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November  2019, 18(6): 3161-3179. doi: 10.3934/cpaa.2019142

## Time discretization of a nonlinear phase field system in general domains

 1 Dipartimento di Matematica "F. Casorati", Università di Pavia, Research Associate at the IMATI – C.N.R. Pavia, Via Ferrata 5, 27100 Pavia, Italy 2 Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

* Corresponding author

Received  November 2018 Revised  January 2019 Published  May 2019

This paper deals with the nonlinear phase field system
 $\begin{equation*} \begin{cases} \partial_t (\theta +\ell \varphi) - \Delta\theta = f & \mbox{in}\ \Omega\times(0, T), \\ \partial_t \varphi - \Delta\varphi + \xi + \pi(\varphi) = \ell \theta,\ \xi\in\beta(\varphi) & \mbox{in}\ \Omega\times(0, T) \end{cases} \end{equation*}$
in a general domain
 $\Omega\subseteq\mathbb{R}^{d}$
. Here
 $d \in \mathbb{N}$
,
 $T>0$
,
 $\ell>0$
,
 $f$
is a source term,
 $\beta$
is a maximal monotone graph and
 $\pi$
is a Lipschitz continuous function. We note that in the above system the nonlinearity
 $\beta+\pi$
replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that
 $\Omega$
is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step
 $h$
goes to
 $0$
. In the limit procedure we face with the difficulty that the embedding
 $H^1(\Omega) \hookrightarrow L^2(\Omega)$
is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order
 $h^{1/2}$
for the difference between continuous and discrete solutions.
Citation: Pierluigi Colli, Shunsuke Kurima. Time discretization of a nonlinear phase field system in general domains. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3161-3179. doi: 10.3934/cpaa.2019142
##### References:
 [1] T. G. Amler, N. D. Botkin, K.-H. Hoffmann and K. A. Ruf, Regularity of solutions of a phase field model, Dyn. Partial Differ. Equ., 10 (2013), 353-365.  doi: 10.4310/DPDE.2013.v10.n4.a3. [2] B. D. Bangola, Global and exponential attractors for a Caginalp type phase-field problem, Cent. Eur. J. Math., 11 (2013), 1651-1676.  doi: 10.2478/s11533-013-0258-0. [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [5] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133.  doi: 10.1137/15M102424X. [6] S. Benzoni-Gavage, L. Chupin, D. Jamet and J. Vovelle, On a phase field model for solid-liquid phase transitions, Discrete Contin. Dyn. Syst., 32 (2012), 1997-2025.  doi: 10.3934/dcds.2012.32.1997. [7] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [8] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827. [9] G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534.  doi: 10.1137/070680965. [10] G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162. [11] G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982.  doi: 10.3934/cpaa.2012.11.1959. [12] O. Cârjă, A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Anal., 113 (2015), 190-208.  doi: 10.1016/j.na.2014.10.003. [13] X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Ration. Mech. Anal., 202 (2011), 349-372.  doi: 10.1007/s00205-011-0429-8. [14] L. Cherfils, S. Gatti and A. Miranville, Long time behavior of 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. [15] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6. [16] P. Colli, G. Gilardi and G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl., 434 (2016), 432-463.  doi: 10.1016/j.jmaa.2015.09.011. [17] P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95. [18] P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81.  doi: 10.3934/dcds.2009.25.63. [19] M. Conti, S. Gatti, A. Miranville and R. Quintanilla, On a Caginalp phase-field system with two temperatures and memory, Milan J. Math., 85 (2017), 1-27.  doi: 10.1007/s00032-017-0263-z. [20] 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. doi: 10.1142/S0219530513500243. [21] G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sr. A-B, 276 (1973), A1461–A1463. [22] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations, in "Free Boundary Problems", Internat. Ser. Numer. Math., 95, 46-58, Birkhäuser Verlag, Basel, (1990). [23] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [24] T. Fukao, S. Kurima and T. Yokota, Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems on unbounded domains via Cauchy's criterion, Math. Methods Appl. Sci., 41 (2018), 2590-2601.  doi: 10.1002/mma.4760. [25] M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.  doi: 10.4171/ZAA/1277. [26] J.W. Jerome, Approximation of Nonlinear Evolution Systems, Mathematics in Science and Engineering 164, Academic Press Inc., Orlando, 1983. [27] K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13 (1992), 11-27.  doi: 10.1080/01630569208816458. [28] K.-H. Hoffmann, N. Kenmochi, M. Kubo and N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl., 17 (2007), 305-336. [29] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., 22 (1994), 1163-1180.  doi: 10.1016/0362-546X(94)90235-6. [30] S. Kurima, Existence and energy estimates of weak solutions for nonlocal Cahn–Hilliard equations on unbounded domains, preprint, arXiv: 1806.06361, (2018). [31] S. Kurima, Asymptotic analysis for Cahn–Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci, to appear. [32] S. Kurima and T. Yokota, A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operators, Adv. Math. Sci. Appl., 26 (2017), 221-242. [33] S. Kurima and T. Yokota, Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates, J. Differential Equations, 263 (2017), 2024-2050.  doi: 10.1016/j.jde.2017.03.040. [34] A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 271-306.  doi: 10.3934/dcdss.2014.7.271. [35] 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. [36] A. Miranville and A. J. Ntsokongo, On anisotropic Caginalp phase-field type models with singular nonlinear terms, J. Appl. Anal. Comput., 8 (2018), 655-674. [37] A. Miranville and R. Quintanilla, Spatial decay for several phase-field models, ZAMM Z. Angew. Math. Mech., 93 (2013), 801-810.  doi: 10.1002/zamm.201200131. [38] A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.  doi: 10.1007/s00245-010-9114-9. [39] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.  doi: 10.1080/00036810903042182. [40] T. Miyasita, Global existence and exponential attractor of solutions of Fix-Caginalp equation, Sci. Math. Jpn., 77 (2015), 339-355. [41] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.  doi: 10.1051/cocv:2006013. [42] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360. [43] A. Visintin, Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4078-5.

show all references

##### References:
 [1] T. G. Amler, N. D. Botkin, K.-H. Hoffmann and K. A. Ruf, Regularity of solutions of a phase field model, Dyn. Partial Differ. Equ., 10 (2013), 353-365.  doi: 10.4310/DPDE.2013.v10.n4.a3. [2] B. D. Bangola, Global and exponential attractors for a Caginalp type phase-field problem, Cent. Eur. J. Math., 11 (2013), 1651-1676.  doi: 10.2478/s11533-013-0258-0. [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [5] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133.  doi: 10.1137/15M102424X. [6] S. Benzoni-Gavage, L. Chupin, D. Jamet and J. Vovelle, On a phase field model for solid-liquid phase transitions, Discrete Contin. Dyn. Syst., 32 (2012), 1997-2025.  doi: 10.3934/dcds.2012.32.1997. [7] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [8] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827. [9] G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534.  doi: 10.1137/070680965. [10] G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162. [11] G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982.  doi: 10.3934/cpaa.2012.11.1959. [12] O. Cârjă, A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Anal., 113 (2015), 190-208.  doi: 10.1016/j.na.2014.10.003. [13] X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Ration. Mech. Anal., 202 (2011), 349-372.  doi: 10.1007/s00205-011-0429-8. [14] L. Cherfils, S. Gatti and A. Miranville, Long time behavior of 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. [15] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6. [16] P. Colli, G. Gilardi and G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl., 434 (2016), 432-463.  doi: 10.1016/j.jmaa.2015.09.011. [17] P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95. [18] P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81.  doi: 10.3934/dcds.2009.25.63. [19] M. Conti, S. Gatti, A. Miranville and R. Quintanilla, On a Caginalp phase-field system with two temperatures and memory, Milan J. Math., 85 (2017), 1-27.  doi: 10.1007/s00032-017-0263-z. [20] 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. doi: 10.1142/S0219530513500243. [21] G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sr. A-B, 276 (1973), A1461–A1463. [22] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations, in "Free Boundary Problems", Internat. Ser. Numer. Math., 95, 46-58, Birkhäuser Verlag, Basel, (1990). [23] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [24] T. Fukao, S. Kurima and T. Yokota, Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems on unbounded domains via Cauchy's criterion, Math. Methods Appl. Sci., 41 (2018), 2590-2601.  doi: 10.1002/mma.4760. [25] M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.  doi: 10.4171/ZAA/1277. [26] J.W. Jerome, Approximation of Nonlinear Evolution Systems, Mathematics in Science and Engineering 164, Academic Press Inc., Orlando, 1983. [27] K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13 (1992), 11-27.  doi: 10.1080/01630569208816458. [28] K.-H. Hoffmann, N. Kenmochi, M. Kubo and N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl., 17 (2007), 305-336. [29] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., 22 (1994), 1163-1180.  doi: 10.1016/0362-546X(94)90235-6. [30] S. Kurima, Existence and energy estimates of weak solutions for nonlocal Cahn–Hilliard equations on unbounded domains, preprint, arXiv: 1806.06361, (2018). [31] S. Kurima, Asymptotic analysis for Cahn–Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci, to appear. [32] S. Kurima and T. Yokota, A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operators, Adv. Math. Sci. Appl., 26 (2017), 221-242. [33] S. Kurima and T. Yokota, Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates, J. Differential Equations, 263 (2017), 2024-2050.  doi: 10.1016/j.jde.2017.03.040. [34] A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 271-306.  doi: 10.3934/dcdss.2014.7.271. [35] 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. [36] A. Miranville and A. J. Ntsokongo, On anisotropic Caginalp phase-field type models with singular nonlinear terms, J. Appl. Anal. Comput., 8 (2018), 655-674. [37] A. Miranville and R. Quintanilla, Spatial decay for several phase-field models, ZAMM Z. Angew. Math. Mech., 93 (2013), 801-810.  doi: 10.1002/zamm.201200131. [38] A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.  doi: 10.1007/s00245-010-9114-9. [39] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.  doi: 10.1080/00036810903042182. [40] T. Miyasita, Global existence and exponential attractor of solutions of Fix-Caginalp equation, Sci. Math. Jpn., 77 (2015), 339-355. [41] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.  doi: 10.1051/cocv:2006013. [42] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  doi: 10.1007/BF01762360. [43] A. Visintin, Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4078-5.
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