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

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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

Abstract
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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 - A, 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., 154* (2007), 45. doi: 10.1007/978-3-7643-7719-9_5. Google Scholar*
[2]	H. Attuoch, Variational Convergence for Functions and Operators,, Pitman Advanced Publishing Program, (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. Google Scholar
[4]	H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland, (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. 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. 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. 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. 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. 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. Google Scholar
[11]	N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar
[12]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Edu., 30 (1981), 1. Google Scholar
[13]	N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, Handbook of Differential Equations: Stationary Partial Differential Equations, 4 (2007), 203. 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. 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. 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. Google Scholar
[17]	Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491. 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., 154* (2007), 45. doi: 10.1007/978-3-7643-7719-9_5. Google Scholar*
[2]	H. Attuoch, Variational Convergence for Functions and Operators,, Pitman Advanced Publishing Program, (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. Google Scholar
[4]	H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland, (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. 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. 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. 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. 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. 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. Google Scholar
[11]	N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar
[12]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Edu., 30 (1981), 1. Google Scholar
[13]	N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, Handbook of Differential Equations: Stationary Partial Differential Equations, 4 (2007), 203. 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. 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. 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. Google Scholar
[17]	Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491. Google Scholar

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