Our aim in this paper is to study a modified higher-order (in space) phase field crystal model taking anisotropy into account. In particular, we deduce a priori estimates to prove well-posedness results and the dissipativity of the semigroup.
S.Agmon, Lectures on Elliptic Boundary Value Problems, Mathematical Studies, Van Nostrand, New York, 1965.
2.
S.Agmon, A.Douglis and L.Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations, I, Commun. Pure Appl. Math.12 (1959), 623–727. doi:10.1002/cpa.3160120405.
3.
S.Agmon, A.Douglis and L.Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations, II, Commun. Pure Appl. Math.17 (1964), 35–92. doi:10.1002/cpa.3160170104.
4.
G.Caginalp and E.Esenturk, Anisotropic phase field equations of arbitrary order, Discrete Contin. Dyn. Systems S4 (2011), 311–350. doi:10.3934/dcdss.2011.4.311.
J.W.Cahn and J.E.Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys.2 (1958), 258–267. doi:10.1063/1.1744102.
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. doi:10.4208/cicp.101111.110512a.
8.
X.Chen, G.Caginalp and E.Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Rational Mech. Anal.202 (2011), 349–372. doi:10.1007/s00205-011-0429-8.
9.
L.Cherfils, A.Miranville and S.Peng, Higher-order models in phase separation, J. Appl. Anal. Comput.7(1) (2016), 39–56.
10.
L.Cherfils, A.Miranville and S.Peng, Higher-order Allen–Cahn models with logarithmic nonlinear terms, in: Advances in Dynamical Systems and Control, Springer International Publishing, 2016, pp. 247–263. doi:10.1007/978-3-319-40673-2_12.
11.
L.Cherfils, A.Miranville and S.Peng, Higher-order anisotropic models in phase separation, in: Advances in Nonlinear Analysis, 2017.
12.
L.Cherfils, A.Miranville, S.Peng and W.Zhang, Higher-order generalized Cahn–Hilliard equations, E. Jou. of Qualitative Theory of Differential Equations9 (2017), 1–22.
13.
L.Cherfils, A.Miranville and S.Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math79 (2011), 561–596. doi:10.1007/s00032-011-0165-4.
14.
M.Conti, S.Gatti and A.Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions, Anal. Appl.11 (2013), 1350024. doi:10.1142/S0219530513500243.
15.
P.G.de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys.72 (1980), 4756–4763. doi:10.1063/1.439809.
16.
A.Debussche and L.Dettori, On the Cahn–Hilliard equation with a logarithmic free energy, Nonlinear Anal. TMA24 (1995), 1491–1514. doi:10.1016/0362-546X(94)00205-V.
17.
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. doi:10.1080/00018732.2012.737555.
18.
P.Galenko, Phase-field models with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A287 (2001), 190–197. doi:10.1016/S0375-9601(01)00489-3.
19.
P.Galenko, D.Danilov and V.Lebedev, Phase-field-crystal and Swift–Hohenberg equations with fast dynamics, Phys. Rev. E79 (2009), 051110. doi:10.1103/PhysRevE.79.051110.
20.
S.Gatti, M.Grasselli, A.Miranville and V.Pata, On the hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation, J. Math. Anal. Appl.312 (2005), 230–247. doi:10.1016/j.jmaa.2005.03.029.
21.
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. doi:10.1007/BF02181479.
22.
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. doi:10.1137/S0036139996313046.
23.
G.Gompper and M.Kraus, Ginzburg–Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E47 (1993), 4289–4300. doi:10.1103/PhysRevE.47.4289.
24.
G.Gompper and M.Kraus, Ginzburg–Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E47 (1993), 4301–4312. doi:10.1103/PhysRevE.47.4301.
25.
M.Grasselli, A.Miranville and G.Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems28 (2010), 67–98. doi:10.3934/dcds.2010.28.67.
26.
M.Grasselli and M.Pierre, Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM: M2AN, 50 (2016), 1523–1560.
27.
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. doi:10.1142/S0218202514500365.
28.
M.Grasselli and H.Wu, Robust exponential attractors for the modified phase-field crystal equation, Discrete Contin. Dyn. Systems35 (2015), 2539–2564. doi:10.3934/dcds.2015.35.2539.
29.
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. doi:10.1016/j.jcp.2009.04.020.
30.
R.Kobayashi, Modelling and numerical simulations of dendritic crystal growth, Phys. D63 (1993), 410–423. doi:10.1016/0167-2789(93)90120-P.
31.
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. doi:10.1137/100817590.
32.
M.Korzec and P.Rybka, On a higher order convective Cahn–Hilliard type equation, SIAM J. Appl. Math.72 (2012), 1343–1360. doi:10.1137/110834123.
33.
A.Miranville, Asymptotic behavior of a sixth-order Cahn–Hilliard system, Central Europ. J. Math.12 (2014), 141–154.
A.Miranville, Sixth-order Cahn–Hilliard systems with dynamic boundary conditions, Math. Methods Appl. Sci.38 (2015), 1127–1145. doi:10.1002/mma.3134.
36.
A.Miranville, On the phase-field-crystal model with logarithmic nonlinear terms, RACSAM, to appear.
37.
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, pp. 103–200.
38.
A.Miranville and S.Zelik, The Cahn–Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Systems28 (2010), 275–310. doi:10.3934/dcds.2010.28.275.
39.
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, pp. 201–228.
40.
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. doi:10.3934/cpaa.2011.10.1823.
41.
I.Pawlow and W.Zajaczkowski, On a class of sixth order viscous Cahn–Hilliard type equations, Discrete Contin. Dyn. Systems S6 (2013), 517–546. doi:10.3934/dcdss.2013.6.517.
42.
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. E67 (2003), 021606. doi:10.1103/PhysRevE.67.021606.
43.
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.
44.
P.Stefanovic, M.Haataja and N.Provatas, Phase-field crystals study of deformation and plasticity in nanocrystalline materials, Phys. Rev. E.80 (2009), 046107. doi:10.1103/PhysRevE.80.046107.
45.
J.E.Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater.40 (1992), 1475–1495. doi:10.1016/0956-7151(92)90091-R.
46.
R.Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.
47.
S.Torabi, J.Lowengrub, A.Voigt and S.Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A465 (2009), 1337–1359. doi:10.1098/rspa.2008.0385.
48.
C.Wang and S.M.Wise, Global smooth solutions of the modified phase field crystal equation, Methods Appl. Anal.17 (2010), 191–212.
49.
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.
50.
A.A.Wheeler and G.B.McFadden, On the notion of ξ-vector and stress tensor for a general class of anisotropic diffuse interface models, Proc. R. Soc. London Ser. A453 (1997), 1611–1630. doi:10.1098/rspa.1997.0086.
51.
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.
52.
H.Zhu, L.Cherfils, A.Miranville, S.Peng and W.Zhang, Energy stable finite element/spectral method for modified higher-order generalized Cahn–Hilliard Equations, J. Math. Study.51 (2018), 253–293.
