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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

Takeshi Fukao 1, and  Nobuyuki Kenmochi 2,

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.
Keywords: Quasi-variational inequality, velocity constraint, Navier-Stokes equation, evolution equation., heat convection.
Mathematics Subject Classification: Primary: 35M86, 47J35; Secondary: 47J2.
Citation: Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & 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.  Google Scholar

[2]

H. Attuoch, Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program, Pitman, Boston, 1984.  Google Scholar

[3]

A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature, Adv. Math. Sci. Appl., 20 (2010), 153-168.  Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[7]

T. Fukao and N. Kenmochi, Parabolic variational inequalities with weakly time-dependent constraints, Adv. Math. Sci. Appl., 23 (2013), 365-395.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with non-local constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583.  Google Scholar

[11]

N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331. doi: 10.1007/BF02761596.  Google Scholar

[12]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Edu., Chiba Univ., 30 (1981), 1-87. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491-515.  Google Scholar

show all references

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.  Google Scholar

[2]

H. Attuoch, Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program, Pitman, Boston, 1984.  Google Scholar

[3]

A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature, Adv. Math. Sci. Appl., 20 (2010), 153-168.  Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[7]

T. Fukao and N. Kenmochi, Parabolic variational inequalities with weakly time-dependent constraints, Adv. Math. Sci. Appl., 23 (2013), 365-395.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with non-local constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583.  Google Scholar

[11]

N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331. doi: 10.1007/BF02761596.  Google Scholar

[12]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Edu., Chiba Univ., 30 (1981), 1-87. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491-515.  Google Scholar

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