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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[33]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[33]

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

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