On singular Navier-Stokes equations and irreversible phase transitions

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September 2012, 11(5): 2055-2078. doi: 10.3934/cpaa.2012.11.2055

On singular Navier-Stokes equations and irreversible phase transitions

José Luiz Boldrini ^1, , Luís H. de Miranda ^2, and Gabriela Planas ^3,

1.	Departamento de Matemática, Instituto de Matem, Brazil
2.	Departamento de Matem, Brazil
3.	Departamento de Matemática, IMECC - UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859 Campinas, SP

Received April 2010 Revised December 2011 Published March 2012

Abstract
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We analyze a singular system of partial differential equations corresponding to a model for the evolution of an irreversible solidification of certain pure materials by taking into account the effects of fluid flow in the molten regions. The model consists of a system of highly non-linear free-boundary parabolic equations and includes: a heat equation, a doubly nonlinear inclusion for the phase change and Navier-Stokes equations singularly perturbed by a Carman-Kozeny type term to take care of the flow in the mushy region and a Boussinesq term for the buoyancy forces due to thermal differences. Our approach to show existence of generalized solutions of this system involves time-discretization, a suitable regularization procedure and fixed point arguments.

Keywords: singular Navier Stokes equations, free-boundary problem., Irreversible phase transitions.

Mathematics Subject Classification: Primary: 35K55, 35K67, 35Q30, 35R35; Secondary: 47H05, 80A2.

Citation: José Luiz Boldrini, Luís H. de Miranda, Gabriela Planas. On singular Navier-Stokes equations and irreversible phase transitions. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2055-2078. doi: 10.3934/cpaa.2012.11.2055

References:

[1]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.
[2]	M. Aso, M. Frémond and N. Kenmochi, Parabolic systems with the unknown dependent constraints arising in phase transitions, "Free Boundary Problems: Theory and Applications" (eds. I. N. Figueiredo, J. N. Rodrigues and L. Santos), doi: 10.1007/978-3-7643-7719-9_5.
[3]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.
[4]	J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Continuous Dynam. Systems - A, 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429.
[5]	J. L. Boldrini and G. Planas, Some thoughts on mathematical modeling of solidification and melting, Boletín de la Sociedad Española de Matemática Aplicada, 41 (2007), 77-90.
[6]	E. Bonetti, Global solution to a nonlinear phase transition model with dissipation, Adv. Math. Sci. Appl., 12 (2002), 355-376.
[7]	G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.
[8]	H. Brezis, "Opératours Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North Holland, Amsterdan, 1973.
[9]	P. Colli, F. Luterotti, G. Schimperna and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA, Nonlinear Differ. Equ. Appl., 9 (2002), 255-276. doi: 10.1007/s00030-002-8127-8.
[10]	K-H. Hoffmann and L. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. and Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458.
[11]	P. Laurençot, G. Schimperma and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0.
[12]	F. Luterotti, G. Schimperma and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316.
[13]	G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068.
[14]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

show all references

References:

[1]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.
[2]	M. Aso, M. Frémond and N. Kenmochi, Parabolic systems with the unknown dependent constraints arising in phase transitions, "Free Boundary Problems: Theory and Applications" (eds. I. N. Figueiredo, J. N. Rodrigues and L. Santos), doi: 10.1007/978-3-7643-7719-9_5.
[3]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.
[4]	J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Continuous Dynam. Systems - A, 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429.
[5]	J. L. Boldrini and G. Planas, Some thoughts on mathematical modeling of solidification and melting, Boletín de la Sociedad Española de Matemática Aplicada, 41 (2007), 77-90.
[6]	E. Bonetti, Global solution to a nonlinear phase transition model with dissipation, Adv. Math. Sci. Appl., 12 (2002), 355-376.
[7]	G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.
[8]	H. Brezis, "Opératours Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North Holland, Amsterdan, 1973.
[9]	P. Colli, F. Luterotti, G. Schimperna and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA, Nonlinear Differ. Equ. Appl., 9 (2002), 255-276. doi: 10.1007/s00030-002-8127-8.
[10]	K-H. Hoffmann and L. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. and Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458.
[11]	P. Laurençot, G. Schimperma and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0.
[12]	F. Luterotti, G. Schimperma and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316.
[13]	G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068.
[14]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

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