February  2014, 7(1): 17-34. doi: 10.3934/dcdss.2014.7.17

A thermohydraulics model with temperature dependent constraint on velocity fields

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  February 2012 Revised  March 2013 Published  July 2013

In this paper, the Navier-Stokes variational inequality with the temperature dependent constraint is considered in 3-dimensional space. This problem is motivated by an initial-boundary value problem for a thermohydraulics model in which the absolute value of the velocity field is constrained, depending on the unknown temperature. The abstract theory of nonlinear evolution equations governed by subdifferentials of time-dependent convex functionals is useful in showing the existence of a solution. In the mathematical treatment, the point of emphasis is to specify a class of time-dependence of convex constraints.
Citation: Takeshi Fukao, Nobuyuki Kenmochi. A thermohydraulics model with temperature dependent constraint on velocity fields. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 17-34. doi: 10.3934/dcdss.2014.7.17
References:
[1]

T. Aiki and E. Minchev, A prey-predator model with hysteresis effect, SIAM J. Math. Anal., 36 (2005), 2020-2032. doi: 10.1137/S0036141004440186.  Google Scholar

[2]

C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems," A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.  Google Scholar

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei Republicii Socialiste Romnia, Bucharest; Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[4]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307. doi: 10.1006/jmaa.2000.7256.  Google Scholar

[5]

M. Biroli, Sur l'inéquation d'évolution de Navier-Stokes. I, II, III, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8), 52 (1972), 457-460, 591-598, 811-820.  Google Scholar

[6]

M. Biroli, Sur la solution faible des inéquations d'évolution du type de Navier-Stokes avec convexe dépendant du temps, Boll. Unione Mat. Ital. (4), 11 (1975), 309-321.  Google Scholar

[7]

H. Brézis, Inéquations variationnellers relatives à l'opérateur de Navier-Stokes, J. Math. Anal. Appl., 39 (1972), 159-165. doi: 10.1016/0022-247X(72)90231-4.  Google Scholar

[8]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert," North-Holland Mathematics Studies, No. 5, Notas de Matemática (50), North-Holland Publishing Co., Inc., New York, 1973.  Google Scholar

[9]

P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685. doi: 10.1006/jmaa.2000.7338.  Google Scholar

[10]

T. Fukao, Variational inequality for the Stokes equations with constraint, Discrete Contin. Dyn. Syst., suppl., (2011), 437-446.  Google Scholar

[11]

T. Fukao and N. Kenmochi, Variational inequality for the Navier-Stokes equations with time dependent constraint, in "International Symposium on Computational Science 2011," GAKUTO Internat. Ser. Math. Sci. Appl., 34, (2011), 273-287. Google Scholar

[12]

T. Fukao and N. Kenmochi, Weak solvabilty of a class of parabolic variational iequalities with time-dependent constraints,, in preparation., ().   Google Scholar

[13]

R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, in "Nonlocal and Abstract Parabolic Equations and their Applications, Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 175-194. doi: 10.4064/bc86-0-11.  Google Scholar

[14]

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

[15]

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

[16]

M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints, Adv. Math. Sci. Appl., 15 (2005), 335-354.  Google Scholar

[17]

M. Kunze and J. F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications, Math. Meth. Appl. Sci., 23 (2000), 897-908. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H.  Google Scholar

[18]

A. I. Lefter, Navier-Stokes equations with potentials, Abstr. Appl. Anal., 2007 (2007), ID 79406, 1-30. doi: 10.1155/2007/79406.  Google Scholar

[19]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Études Mathématiques, Dunod Gauthier-Villas, Paris, 1968. Google Scholar

[20]

F. Mignot and J. P. Puel, Inéquations d'evolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d'evolution, Arch. Rational Mech. Anal., 64 (1977), 59-91. doi: 10.1007/BF00280179.  Google Scholar

[21]

E. Minchev, On a system of nonlinear PDE's for phase transitions with vector order parameter, Adv. Math. Sci. Appl., 14 (2004), 187-209.  Google Scholar

[22]

U. Mosco, Convergence of convex set and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[23]

J. Naumann, On evolution inequalities of Navier-Stokes type in three dimensions, Ann. Mat. Pure Appl.(4), 124 (1980), 107-125. doi: 10.1007/BF01795388.  Google Scholar

[24]

M. ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), 268-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar

[25]

G. Prouse, On an inequality related to the motion, in any dimension, of viscous, incompressible fluids Nota I, II, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (8), 67 (1979), 191-196, 282-288.  Google Scholar

[26]

J. F. Rodrigues, On the evolution Boussinesq-Stefan problem for non-Newtonian fluids, in "Free Boundary Problems: Theory and Applications, II," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 390-397.  Google Scholar

[27]

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

[28]

J. F. Rodrigues and J. M. Urbano, On the stationary Boussinesq-Stefan problem with constitutive power-laws, Int. J. Non-Linear Mechanics, 33 (1998), 555-566. doi: 10.1016/S0020-7462(97)00041-3.  Google Scholar

[29]

J. F. Rodrigues and J. M. Urbano, On a three-dimensional convective Stefan problem for a non-Newtonian fluid, in "Applied Nonlinear Analysis," Kluwer/Plenum, New York, (1999), 457-468.  Google Scholar

[30]

R. Rossi and U. Stefanelli, An order approach to a class of quasivariational sweeping process, Adv. Differential Equations, 10 (2005), 527-552.  Google Scholar

[31]

U. Stefanelli, Nonlocal quasivariational evolution problems, J. Differential Equations, 229 (2006), 204-228. doi: 10.1016/j.jde.2006.05.004.  Google Scholar

[32]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[33]

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]

T. Aiki and E. Minchev, A prey-predator model with hysteresis effect, SIAM J. Math. Anal., 36 (2005), 2020-2032. doi: 10.1137/S0036141004440186.  Google Scholar

[2]

C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems," A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.  Google Scholar

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei Republicii Socialiste Romnia, Bucharest; Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[4]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307. doi: 10.1006/jmaa.2000.7256.  Google Scholar

[5]

M. Biroli, Sur l'inéquation d'évolution de Navier-Stokes. I, II, III, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8), 52 (1972), 457-460, 591-598, 811-820.  Google Scholar

[6]

M. Biroli, Sur la solution faible des inéquations d'évolution du type de Navier-Stokes avec convexe dépendant du temps, Boll. Unione Mat. Ital. (4), 11 (1975), 309-321.  Google Scholar

[7]

H. Brézis, Inéquations variationnellers relatives à l'opérateur de Navier-Stokes, J. Math. Anal. Appl., 39 (1972), 159-165. doi: 10.1016/0022-247X(72)90231-4.  Google Scholar

[8]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert," North-Holland Mathematics Studies, No. 5, Notas de Matemática (50), North-Holland Publishing Co., Inc., New York, 1973.  Google Scholar

[9]

P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685. doi: 10.1006/jmaa.2000.7338.  Google Scholar

[10]

T. Fukao, Variational inequality for the Stokes equations with constraint, Discrete Contin. Dyn. Syst., suppl., (2011), 437-446.  Google Scholar

[11]

T. Fukao and N. Kenmochi, Variational inequality for the Navier-Stokes equations with time dependent constraint, in "International Symposium on Computational Science 2011," GAKUTO Internat. Ser. Math. Sci. Appl., 34, (2011), 273-287. Google Scholar

[12]

T. Fukao and N. Kenmochi, Weak solvabilty of a class of parabolic variational iequalities with time-dependent constraints,, in preparation., ().   Google Scholar

[13]

R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, in "Nonlocal and Abstract Parabolic Equations and their Applications, Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 175-194. doi: 10.4064/bc86-0-11.  Google Scholar

[14]

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

[15]

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

[16]

M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints, Adv. Math. Sci. Appl., 15 (2005), 335-354.  Google Scholar

[17]

M. Kunze and J. F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications, Math. Meth. Appl. Sci., 23 (2000), 897-908. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H.  Google Scholar

[18]

A. I. Lefter, Navier-Stokes equations with potentials, Abstr. Appl. Anal., 2007 (2007), ID 79406, 1-30. doi: 10.1155/2007/79406.  Google Scholar

[19]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Études Mathématiques, Dunod Gauthier-Villas, Paris, 1968. Google Scholar

[20]

F. Mignot and J. P. Puel, Inéquations d'evolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d'evolution, Arch. Rational Mech. Anal., 64 (1977), 59-91. doi: 10.1007/BF00280179.  Google Scholar

[21]

E. Minchev, On a system of nonlinear PDE's for phase transitions with vector order parameter, Adv. Math. Sci. Appl., 14 (2004), 187-209.  Google Scholar

[22]

U. Mosco, Convergence of convex set and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[23]

J. Naumann, On evolution inequalities of Navier-Stokes type in three dimensions, Ann. Mat. Pure Appl.(4), 124 (1980), 107-125. doi: 10.1007/BF01795388.  Google Scholar

[24]

M. ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), 268-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar

[25]

G. Prouse, On an inequality related to the motion, in any dimension, of viscous, incompressible fluids Nota I, II, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (8), 67 (1979), 191-196, 282-288.  Google Scholar

[26]

J. F. Rodrigues, On the evolution Boussinesq-Stefan problem for non-Newtonian fluids, in "Free Boundary Problems: Theory and Applications, II," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 390-397.  Google Scholar

[27]

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

[28]

J. F. Rodrigues and J. M. Urbano, On the stationary Boussinesq-Stefan problem with constitutive power-laws, Int. J. Non-Linear Mechanics, 33 (1998), 555-566. doi: 10.1016/S0020-7462(97)00041-3.  Google Scholar

[29]

J. F. Rodrigues and J. M. Urbano, On a three-dimensional convective Stefan problem for a non-Newtonian fluid, in "Applied Nonlinear Analysis," Kluwer/Plenum, New York, (1999), 457-468.  Google Scholar

[30]

R. Rossi and U. Stefanelli, An order approach to a class of quasivariational sweeping process, Adv. Differential Equations, 10 (2005), 527-552.  Google Scholar

[31]

U. Stefanelli, Nonlocal quasivariational evolution problems, J. Differential Equations, 229 (2006), 204-228. doi: 10.1016/j.jde.2006.05.004.  Google Scholar

[32]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[33]

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

[1]

Masahiro Kubo, Noriaki Yamazaki. Elliptic-parabolic variational inequalities with time-dependent constraints. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 335-359. doi: 10.3934/dcds.2007.19.335

[2]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015

[3]

Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418

[4]

Masahiro Kubo, Noriaki Yamazaki. Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. Conference Publications, 2007, 2007 (Special) : 614-623. doi: 10.3934/proc.2007.2007.614

[5]

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

[6]

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

[7]

Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control & Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143

[8]

Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020088

[9]

Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659

[10]

Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021168

[11]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

[12]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749

[13]

Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240

[14]

Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053

[15]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141

[16]

Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061

[17]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1

[18]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969

[19]

Liping Pang, Fanyun Meng, Jinhe Wang. Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1653-1675. doi: 10.3934/jimo.2018116

[20]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]