Asymptotic behavior of a sixth-order Cahn-Hilliard system

 Access to full text

 Full (PDF)

1. [1] Babin A.V., Vishik M.I., Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland, Amsterdam, 1992 Zbl0778.58002
2. [2] Berry J., Elder K.R., Grant M., Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 2008, 77(6), #061506 
3. [3] Berry J., Grant M., Elder K.R., Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 2006, 73(3), #031609 
4. [4] Caginalp G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 1986, 92(3), 205–245 Zbl0608.35080
5. [5] Cahn J.W., Hilliard J.E., Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 1958, 28(2), 258–267 http://dx.doi.org/10.1063/1.1744102 
6. [6] Chen F., Shen J., Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 2013, 13(5), 1189–1208 
7. [7] Eden A., Foias C., Nicolaenko B., Temam R., Exponential Attractors for Dissipative Evolution Equations, RAM Res. Appl. Math., 37, John Wiley & Sons, Chichester, 1994 
8. [8] Efendiev M., Miranville A., Zelik S., Exponential attractors for a nonlinear reaction-diffusion system in ℝ3, C. R. Acad. Sci. Paris Sér. I Math., 2000, 330(8), 713–718 http://dx.doi.org/10.1016/S0764-4442(00)00259-7 Zbl1151.35315
9. [9] Efendiev M., Miranville A., Zelik S., Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 2004, 272, 11–31 http://dx.doi.org/10.1002/mana.200310186 Zbl1046.37047
10. [10] Galenko P., Danilov D., Lebedev V., Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 2009, 79(5), #051110 
11. [11] de Gennes P.G., Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 1980, 72(9), 4756–4763 http://dx.doi.org/10.1063/1.439809 Zbl1110.82310
12. [12] Gompper G., Kraus M., Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 1993, 47(6), 4289–4300 http://dx.doi.org/10.1103/PhysRevE.47.4289 
13. [13] Gompper G., Kraus M., Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 1993, 47(6), 4301–4312 http://dx.doi.org/10.1103/PhysRevE.47.4301 
14. [14] Hu Z., Wise S.M., Wang C., Lowengrub J.S., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 2009, 228(15), 5323–5339 http://dx.doi.org/10.1016/j.jcp.2009.04.020 Zbl1171.82015
15. [15] Korzec M.D., Nayar P., Rybka P., Global weak solutions to a sixth order Cahn-Hilliard type equation, SIAM J. Math. Anal., 2012, 44(5), 3369–3387 http://dx.doi.org/10.1137/100817590 Zbl1270.35252
16. [16] Korzec M.D., Rybka P., On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 2012, 72(4), 1343–1360 http://dx.doi.org/10.1137/110834123 Zbl1257.35109
17. [17] Miranville A., Zelik S., Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations, 4, Elsevier, Amsterdam, 2008, 103–200 Zbl1221.37158
18. [18] Pawłow I., Zajaczkowski W.M., A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal., 2011, 10(6), 1823–1847 http://dx.doi.org/10.3934/cpaa.2011.10.1823 Zbl1229.35108
19. [19] Pawłow I., Zajaczkowski W.M., On a class of sixth order viscous Cahn-Hilliard type equations, Discrete Contin. Dyn. Syst. Ser. S, 2013, 6(2), 517–546 Zbl1276.35100
20. [20] Promislow K., Zhang H., Critical points of functionalized Lagrangians, Discrete Contin. Dyn. Syst., 2013, 33(4), 1231–1246 http://dx.doi.org/10.3934/dcds.2013.33.1231 Zbl1262.49017
21. [21] Savina T.V., Golovin A.A., Davis S.H., Nepomnyashchy A.A., Voorhees P.W., Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 2003, 67(2), #021606 
22. [22] Schimperna G., Pawłow I., A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 2013, 254(2), 779–803 http://dx.doi.org/10.1016/j.jde.2012.09.018 Zbl1272.35115
23. [23] Schimperna G., Pawłow I., On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 2013, 45(1), 31–63 http://dx.doi.org/10.1137/110835608 Zbl1276.35101
24. [24] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., 68, Springer, New York, 1997 http://dx.doi.org/10.1007/978-1-4612-0645-3 
25. [25] Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359 http://dx.doi.org/10.1098/rspa.2008.0385 Zbl1186.80014
26. [26] Wang C., Wise S.M., Global smooth solutions of the three-dimensional modified phase field crystal equation, Methods Appl. Anal., 2010, 17(2), 191–211 Zbl1217.35040
27. [27] Wang C., Wise S.M., An energy stable and convergent finite difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 2011, 49(3), 945–969 http://dx.doi.org/10.1137/090752675 Zbl1230.82005
28. [28] Wise S.M., Wang C., Lowengrub J.S., An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 2009, 47(3), 2269–2288 http://dx.doi.org/10.1137/080738143 Zbl1201.35027

1. Ahmad Makki, On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization