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Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions
1. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex |
2. | University "Al. I. Cuza" of Iasi, 700506 Iaşi, Romania |
References:
[1] |
S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1084-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system, Intern. J. Com. Math., 62 (1996), 209-221.
doi: 10.1080/00207169608804538. |
[3] |
T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. and Optimiz., 30 (2009), 199-213.
doi: 10.1080/01630560902841120. |
[4] |
T. Benincasa, A. Favini and C. Moroşanu, A Product Formula Approach to a Non-homogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System. PART I: A Phase-field Model, J. Optim. Theory and Appl., 148 (2011), 14-30.
doi: 10.1007/s10957-010-9742-x. |
[5] |
J. L. Boldrini, B. M. C. Caretta and E. Fernández-Cara, Analysis of a two-phase field model for the solidification of an alloy, J. Math. Anal. Appl., 357 (2009), 25-44.
doi: 10.1016/j.jmaa.2009.03.063. |
[6] |
G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. Jnl of Applied Mathematics, 9 (1998), 417-445.
doi: 10.1017/S0956792598003520. |
[7] |
L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 79 (2013), 12-27, arXiv:1206.6738
doi: 10.1016/j.na.2012.11.010. |
[8] |
C. Cavaterra, C. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal. TMA, 72 (2010), 2375-2399.
doi: 10.1016/j.na.2009.11.002. |
[9] |
L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase field system with dynamic boundary conditions and singular potentials, J. Math. Anal. Appl., 343 (2008), 557-566.
doi: 10.1016/j.jmaa.2008.01.077. |
[10] |
L. Cherfils, S. Gatti and A. Miranville, Long time behavior to the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290.
doi: 10.3934/cpaa.2012.11.2261. |
[11] |
M. Conti, S. Gatti and A. Miranville, Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 485-505.
doi: 10.3934/dcdss.2012.5.485. |
[12] |
M. Conti, S. Gatti and A. Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions, Anal. Appl. (Singap.), 11 (2013), 1350024, 31 pp. |
[13] |
C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in Internat. Ser. Numer. Math., 95, Birkhauser, Basel, (1990), 46-58. |
[14] |
I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon, Oxford, 1995. |
[15] |
C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.
doi: 10.3934/dcds.2008.22.1009. |
[16] |
C. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710.
doi: 10.3934/cpaa.2009.8.689. |
[17] |
C. Gal, M. Grasselli, A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535-556.
doi: 10.1007/s00030-008-7029-9. |
[18] |
C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions, Nonlinear phenomena with energy dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 29 (2008), 117-139. |
[19] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, Differential equations: inverse and direct problems, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 251 (2006), 149-170.
doi: 10.1201/9781420011135.ch9. |
[20] |
K.-H. Hoffman and L. Jiang, Optimal control problem of a phase field model for solidification, Numer. Funct. Anal. and Optimiz., 13 (1992), 11-27.
doi: 10.1080/01630569208816458. |
[21] |
Gh. Iorga, C. Moroşanu and I. Tofan, Numerical simulation of the thickness accretions in the secondary cooling zone of a continuous casting machine, Metalurgia International, XIV (2009), 72-75. |
[22] |
H. Israel, Long time behavior of an Allen-Cahn type equation with singular potential and dynamic boundary conditions, Journal of Applied Analysis and Computation, 2 (2012), 29-56. |
[23] |
N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal. TMA, 22 (1994), 1163-1180.
doi: 10.1016/0362-546X(94)90235-6. |
[24] |
O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type, Prov. Amer. Math. Soc., 1968. |
[25] |
J. L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985. |
[26] |
A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Model., 40 (2016), 192-207.
doi: 10.1016/j.apm.2015.04.039. |
[27] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[28] |
C. Moroşanu, Approximation of the phase-field transition system via fractional steps method, Numer. Funct. Anal. and Optimiz., 18 (1997), 623-648.
doi: 10.1080/01630569708816782. |
[29] |
C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers, 2012.
doi: 10.2174/97816080535061120101. |
[30] |
C. Moroşanu and D. Motreanu, A generalized phase field system, J. Math. Anal. Appl., 237 (1999), 515-540.
doi: 10.1006/jmaa.1999.6467. |
[31] |
C. Moroşanu and D. Motreanu, Uniqueness and approximation for the phase field equation in caginalp's model, Intern. J. of Appl. Math., 2 (2000), 113-129. |
[32] |
C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity, International Journal of Differential Equations and Applications, 1 (2000), 187-204. |
[33] |
O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions, Phys. D., 43 (1990), 44-62.
doi: 10.1016/0167-2789(90)90015-H. |
show all references
References:
[1] |
S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1084-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system, Intern. J. Com. Math., 62 (1996), 209-221.
doi: 10.1080/00207169608804538. |
[3] |
T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. and Optimiz., 30 (2009), 199-213.
doi: 10.1080/01630560902841120. |
[4] |
T. Benincasa, A. Favini and C. Moroşanu, A Product Formula Approach to a Non-homogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System. PART I: A Phase-field Model, J. Optim. Theory and Appl., 148 (2011), 14-30.
doi: 10.1007/s10957-010-9742-x. |
[5] |
J. L. Boldrini, B. M. C. Caretta and E. Fernández-Cara, Analysis of a two-phase field model for the solidification of an alloy, J. Math. Anal. Appl., 357 (2009), 25-44.
doi: 10.1016/j.jmaa.2009.03.063. |
[6] |
G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. Jnl of Applied Mathematics, 9 (1998), 417-445.
doi: 10.1017/S0956792598003520. |
[7] |
L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 79 (2013), 12-27, arXiv:1206.6738
doi: 10.1016/j.na.2012.11.010. |
[8] |
C. Cavaterra, C. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal. TMA, 72 (2010), 2375-2399.
doi: 10.1016/j.na.2009.11.002. |
[9] |
L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase field system with dynamic boundary conditions and singular potentials, J. Math. Anal. Appl., 343 (2008), 557-566.
doi: 10.1016/j.jmaa.2008.01.077. |
[10] |
L. Cherfils, S. Gatti and A. Miranville, Long time behavior to the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290.
doi: 10.3934/cpaa.2012.11.2261. |
[11] |
M. Conti, S. Gatti and A. Miranville, Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 485-505.
doi: 10.3934/dcdss.2012.5.485. |
[12] |
M. Conti, S. Gatti and A. Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions, Anal. Appl. (Singap.), 11 (2013), 1350024, 31 pp. |
[13] |
C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in Internat. Ser. Numer. Math., 95, Birkhauser, Basel, (1990), 46-58. |
[14] |
I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon, Oxford, 1995. |
[15] |
C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.
doi: 10.3934/dcds.2008.22.1009. |
[16] |
C. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710.
doi: 10.3934/cpaa.2009.8.689. |
[17] |
C. Gal, M. Grasselli, A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535-556.
doi: 10.1007/s00030-008-7029-9. |
[18] |
C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions, Nonlinear phenomena with energy dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 29 (2008), 117-139. |
[19] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, Differential equations: inverse and direct problems, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 251 (2006), 149-170.
doi: 10.1201/9781420011135.ch9. |
[20] |
K.-H. Hoffman and L. Jiang, Optimal control problem of a phase field model for solidification, Numer. Funct. Anal. and Optimiz., 13 (1992), 11-27.
doi: 10.1080/01630569208816458. |
[21] |
Gh. Iorga, C. Moroşanu and I. Tofan, Numerical simulation of the thickness accretions in the secondary cooling zone of a continuous casting machine, Metalurgia International, XIV (2009), 72-75. |
[22] |
H. Israel, Long time behavior of an Allen-Cahn type equation with singular potential and dynamic boundary conditions, Journal of Applied Analysis and Computation, 2 (2012), 29-56. |
[23] |
N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal. TMA, 22 (1994), 1163-1180.
doi: 10.1016/0362-546X(94)90235-6. |
[24] |
O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type, Prov. Amer. Math. Soc., 1968. |
[25] |
J. L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985. |
[26] |
A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Model., 40 (2016), 192-207.
doi: 10.1016/j.apm.2015.04.039. |
[27] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[28] |
C. Moroşanu, Approximation of the phase-field transition system via fractional steps method, Numer. Funct. Anal. and Optimiz., 18 (1997), 623-648.
doi: 10.1080/01630569708816782. |
[29] |
C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers, 2012.
doi: 10.2174/97816080535061120101. |
[30] |
C. Moroşanu and D. Motreanu, A generalized phase field system, J. Math. Anal. Appl., 237 (1999), 515-540.
doi: 10.1006/jmaa.1999.6467. |
[31] |
C. Moroşanu and D. Motreanu, Uniqueness and approximation for the phase field equation in caginalp's model, Intern. J. of Appl. Math., 2 (2000), 113-129. |
[32] |
C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity, International Journal of Differential Equations and Applications, 1 (2000), 187-204. |
[33] |
O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions, Phys. D., 43 (1990), 44-62.
doi: 10.1016/0167-2789(90)90015-H. |
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