American Institute of Mathematical Sciences

June  2015, 35(6): 2523-2538. doi: 10.3934/dcds.2015.35.2523

Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint

 1 Department of Mathematics, Kyoto University of Education, Fuji 1, Fukakusa Fushimi-ku, Kyoto 612-8522 2 Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301

Received  December 2013 Revised  April 2014 Published  December 2014

This paper is concerned with a heat convection problem. We discuss it in the framework of parabolic variational inequalities. The problem is a system of a heat equation with convection and a Navier-Stokes variational inequality with temperature-dependent velocity constraint. Our problem is a sort of parabolic quasi-variational inequalities in the sense that the constraint set for the velocity field depends on the unknown temperature. We shall give an existence result of the heat convection problem in a weak sense, and show that under some additional constraint for temperature there exists a strong solution of the problem.
Citation: Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523
References:
 [1] M. Aso, M. Frémond and N. Kenmochi, Parabolic systems with the unknown dependent constraints arising in phase transitions, Internat. Ser. Numer. Math., Birkhäuser Verlag, Basel, 154 (2007), 45-54. doi: 10.1007/978-3-7643-7719-9_5. [2] H. Attuoch, Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program, Pitman, Boston, 1984. [3] A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature, Adv. Math. Sci. Appl., 20 (2010), 153-168. [4] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] T. Fukao and N. Kenmochi, Variational inequality for the Navier-Stokes equations with time-dependent constraint, GAKUTO Internat. Ser. Math. Sci. Appl., 34 (2011), 87-102. [6] T. Fukao and N. Kenmochi, A thermohydraulics model with temperature dependent constraint on velocity fields, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 17-34. doi: 10.3934/dcdss.2014.7.17. [7] T. Fukao and N. Kenmochi, Parabolic variational inequalities with weakly time-dependent constraints, Adv. Math. Sci. Appl., 23 (2013), 365-395. [8] A. Kadoya, Y. Murase and N. Kenmochi, A class of nonlinear parabolic systems with environmental constraints, Adv. Math. Sci. Appl., 20 (2010), 281-313. [9] R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86 (2009), 175-194. doi: 10.4064/bc86-0-11. [10] R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with non-local constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583. [11] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331. doi: 10.1007/BF02761596. [12] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Edu., Chiba Univ., 30 (1981), 1-87. [13] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of Differential Equations: Stationary Partial Differential Equations, North-Holland, Amsterdam, 4 (2007), 203-298. doi: 10.1016/S1874-5733(07)80007-6. [14] N. Kenmochi, Parabolic quasi-variational diffusion problems with gradient constraints, Discrete Contin. Dyn. Syst. Ser. S., 6 (2013), 423-438. doi: 10.3934/dcdss.2013.6.423. [15] J. F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 29 (2000), 153-169. [16] K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials, GAKUTO Internat. Ser. Math. Sci. Appl., 11 (1998), 287-310. [17] Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491-515.

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References:
 [1] M. Aso, M. Frémond and N. Kenmochi, Parabolic systems with the unknown dependent constraints arising in phase transitions, Internat. Ser. Numer. Math., Birkhäuser Verlag, Basel, 154 (2007), 45-54. doi: 10.1007/978-3-7643-7719-9_5. [2] H. Attuoch, Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program, Pitman, Boston, 1984. [3] A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature, Adv. Math. Sci. Appl., 20 (2010), 153-168. [4] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] T. Fukao and N. Kenmochi, Variational inequality for the Navier-Stokes equations with time-dependent constraint, GAKUTO Internat. Ser. Math. Sci. Appl., 34 (2011), 87-102. [6] T. Fukao and N. Kenmochi, A thermohydraulics model with temperature dependent constraint on velocity fields, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 17-34. doi: 10.3934/dcdss.2014.7.17. [7] T. Fukao and N. Kenmochi, Parabolic variational inequalities with weakly time-dependent constraints, Adv. Math. Sci. Appl., 23 (2013), 365-395. [8] A. Kadoya, Y. Murase and N. Kenmochi, A class of nonlinear parabolic systems with environmental constraints, Adv. Math. Sci. Appl., 20 (2010), 281-313. [9] R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86 (2009), 175-194. doi: 10.4064/bc86-0-11. [10] R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with non-local constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583. [11] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331. doi: 10.1007/BF02761596. [12] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Edu., Chiba Univ., 30 (1981), 1-87. [13] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of Differential Equations: Stationary Partial Differential Equations, North-Holland, Amsterdam, 4 (2007), 203-298. doi: 10.1016/S1874-5733(07)80007-6. [14] N. Kenmochi, Parabolic quasi-variational diffusion problems with gradient constraints, Discrete Contin. Dyn. Syst. Ser. S., 6 (2013), 423-438. doi: 10.3934/dcdss.2013.6.423. [15] J. F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 29 (2000), 153-169. [16] K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials, GAKUTO Internat. Ser. Math. Sci. Appl., 11 (1998), 287-310. [17] Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491-515.
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