# American Institute of Mathematical Sciences

April  2016, 9(2): 537-556. doi: 10.3934/dcdss.2016011

## 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 Iasi, 700506 Iaşi, Romania

Received  August 2014 Revised  November 2014 Published  March 2016

The paper concerns with the existence, uniqueness, regularity and the approximation of solutions to the nonlinear phase-field (Allen-Cahn) equation, endowed with non-homogeneous dynamic boundary conditions (depending both on time and space variables). It extends the already studied types of boundary conditions, which makes the problem to be more able to describe many important phenomena of two-phase systems, in particular, the interactions with the walls in confined systems. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual numerical algorithm is formulated in the end.
Citation: Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011
##### 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. doi: 10.1016/0001-6160(79)90196-2. Google Scholar [2] V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system,, Intern. J. Com. Math., 62 (1996), 209. doi: 10.1080/00207169608804538. Google Scholar [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. doi: 10.1080/01630560902841120. Google Scholar [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. doi: 10.1007/s10957-010-9742-x. Google Scholar [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. doi: 10.1016/j.jmaa.2009.03.063. Google Scholar [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. doi: 10.1017/S0956792598003520. Google Scholar [7] L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions,, Nonlinear Analysis: Theory, 79 (2013), 12. doi: 10.1016/j.na.2012.11.010. Google Scholar [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. doi: 10.1016/j.na.2009.11.002. Google Scholar [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. doi: 10.1016/j.jmaa.2008.01.077. Google Scholar [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. doi: 10.3934/cpaa.2012.11.2261. Google Scholar [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. doi: 10.3934/dcdss.2012.5.485. Google Scholar [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). Google Scholar [13] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in Internat. Ser. Numer. Math., 95 (1990), 46. Google Scholar [14] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Clarendon, (1995). Google Scholar [15] C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions,, Discrete Contin. Dyn. Syst., 22 (2008), 1009. doi: 10.3934/dcds.2008.22.1009. Google Scholar [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. doi: 10.3934/cpaa.2009.8.689. Google Scholar [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. doi: 10.1007/s00030-008-7029-9. Google Scholar [18] C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions,, Nonlinear phenomena with energy dissipation, 29 (2008), 117. Google Scholar [19] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, Differential equations: inverse and direct problems, 251 (2006), 149. doi: 10.1201/9781420011135.ch9. Google Scholar [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. doi: 10.1080/01630569208816458. Google Scholar [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. Google Scholar [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. Google Scholar [23] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems,, Nonlinear Anal. TMA, 22 (1994), 1163. doi: 10.1016/0362-546X(94)90235-6. Google Scholar [24] O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type,, Prov. Amer. Math. Soc., (1968). Google Scholar [25] J. L. Lions, Control of Distributed Singular Systems,, Gauthier-Villars, (1985). Google Scholar [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. doi: 10.1016/j.apm.2015.04.039. Google Scholar [27] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709. doi: 10.1002/mma.590. Google Scholar [28] C. Moroşanu, Approximation of the phase-field transition system via fractional steps method,, Numer. Funct. Anal. and Optimiz., 18 (1997), 623. doi: 10.1080/01630569708816782. Google Scholar [29] C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods,, Bentham Science Publishers, (2012). doi: 10.2174/97816080535061120101. Google Scholar [30] C. Moroşanu and D. Motreanu, A generalized phase field system,, J. Math. Anal. Appl., 237 (1999), 515. doi: 10.1006/jmaa.1999.6467. Google Scholar [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. Google Scholar [32] C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity,, International Journal of Differential Equations and Applications, 1 (2000), 187. Google Scholar [33] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions,, Phys. D., 43 (1990), 44. doi: 10.1016/0167-2789(90)90015-H. Google Scholar

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##### 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. doi: 10.1016/0001-6160(79)90196-2. Google Scholar [2] V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system,, Intern. J. Com. Math., 62 (1996), 209. doi: 10.1080/00207169608804538. Google Scholar [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. doi: 10.1080/01630560902841120. Google Scholar [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. doi: 10.1007/s10957-010-9742-x. Google Scholar [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. doi: 10.1016/j.jmaa.2009.03.063. Google Scholar [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. doi: 10.1017/S0956792598003520. Google Scholar [7] L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions,, Nonlinear Analysis: Theory, 79 (2013), 12. doi: 10.1016/j.na.2012.11.010. Google Scholar [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. doi: 10.1016/j.na.2009.11.002. Google Scholar [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. doi: 10.1016/j.jmaa.2008.01.077. Google Scholar [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. doi: 10.3934/cpaa.2012.11.2261. Google Scholar [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. doi: 10.3934/dcdss.2012.5.485. Google Scholar [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). Google Scholar [13] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in Internat. Ser. Numer. Math., 95 (1990), 46. Google Scholar [14] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Clarendon, (1995). Google Scholar [15] C. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions,, Discrete Contin. Dyn. Syst., 22 (2008), 1009. doi: 10.3934/dcds.2008.22.1009. Google Scholar [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. doi: 10.3934/cpaa.2009.8.689. Google Scholar [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. doi: 10.1007/s00030-008-7029-9. Google Scholar [18] C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions,, Nonlinear phenomena with energy dissipation, 29 (2008), 117. Google Scholar [19] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, Differential equations: inverse and direct problems, 251 (2006), 149. doi: 10.1201/9781420011135.ch9. Google Scholar [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. doi: 10.1080/01630569208816458. Google Scholar [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. Google Scholar [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. Google Scholar [23] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems,, Nonlinear Anal. TMA, 22 (1994), 1163. doi: 10.1016/0362-546X(94)90235-6. Google Scholar [24] O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type,, Prov. Amer. Math. Soc., (1968). Google Scholar [25] J. L. Lions, Control of Distributed Singular Systems,, Gauthier-Villars, (1985). Google Scholar [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. doi: 10.1016/j.apm.2015.04.039. Google Scholar [27] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709. doi: 10.1002/mma.590. Google Scholar [28] C. Moroşanu, Approximation of the phase-field transition system via fractional steps method,, Numer. Funct. Anal. and Optimiz., 18 (1997), 623. doi: 10.1080/01630569708816782. Google Scholar [29] C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods,, Bentham Science Publishers, (2012). doi: 10.2174/97816080535061120101. Google Scholar [30] C. Moroşanu and D. Motreanu, A generalized phase field system,, J. Math. Anal. Appl., 237 (1999), 515. doi: 10.1006/jmaa.1999.6467. Google Scholar [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. Google Scholar [32] C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity,, International Journal of Differential Equations and Applications, 1 (2000), 187. Google Scholar [33] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions,, Phys. D., 43 (1990), 44. doi: 10.1016/0167-2789(90)90015-H. Google Scholar
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