American Institute of Mathematical Sciences

Advanced
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 - 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

[1]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[2]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583
[3]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[4]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[5]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[6]

I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191
[7]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[8]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020105
[9]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
[10]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[11]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[12]

Stanislaw Migórski, Anna Ochal. Navier-Stokes problems modeled by evolution hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 731-740. doi: 10.3934/proc.2007.2007.731
[13]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[14]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[15]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[16]

Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1
[17]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[18]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
[19]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[20]

Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences