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June  2015, 35(6): 2539-2564. doi: 10.3934/dcds.2015.35.2539

Robust exponential attractors for the modified phase-field crystal equation

Maurizio Grasselli 1, and  Hao Wu 2,

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

Received  December 2013 Revised  April 2014 Published  December 2014

We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_\beta$ that are Hölder continuous with respect to $\beta$.
Keywords: Phase-field crystal equation, exponential attractors, Hölder continuity., regular attracting sets, robustness.
Mathematics Subject Classification: Primary: 35Q82, 37L25; Secondary: 74N05, 82C2.
Citation: Maurizio Grasselli, Hao Wu. Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2539-2564. doi: 10.3934/dcds.2015.35.2539
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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