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Robust exponential attractors for the modified phase-field crystal equation
1. | Dipartimento di Matematica, Politecnico di Milano, 20133 Milano |
2. | School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, 200433 Shanghai |
References:
[1] |
R. Backofen, A. Rätz and A. Voigt, Nucleation and growth by a phase field crystal (PFC) model, Phil. Mag. Lett., 87 (2007), 813-820.
doi: 10.1080/09500830701481737. |
[2] |
A. Baskaran, Z. Hu, J. S. Lowengrub, C. Wang, S. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear-multigrid schemes for the modified phase-field crystal equation, J. Comput. Phys., 250 (2013) 270-292.
doi: 10.1016/j.jcp.2013.04.024. |
[3] |
A. Baskaran, J. S. Lowengrub, C. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873.
doi: 10.1137/120880677. |
[4] |
M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227 (2008), 6241-6248.
doi: 10.1016/j.jcp.2008.03.012. |
[5] |
M. Cheng, J. Kundin, D. Li and H. Emmerich, Thermodynamic consistency and fast dynamics in phase-field crystal modeling, Phil. Mag. Lett., 92 (2012), 517-526.
doi: 10.1080/09500839.2012.691215. |
[6] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701.
doi: 10.1103/PhysRevLett.88.245701. |
[7] |
K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase-field crystal, Phys. Rev. E, 70 (2004), 051605.
doi: 10.1103/PhysRevE.70.051605. |
[8] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[9] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[10] |
H. Emmerich, L. Gránásy and H. Löwen, Selected issues of phase-field crystal simulations, Eur. Phys. J. Plus, 126 (2011), p102.
doi: 10.1140/epjp/i2011-11102-1. |
[11] |
H. Emmerich, H. Löwen, R. Wittkowskib, T. Gruhn, G. I. Tóth, G. Tegze and L. Gránásy, Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview, Adv. Phys., 61 (2012), 665-743.
doi: 10.1080/00018732.2012.737555. |
[12] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[13] |
P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125.
doi: 10.1103/PhysRevE.71.046125. |
[14] |
P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110, 11pp.
doi: 10.1103/PhysRevE.79.051110. |
[15] |
P. Galenko and K. Elder, Marginal stability analysis of the phase field crystal model in one spatial dimension, Phys. Rev. B, 83 (2011), 064113.
doi: 10.1103/PhysRevB.83.064113. |
[16] |
P. Galenko, H. Gomez, N. V. Kropotin and K. R. Elder, Unconditionally stable method and numerical solution of the hyperbolic (modified) phase-field crystal equation, Phys. Rev. E, 88 (2013), 013310. |
[17] |
H. Gomez and X. Nogueira, An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 52-61.
doi: 10.1016/j.cma.2012.03.002. |
[18] |
M. Grasselli and H. Wu, Well-posedness and long-time behavior for the modified phase-field crystal equation, Math. Models Methods Appl. Sci., 24 (2014), 2743-2783.
doi: 10.1142/S0218202514500365. |
[19] |
Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.
doi: 10.1016/j.jcp.2009.04.020. |
[20] |
A. Miranville, V. Pata and S. Zelik, Exponential attractors for singularly perturbed damped wave equations: a simple construction, Asymptot. Anal., 53 (2007), 1-12. |
[21] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of differential equations: evolutionary equations, IV, Elsevier/North-Holland, Amsterdam, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[22] |
H. Ohnogi and Y. Shiwa, Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation, Phys. D, 237 (2008), 3046-3052.
doi: 10.1016/j.physd.2008.06.011. |
[23] |
N. Provatas, J. A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution, Journal of the Minerals, Metals and Materials Society, 59 (2007), 83-90.
doi: 10.1007/s11837-007-0095-3. |
[24] |
P. Stefanovic, M. Haataja and N. Provatas, Phase-field crystals with elastic interactions, Phys. Rev. Lett., 96 (2006), 225504.
doi: 10.1103/PhysRevLett.96.225504. |
[25] |
P. Stefanovic, M. Haataja and N. Provatas, Phase-field crystal study of deformation and plasticity in nanocrystalline materials, Phys. Rev. E, 80 (2009), 046107.
doi: 10.1103/PhysRevE.80.046107. |
[26] |
C. Wang and S. M. Wise, Global smooth solutions of the three dimensional modified phase field crystal equation, Methods Appl. Anal., 17 (2010), 191-211.
doi: 10.4310/MAA.2010.v17.n2.a4. |
[27] |
C. Wang and S. M. Wise, An energy stable and convergent finite difference scheme for the modified phase-field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.
doi: 10.1137/090752675. |
[28] |
S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase-field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288.
doi: 10.1137/080738143. |
show all references
References:
[1] |
R. Backofen, A. Rätz and A. Voigt, Nucleation and growth by a phase field crystal (PFC) model, Phil. Mag. Lett., 87 (2007), 813-820.
doi: 10.1080/09500830701481737. |
[2] |
A. Baskaran, Z. Hu, J. S. Lowengrub, C. Wang, S. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear-multigrid schemes for the modified phase-field crystal equation, J. Comput. Phys., 250 (2013) 270-292.
doi: 10.1016/j.jcp.2013.04.024. |
[3] |
A. Baskaran, J. S. Lowengrub, C. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873.
doi: 10.1137/120880677. |
[4] |
M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227 (2008), 6241-6248.
doi: 10.1016/j.jcp.2008.03.012. |
[5] |
M. Cheng, J. Kundin, D. Li and H. Emmerich, Thermodynamic consistency and fast dynamics in phase-field crystal modeling, Phil. Mag. Lett., 92 (2012), 517-526.
doi: 10.1080/09500839.2012.691215. |
[6] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701.
doi: 10.1103/PhysRevLett.88.245701. |
[7] |
K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase-field crystal, Phys. Rev. E, 70 (2004), 051605.
doi: 10.1103/PhysRevE.70.051605. |
[8] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[9] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[10] |
H. Emmerich, L. Gránásy and H. Löwen, Selected issues of phase-field crystal simulations, Eur. Phys. J. Plus, 126 (2011), p102.
doi: 10.1140/epjp/i2011-11102-1. |
[11] |
H. Emmerich, H. Löwen, R. Wittkowskib, T. Gruhn, G. I. Tóth, G. Tegze and L. Gránásy, Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview, Adv. Phys., 61 (2012), 665-743.
doi: 10.1080/00018732.2012.737555. |
[12] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[13] |
P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125.
doi: 10.1103/PhysRevE.71.046125. |
[14] |
P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110, 11pp.
doi: 10.1103/PhysRevE.79.051110. |
[15] |
P. Galenko and K. Elder, Marginal stability analysis of the phase field crystal model in one spatial dimension, Phys. Rev. B, 83 (2011), 064113.
doi: 10.1103/PhysRevB.83.064113. |
[16] |
P. Galenko, H. Gomez, N. V. Kropotin and K. R. Elder, Unconditionally stable method and numerical solution of the hyperbolic (modified) phase-field crystal equation, Phys. Rev. E, 88 (2013), 013310. |
[17] |
H. Gomez and X. Nogueira, An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 52-61.
doi: 10.1016/j.cma.2012.03.002. |
[18] |
M. Grasselli and H. Wu, Well-posedness and long-time behavior for the modified phase-field crystal equation, Math. Models Methods Appl. Sci., 24 (2014), 2743-2783.
doi: 10.1142/S0218202514500365. |
[19] |
Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.
doi: 10.1016/j.jcp.2009.04.020. |
[20] |
A. Miranville, V. Pata and S. Zelik, Exponential attractors for singularly perturbed damped wave equations: a simple construction, Asymptot. Anal., 53 (2007), 1-12. |
[21] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of differential equations: evolutionary equations, IV, Elsevier/North-Holland, Amsterdam, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[22] |
H. Ohnogi and Y. Shiwa, Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation, Phys. D, 237 (2008), 3046-3052.
doi: 10.1016/j.physd.2008.06.011. |
[23] |
N. Provatas, J. A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution, Journal of the Minerals, Metals and Materials Society, 59 (2007), 83-90.
doi: 10.1007/s11837-007-0095-3. |
[24] |
P. Stefanovic, M. Haataja and N. Provatas, Phase-field crystals with elastic interactions, Phys. Rev. Lett., 96 (2006), 225504.
doi: 10.1103/PhysRevLett.96.225504. |
[25] |
P. Stefanovic, M. Haataja and N. Provatas, Phase-field crystal study of deformation and plasticity in nanocrystalline materials, Phys. Rev. E, 80 (2009), 046107.
doi: 10.1103/PhysRevE.80.046107. |
[26] |
C. Wang and S. M. Wise, Global smooth solutions of the three dimensional modified phase field crystal equation, Methods Appl. Anal., 17 (2010), 191-211.
doi: 10.4310/MAA.2010.v17.n2.a4. |
[27] |
C. Wang and S. M. Wise, An energy stable and convergent finite difference scheme for the modified phase-field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.
doi: 10.1137/090752675. |
[28] |
S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase-field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288.
doi: 10.1137/080738143. |
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