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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 |
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 $.
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. Bai, C. M. Elliott, A. Gardiner, A. 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. Brochet, D. 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. Chill, E. 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. Dupaix, D. 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. Efendiev, A. 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. Efendiev, A. 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. Fabrie, C. 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. Grasselli, A. 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. 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. |
[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. Bai, C. M. Elliott, A. Gardiner, A. 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. Brochet, D. 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. Chill, E. 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. Dupaix, D. 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. Efendiev, A. 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. Efendiev, A. 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. Fabrie, C. 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. Grasselli, A. 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. 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. |
[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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