# 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [14] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Clarendon, (1995). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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, (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. 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. 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. 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. 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. [32] C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity,, International Journal of Differential Equations and Applications, 1 (2000), 187. [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.

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. 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. 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. 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. 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. 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. 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, 79 (2013), 12. 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. 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. 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. 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. 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). [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. [14] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Clarendon, (1995). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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, (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. 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. 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. 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. 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. [32] C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity,, International Journal of Differential Equations and Applications, 1 (2000), 187. [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.
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