global existence for a two-dimensional thermoelastic system of shape memory materials, in "Mathematical Approach to Nonlinear Phenomena," GAKUTO Internat. Ser. Math. Sci. Appl., 23, Gakkotosho, Tokyo, (2005), 297-306.

[22]

S. Yoshikawa, Global solutions for shape memory alloy systems, Tohoku Math. Publ., 32 (2007), 105. doi: 10.2748/tmpub.32.1.

[23]

S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials, SIAM J. Math. Anal., 38 (2007), 1733-1759. doi: 10.1137/060653159.

[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), 1093-1115. doi: 10.3934/cpaa.2009.8.1093.

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