# 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 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.
 [1] Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. [2] Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure and Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041 [3] Long Huang, Jun Liu, Dachun Yang, Wen Yuan. Real-variable characterizations of new anisotropic mixed-norm Hardy spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3033-3082. doi: 10.3934/cpaa.2020132 [4] Ferenc Weisz. Dual spaces of mixed-norm martingale Hardy spaces. Communications on Pure and Applied Analysis, 2021, 20 (2) : 681-695. doi: 10.3934/cpaa.2020285 [5] Valerii Los, Vladimir Mikhailets, Aleksandr Murach. Parabolic problems in generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3605-3636. doi: 10.3934/cpaa.2021123 [6] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [7] Sergey Degtyarev. Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022029 [8] Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629 [9] Doyoon Kim, Kyeong-Hun Kim, Kijung Lee. Parabolic Systems with measurable coefficients in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2587-2613. doi: 10.3934/cpaa.2022062 [10] Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034 [11] P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151 [12] Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893 [13] Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013 [14] Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097 [15] Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201 [16] Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389 [17] Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047 [18] Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1 [19] Frédéric Bernicot, Vjekoslav Kovač. Sobolev norm estimates for a class of bilinear multipliers. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1305-1315. doi: 10.3934/cpaa.2014.13.1305 [20] Michel Frémond, Elisabetta Rocca. A model for shape memory alloys with the possibility of voids. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1633-1659. doi: 10.3934/dcds.2010.27.1633

2021 Impact Factor: 1.865