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 & 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]

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

[10]

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

[11]

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

[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

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]