2018, 17(1): 1-19. doi: 10.3934/cpaa.2018001

Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system

1. 

Faculté des Sciences et Techniques, Université Marien Ngouabi, BP 69 Brazzaville, République du Congo, France

2. 

Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86962 Chasseneuil, France

* Corresponding author: M. Pierre

Received  June 2017 Revised  August 2017 Published  September 2017

Fund Project: This work has been partially supported by the Fédération MIRES

We consider a time semi-discretization of the Caginalp phase-field model based on an operator splitting method. For every time-step parameter $ \tau $, we build an exponential attractor $ \mathcal{M}_\tau $ of the discrete-in-time dynamical system. We prove that $ \mathcal{M}_\tau $ converges to an exponential attractor $\mathcal{M}_0$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as $ \tau $ tends to $0$. We also provide an explicit estimate of this distance and we prove that the fractal dimension of $ \mathcal{M}_\tau $ is bounded by a constant independent of $ \tau $.

Citation: Narcisse Batangouna, Morgan Pierre. Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis, 2018, 17 (1) : 1-19. doi: 10.3934/cpaa.2018001
References:
[1]

Y. B. Altundas and G. Caginalp, Velocity selection in 3D dendrites: phase field computations and microgravity experiments, Nonlinear Anal., 62 (2005), 467-481. doi: 10.1016/j.na.2005.02.122.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, vol. 25 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1992.

[3]

F. BaiC. M. ElliottA. GardinerA. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Ⅰ. Computations, Nonlinearity, 8 (1995), 131-160.

[4]

P. W. Bates and S. M. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dynam. Differential Equations, 4 (1992), 375-398. doi: 10.1007/BF01049391.

[5]

H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[6]

D. BrochetD. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.

[7]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, vol. 121 of Applied Mathematical Sciences Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[8]

G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods, Appl. Math. Lett., 2 (1989), 117-120. doi: 10.1016/0893-9659(89)90002-5.

[9]

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.

[10]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002.

[12]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.

[13]

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.

[14]

R. ChillE. Fašangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.

[15]

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762. doi: 10.2307/2004575.

[16]

C. DupaixD. Hilhorst and I. N. Kostin, The viscous Cahn-Hilliard equation as a limit of the phase field model: lower semicontinuity of the attractor, J. Dynam. Differential Equations, 11 (1999), 333-353. doi: 10.1023/A:1021985631123.

[17]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[18]

M. Efendiev and A. Miranville, The dimension of the global attractor for dissipative reaction-diffusion systems, Appl. Math. Lett., 16 (2003), 351-355. doi: 10.1016/S0893-9659(03)80056-3.

[19]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reactiondiffusion system in $ {\bf R}^3 $, C. R. Acad. Sci. Paris Sér. Ⅰ Math, 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[20]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[21]

C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663. doi: 10.1137/0730084.

[22]

D. Eyre, An unconditionnally stable one-step scheme for gradient systems, unpublished, (1998).

[23]

P. FabrieC. Galusinski and A. Miranville, Uniform inertial sets for damped wave equations, Discrete Contin. Dynam. Systems, 6 (2000), 393-418. doi: 10.3934/dcds.2000.6.393.

[24]

C. G. 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.

[25]

C. Galusinski, Perturbations singuliéres de problémes dissipatifs : étude dynamique via l'existence et la continuité d'attracteurs exponentiels, PhD thesis, Université de Bordeaux, 1996.

[26]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.

[27]

M. GrasselliH. 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.

[28]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[29]

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.

[30]

A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electron. J. Differential Equations, No. 63, 28.

[31]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[32]

M. Pierre, Convergence of exponential attractors for a time semi-discrete reaction-diffusion equation, submitted, https://hal.archives-ouvertes.fr/hal-01518790v1.

[33]

G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002,885-982. doi: 10.1016/S1874-575X(02)80038-8.

[34]

J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963.

[35]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1996.

[36]

R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. Ⅱ, Arch. Rational Mech. Anal., 33 (1969), 377-385. doi: 10.1007/BF00247696.

[37]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: time discretization, Math. Comp., 79 (2010), 259-280. doi: 10.1090/S0025-5718-09-02256-X.

[39]

X. Wang, Numerical algorithms for stationary statistical properties of dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 4599-4618. doi: 10.3934/dcds.2016.36.4599.

[40]

W. Xie, The classical Stefan model as the singular limit of phase field equations, Discrete Contin. Dynam. Systems, Added Volume Ⅱ (1998), 288-302, Dynamical systems and differential equations, Vol.

[41]

Y. Yan, Dimensions of attractors for discretizations for Navier-Stokes equations, J. Dynam. Differential Equations, 4 (1992), 275-340. doi: 10.1007/BF01049389.

[42]

Y. Yan, Attractors and dimensions for discretizations of a weakly damped Schrödinger equation and a sine-Gordon equation, Nonlinear Anal., 20 (1993), 1417-1452. doi: 10.1016/0362-546X(93)90168-R.

[43]

Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683.

[44]

C. Zhu, Attractor of a semi-discrete Benjamin-Bona-Mahony equation on $ \mathbb{R}^1 $, Ann. Polon. Math., 115 (2015), 219-234. doi: 10.4064/ap115-3-2.

show all references

References:
[1]

Y. B. Altundas and G. Caginalp, Velocity selection in 3D dendrites: phase field computations and microgravity experiments, Nonlinear Anal., 62 (2005), 467-481. doi: 10.1016/j.na.2005.02.122.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, vol. 25 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1992.

[3]

F. BaiC. M. ElliottA. GardinerA. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Ⅰ. Computations, Nonlinearity, 8 (1995), 131-160.

[4]

P. W. Bates and S. M. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dynam. Differential Equations, 4 (1992), 375-398. doi: 10.1007/BF01049391.

[5]

H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[6]

D. BrochetD. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.

[7]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, vol. 121 of Applied Mathematical Sciences Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[8]

G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods, Appl. Math. Lett., 2 (1989), 117-120. doi: 10.1016/0893-9659(89)90002-5.

[9]

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.

[10]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002.

[12]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.

[13]

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.

[14]

R. ChillE. Fašangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.

[15]

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762. doi: 10.2307/2004575.

[16]

C. DupaixD. Hilhorst and I. N. Kostin, The viscous Cahn-Hilliard equation as a limit of the phase field model: lower semicontinuity of the attractor, J. Dynam. Differential Equations, 11 (1999), 333-353. doi: 10.1023/A:1021985631123.

[17]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[18]

M. Efendiev and A. Miranville, The dimension of the global attractor for dissipative reaction-diffusion systems, Appl. Math. Lett., 16 (2003), 351-355. doi: 10.1016/S0893-9659(03)80056-3.

[19]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reactiondiffusion system in $ {\bf R}^3 $, C. R. Acad. Sci. Paris Sér. Ⅰ Math, 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[20]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[21]

C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663. doi: 10.1137/0730084.

[22]

D. Eyre, An unconditionnally stable one-step scheme for gradient systems, unpublished, (1998).

[23]

P. FabrieC. Galusinski and A. Miranville, Uniform inertial sets for damped wave equations, Discrete Contin. Dynam. Systems, 6 (2000), 393-418. doi: 10.3934/dcds.2000.6.393.

[24]

C. G. 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.

[25]

C. Galusinski, Perturbations singuliéres de problémes dissipatifs : étude dynamique via l'existence et la continuité d'attracteurs exponentiels, PhD thesis, Université de Bordeaux, 1996.

[26]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.

[27]

M. GrasselliH. 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.

[28]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[29]

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.

[30]

A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electron. J. Differential Equations, No. 63, 28.

[31]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[32]

M. Pierre, Convergence of exponential attractors for a time semi-discrete reaction-diffusion equation, submitted, https://hal.archives-ouvertes.fr/hal-01518790v1.

[33]

G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002,885-982. doi: 10.1016/S1874-575X(02)80038-8.

[34]

J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963.

[35]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1996.

[36]

R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. Ⅱ, Arch. Rational Mech. Anal., 33 (1969), 377-385. doi: 10.1007/BF00247696.

[37]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: time discretization, Math. Comp., 79 (2010), 259-280. doi: 10.1090/S0025-5718-09-02256-X.

[39]

X. Wang, Numerical algorithms for stationary statistical properties of dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 4599-4618. doi: 10.3934/dcds.2016.36.4599.

[40]

W. Xie, The classical Stefan model as the singular limit of phase field equations, Discrete Contin. Dynam. Systems, Added Volume Ⅱ (1998), 288-302, Dynamical systems and differential equations, Vol.

[41]

Y. Yan, Dimensions of attractors for discretizations for Navier-Stokes equations, J. Dynam. Differential Equations, 4 (1992), 275-340. doi: 10.1007/BF01049389.

[42]

Y. Yan, Attractors and dimensions for discretizations of a weakly damped Schrödinger equation and a sine-Gordon equation, Nonlinear Anal., 20 (1993), 1417-1452. doi: 10.1016/0362-546X(93)90168-R.

[43]

Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683.

[44]

C. Zhu, Attractor of a semi-discrete Benjamin-Bona-Mahony equation on $ \mathbb{R}^1 $, Ann. Polon. Math., 115 (2015), 219-234. doi: 10.4064/ap115-3-2.

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