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, Moscow, 1975, (in Russian).  Google Scholar

[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.  Google Scholar

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

[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.  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), 61-77. 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), 364-383 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Ser 2, 81 (1969), 257-280.  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), 1647-1669. 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), 1-25, (in Russian).  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), 124-150, (in Russian).  Google Scholar

[11]

S. M. Nikolskij, "Approximation of Functions of Several Variables and Imbedding Theorems," Nauka, Moscow, 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), 551-592. 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), 407-442. 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 "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.  Google Scholar

[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.  Google Scholar

[16]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of general type, Trudy Mat. Inst. Steklov, 83 (1965), 1-162, (in Russian).  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, T., 288 (2002), 204-231.  Google Scholar

[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. 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), 47-52. 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), 603-627.  Google Scholar

[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.  Google Scholar

[22]

S. Yoshikawa, Global solutions for shape memory alloy systems, Tohoku Math. Publ., 32 (2007), 105. 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), 1733-1759. 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), 1093-1115. doi: 10.3934/cpaa.2009.8.1093.  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]

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.  Google Scholar

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

[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.  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), 61-77. 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), 364-383 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Ser 2, 81 (1969), 257-280.  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), 1647-1669. 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), 1-25, (in Russian).  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), 124-150, (in Russian).  Google Scholar

[11]

S. M. Nikolskij, "Approximation of Functions of Several Variables and Imbedding Theorems," Nauka, Moscow, 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), 551-592. 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), 407-442. 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 "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.  Google Scholar

[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.  Google Scholar

[16]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of general type, Trudy Mat. Inst. Steklov, 83 (1965), 1-162, (in Russian).  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, T., 288 (2002), 204-231.  Google Scholar

[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. 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), 47-52. 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), 603-627.  Google Scholar

[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.  Google Scholar

[22]

S. Yoshikawa, Global solutions for shape memory alloy systems, Tohoku Math. Publ., 32 (2007), 105. 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), 1733-1759. 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), 1093-1115. doi: 10.3934/cpaa.2009.8.1093.  Google Scholar

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