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On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition
Analysis and optimal control of some solidification processes
1. | Departamento de Ciencias Básicas, Universidad del Bío-Bío, Campus Fernando May, Casilla 447, Chillán, Chile |
2. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain |
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. |
[2] |
W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194.
doi: 10.1007/BF02648953. |
[3] |
J. Ni and C. Beckermann, A volume-averaged two-phase model for solidication transportphenomena, Metallurgical Transactions B, 22 (1991), 349-361. |
[4] |
W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems: I. Model Formulation, Int. J. Heat Mass Transfer, 30 (1987), 2161-2170.
doi: 10.1016/0017-9310(87)90094-9. |
[5] |
W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid Liquid Phase Change Systems: II. Application to Solidification in a Rectangular Cavity, Int. J. Heat Mass Transfer, 30 (1987), 2171-2187.
doi: 10.1016/0017-9310(87)90095-0. |
[6] |
Ph. Blanc, L. Gasser and J. Rappaz, Existence for a stationary model of binary alloy solidification, RAIRO Modél. Math. Anal. Numér., 29 (1995), 687-699. |
[7] |
J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Contin. Dyn. Syst., 13 (2005), 429-450.
doi: 10.3934/dcds.2005.13.429. |
[8] |
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. |
[9] |
M. Durán, E. Ortega-Torres and J. Rappaz, Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes, Math. Comput. Modelling, 42 (2005), 1269-1286.
doi: 10.1016/j.mcm.2005.01.035. |
[10] |
M. Durán and E. Ortega-Torres, Existence of weak solutions for a non-homogeneous solidification model, Electron. J. Differential Equations, 2006 17 pp. |
[11] |
I. Ekeland and R. Temam, Convex analysis and Variational Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1999.
doi: 10.1137/1.9781611971088. |
[12] |
J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595. |
[13] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967. |
[14] |
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. |
[15] |
J. C. Ramirez and C. Beckermann, Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53 (2005), 1721-1736.
doi: 10.1016/j.actamat.2004.12.021. |
[16] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differ. Equations, 66 (1987), 118-139.
doi: 10.1016/0022-0396(87)90043-X. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 64-96.
doi: 10.1007/BF01762360. |
[18] |
J. Simon, A constructive proof of a theorem of G. de Rham, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172. |
[19] |
R. Temam, Navier Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. |
[2] |
W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194.
doi: 10.1007/BF02648953. |
[3] |
J. Ni and C. Beckermann, A volume-averaged two-phase model for solidication transportphenomena, Metallurgical Transactions B, 22 (1991), 349-361. |
[4] |
W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems: I. Model Formulation, Int. J. Heat Mass Transfer, 30 (1987), 2161-2170.
doi: 10.1016/0017-9310(87)90094-9. |
[5] |
W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid Liquid Phase Change Systems: II. Application to Solidification in a Rectangular Cavity, Int. J. Heat Mass Transfer, 30 (1987), 2171-2187.
doi: 10.1016/0017-9310(87)90095-0. |
[6] |
Ph. Blanc, L. Gasser and J. Rappaz, Existence for a stationary model of binary alloy solidification, RAIRO Modél. Math. Anal. Numér., 29 (1995), 687-699. |
[7] |
J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Contin. Dyn. Syst., 13 (2005), 429-450.
doi: 10.3934/dcds.2005.13.429. |
[8] |
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. |
[9] |
M. Durán, E. Ortega-Torres and J. Rappaz, Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes, Math. Comput. Modelling, 42 (2005), 1269-1286.
doi: 10.1016/j.mcm.2005.01.035. |
[10] |
M. Durán and E. Ortega-Torres, Existence of weak solutions for a non-homogeneous solidification model, Electron. J. Differential Equations, 2006 17 pp. |
[11] |
I. Ekeland and R. Temam, Convex analysis and Variational Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1999.
doi: 10.1137/1.9781611971088. |
[12] |
J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595. |
[13] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967. |
[14] |
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. |
[15] |
J. C. Ramirez and C. Beckermann, Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53 (2005), 1721-1736.
doi: 10.1016/j.actamat.2004.12.021. |
[16] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differ. Equations, 66 (1987), 118-139.
doi: 10.1016/0022-0396(87)90043-X. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 64-96.
doi: 10.1007/BF01762360. |
[18] |
J. Simon, A constructive proof of a theorem of G. de Rham, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172. |
[19] |
R. Temam, Navier Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
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