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. doi: 10.1137/S0036141004440186.

[2]

C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems,", A Wiley-Interscience Publication, (1984).

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste Romnia, (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. 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.

[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.

[7]

H. Brézis, Inéquations variationnellers relatives à l'opérateur de Navier-Stokes,, J. Math. Anal. Appl., 39 (1972), 159. 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, (1973).

[9]

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

[10]

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

[11]

T. Fukao and N. Kenmochi, Variational inequality for the Navier-Stokes equations with time dependent constraint,, in, 34 (2011), 273.

[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, 86 (2009), 175. doi: 10.4064/bc86-0-11.

[14]

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

[15]

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

[16]

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

[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. 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), 1. doi: 10.1155/2007/79406.

[19]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Études Mathématiques, (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. 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.

[22]

U. Mosco, Convergence of convex set and of solutions of variational inequalities,, Adv. Math., 3 (1969), 510. 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. 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. 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.

[26]

J. F. Rodrigues, On the evolution Boussinesq-Stefan problem for non-Newtonian fluids,, in, 14 (2000), 390.

[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.

[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. 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, (1999), 457.

[30]

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

[31]

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

[32]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Studies in Mathematics and its Applications, (1977).

[33]

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

show all references

References:
[1]

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

[2]

C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems,", A Wiley-Interscience Publication, (1984).

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste Romnia, (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. 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.

[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.

[7]

H. Brézis, Inéquations variationnellers relatives à l'opérateur de Navier-Stokes,, J. Math. Anal. Appl., 39 (1972), 159. 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, (1973).

[9]

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

[10]

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

[11]

T. Fukao and N. Kenmochi, Variational inequality for the Navier-Stokes equations with time dependent constraint,, in, 34 (2011), 273.

[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, 86 (2009), 175. doi: 10.4064/bc86-0-11.

[14]

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

[15]

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

[16]

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

[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. 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), 1. doi: 10.1155/2007/79406.

[19]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Études Mathématiques, (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. 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.

[22]

U. Mosco, Convergence of convex set and of solutions of variational inequalities,, Adv. Math., 3 (1969), 510. 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. 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. 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.

[26]

J. F. Rodrigues, On the evolution Boussinesq-Stefan problem for non-Newtonian fluids,, in, 14 (2000), 390.

[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.

[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. 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, (1999), 457.

[30]

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

[31]

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

[32]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Studies in Mathematics and its Applications, (1977).

[33]

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

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