HIGHER-ORDER MODELS IN PHASE SEPARATION

Laurence Cherfils; Alain Miranville; Shuiran Peng

doi:10.11948/2017003

2017 Volume 7 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(1): 39-56

Next Article Previous Article

Laurence Cherfils, Alain Miranville, Shuiran Peng. HIGHER-ORDER MODELS IN PHASE SEPARATION[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 39-56. doi: 10.11948/2017003

Citation:

Laurence Cherfils, Alain Miranville, Shuiran Peng. HIGHER-ORDER MODELS IN PHASE SEPARATION[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 39-56. doi: 10.11948/2017003

HIGHER-ORDER MODELS IN PHASE SEPARATION

1 Université de La Rochelle, Laboratoire Mathématiques, Image et Applications, Avenue Michel Crépeau, F-17042 La Rochelle Cedex, France;
2 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2 MI, Boulevard Marie et Pierre Curie-Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

Abstract

Abstract

Our aim in this paper is to study higher-order (in space) AllenCahn and Cahn-Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor.
- Allen-Cahn model /
- Cahn-Hilliard model /
- higher-order models /
- well-posedness /
- dissipativity /
- global attractor
MSC: 35K55;35J60;80A22

FullText(HTML)

References

[1]	S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27(1979), 1085-1095. Google Scholar
[2]	J. Berry, K.R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids:freezing and glass formation, Phys. Rev. E, 77(2008), 061506. Google Scholar
[3]	J. Berry, M. Grant and K.R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73(2006), 031609. Google Scholar
[4]	G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete Contin. Dyn. Systems S, 4(2011), 311-350. Google Scholar
[5]	J.W. Cahn, On spinodal decomposition, Acta Metall., 9(1961), 795-801. Google Scholar
[6]	J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 2(1958), 258-267. Google Scholar
[7]	F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 13(2013), 1189-1208. Google Scholar
[8]	L. Cherfils, S. Gatti and A. Miranville, A variational approach to a CahnHilliard model in a domain with nonpermeable walls, J. Math. Sci., 189(2013), 604-636. Google Scholar
[9]	L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math, 79(2011), 561-596. Google Scholar
[10]	P.G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72(1980), 4756-4763. Google Scholar
[11]	H. Emmerich, H. Löwen, R. Wittkowski, 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. Google Scholar
[12]	P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and SwiftHohenberg equations with fast dynamics, Phys. Rev. E, 79(2009), 051110. Google Scholar
[13]	G. Giacomin and J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits, J. Statist. Phys., 87(1997), 37-61. Google Scholar
[14]	G. Giacomin and J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction Ⅱ. Interface motion, SIAM J. Appl. Math., 58(1998), 1707-1729. Google Scholar
[15]	G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47(1993), 4289-4300. Google Scholar
[16]	G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. Ⅱ. Monte Carlo simulations, Phys. Rev. E, 47(1993), 4301-4312. Google Scholar
[17]	M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems, 28(2010), 67-98. Google Scholar
[18]	M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation, Math. Models Methods Appl. Sci., 24(2014), 2743-2783. Google Scholar
[19]	M. Grasselli and H. Wu, Robust exponential attractors for the modified phasefield crystal equation, Discrete Contin. Dyn. Systems, 35(2015), 2539-2564. Google Scholar
[20]	Z. Hu, S.M. Wise, C. Wang and J.S. Lowengrub, Stable finite difference, nonlinear multigrid simulation of the phase field crystal equation, J. Comput. Phys., 228(2009), 5323-5339. Google Scholar
[21]	M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, SIAM J. Math. Anal., 44(2012), 3369-3387. Google Scholar
[22]	M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 72(2012), 1343-1360. Google Scholar
[23]	A. Miranville, Asymptotic behavior of a sixth-order Cahn-Hilliard system, Central Europ. J. Math., 12(2014), 141-154. Google Scholar
[24]	A. Miranville, Sixth-order Cahn-Hilliard equations with logarithmic nonlinear terms, Appl. Anal., 94(2015), 2133-2146. Google Scholar
[25]	A. Miranville, Sixth-order Cahn-Hilliard systems with dynamic boundary conditions, Math. Methods Appl. Sci., 38(2015), 1127-1145. Google Scholar
[26]	A. Miranville, On the phase-field-crystal model with logarithmic nonlinear terms, RACSAM, to appear. Google Scholar
[27]	A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations, Vol. 4, C.M. Dafermos and M. Pokorny eds., Elsevier, Amsterdam, 2008, 103-200. Google Scholar
[28]	A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations, Evolutionary Partial Differential Equations, C.M. Dafermos and M. Pokorny eds., Elsevier, Amsterdam, 2008, 201-228. Google Scholar
[29]	I. Pawlow and G. Schimperna, On a Cahn-Hilliard model with nonlinear diffusion, SIAM J. Math. Anal., 45(2013), 31-63. Google Scholar
[30]	I. Pawlow and G. Schimperna, A Cahn-Hilliard equation with singular diffusion, J. Diff. Eqns., 254(2013), 779-803. Google Scholar
[31]	I. Pawlow and W. Zajaczkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal., 10(2011), 1823-1847. Google Scholar
[32]	I. Pawlow and W. Zajaczkowski, On a class of sixth order viscous Cahn-Hilliard type equations, Discrete Contin. Dyn. Systems S, 6(2013), 517-546. Google Scholar
[33]	T.V. Savina, A.A. Golovin, S.H. Davis, A.A. Nepomnyashchy and P.W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67(2003), 021606. Google Scholar
[34]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. Google Scholar
[35]	S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465(2009), 1337-1359. Google Scholar
[36]	C. Wang and S.M. Wise, Global smooth solutions of the modified phase field crystal equation, Methods Appl. Anal., 17(2010), 191-212. Google Scholar
[37]	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. Google Scholar
[38]	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. Google Scholar