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On singular Navier-Stokes equations and irreversible phase transitions

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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.
    Mathematics Subject Classification: Primary: 35K55, 35K67, 35Q30, 35R35; Secondary: 47H05, 80A22.


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  • [1]

    R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Academic Press, New York, 2003.


    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.


    V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.


    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.


    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.


    E. Bonetti, Global solution to a nonlinear phase transition model with dissipation, Adv. Math. Sci. Appl., 12 (2002), 355-376.


    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.


    H. Brezis, "Opératours Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North Holland, Amsterdan, 1973.


    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.


    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.


    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.


    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.


    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.


    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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