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Volumn 6, Issue 2, 2008, Pages 309-329

The willmore functional and instabilities in the Cahn-Hilliard equation

Author keywords

Asymptotics; Cahn hilliard equation; Stability; Transition solutions; Willmore functional

Indexed keywords


EID: 49349117199     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2008.v6.n2.a3     Document Type: Article
Times cited : (9)

References (39)
  • 1
    • 44949275928 scopus 로고
    • Slow motion for the Cahn-Hilliard equation in one space dimension
    • N. Alikakos, P.W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differ. Equations, 90, 1, 81-135, 1991.
    • (1991) J. Differ. Equations , vol.90 , Issue.1 , pp. 81-135
    • Alikakos, N.1    Bates, P.W.2    Fusco, G.3
  • 2
    • 84963254790 scopus 로고
    • Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. I: Spectral estimates
    • N. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. I: Spectral estimates, Commun. Partial Differ. Equations, 19, 9-10, 1397-1447, 1994.
    • (1994) Commun. Partial Differ. Equations , vol.19 , Issue.9-10 , pp. 1397-1447
    • Alikakos, N.1    Fusco, G.2
  • 3
    • 43949159714 scopus 로고
    • Metastable patterns for the Cahn-Hilliard equation. I
    • P.W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation. I., J. Differ. Equations, 111, 2, 421-457, 1994.
    • (1994) J. Differ. Equations , vol.111 , Issue.2 , pp. 421-457
    • Bates, P.W.1    Xun, J.2
  • 4
    • 33847721052 scopus 로고    scopus 로고
    • Inpainting of binary images using the Cahn-Hilliard equation
    • A. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc., 16, 1, 285-291, 2007.
    • (2007) IEEE Trans. Image Proc , vol.16 , Issue.1 , pp. 285-291
    • Bertozzi, A.1    Esedoglu, S.2    Gillette, A.3
  • 6
    • 84972038069 scopus 로고
    • The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. II. Numerical analysis
    • J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. II. Numerical analysis, Eur. J. Appl. Math., 3, 2, 147-179, 1992.
    • (1992) Eur. J. Appl. Math , vol.3 , Issue.2 , pp. 147-179
    • Blowey, J.F.1    Elliott, C.M.2
  • 7
    • 0001196888 scopus 로고
    • On the slow dynamics for the Cahn-Hilliard equation in one space dimension
    • L. Bronsard and D. Hilhorst, - On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. R. Soc. Lond., Ser. A, 439, 1907, 669-682, 1992.
    • (1992) Proc. R. Soc. Lond., Ser. A , vol.439 , Issue.1907 , pp. 669-682
    • Bronsard, L.1    Hilhorst, D.2
  • 8
    • 84990575560 scopus 로고
    • On the slowness of phase boundary motion in one space dimension
    • L. Bronsard and R.V. Kohn, On the slowness of phase boundary motion in one space dimension, Commun. Pure Appl. Math., 43, 8, 983-997, 1990.
    • (1990) Commun. Pure Appl. Math , vol.43 , Issue.8 , pp. 983-997
    • Bronsard, L.1    Kohn, R.V.2
  • 9
    • 33746012315 scopus 로고
    • Free energy of a non-uniform system. I. Interfacial free energy
    • J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28, 258-267, 1958.
    • (1958) J. Chem. Phys , vol.28 , pp. 258-267
    • Cahn, J.W.1    Hilliard, J.E.2
  • 10
    • 0000358405 scopus 로고
    • Structured phase transitions on a finite interval
    • J. Carr, M.E. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal., 86, 317-351, 1984.
    • (1984) Arch. Rat. Mech. Anal , vol.86 , pp. 317-351
    • Carr, J.1    Gurtin, M.E.2    Slemrod, M.3
  • 11
    • 0001514990 scopus 로고    scopus 로고
    • Global asymptotic limit of solutions of the Cahn-Hilliard equation
    • X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differ. Geom., 44, 2, 262-311, 1996.
    • (1996) J. Differ. Geom , vol.44 , Issue.2 , pp. 262-311
    • Chen, X.1
  • 12
    • 0000368416 scopus 로고    scopus 로고
    • Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter
    • X. Chen and M. Kowalczyk, Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter, Commun. Partial Differ. Equations, 21, 7-8, 1207-1233, 1996.
    • (1996) Commun. Partial Differ. Equations , vol.21 , Issue.7-8 , pp. 1207-1233
    • Chen, X.1    Kowalczyk, M.2
  • 13
    • 29144514194 scopus 로고    scopus 로고
    • A phase field formulation of the Willmore problem
    • Q. Du, C. Liu, R. Ryham. and X. Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18, 3, 1249-1267, 2005.
    • (2005) Nonlinearity , vol.18 , Issue.3 , pp. 1249-1267
    • Du, Q.1    Liu, C.2    Ryham, R.3    Wang, X.4
  • 14
    • 0001617217 scopus 로고
    • Numerical studies of the Cahn-Hilliard equation for phase separation
    • C.M. Elliott and D.A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38, 97-128, 1987.
    • (1987) IMA J. Appl. Math , vol.38 , pp. 97-128
    • Elliott, C.M.1    French, D.A.2
  • 16
    • 11144340252 scopus 로고    scopus 로고
    • Error analysis of a mixed finite element method for the Cahn-Hilliard equation
    • X. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99, 1, 47-84, 2004.
    • (2004) Numer. Math , vol.99 , Issue.1 , pp. 47-84
    • Feng, X.1    Prohl, A.2
  • 17
    • 15544366700 scopus 로고    scopus 로고
    • Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem
    • X. Feng and A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Interfaces Free Bound., 7, 1, 1-28, 2005.
    • (2005) Interfaces Free Bound , vol.7 , Issue.1 , pp. 1-28
    • Feng, X.1    Prohl, A.2
  • 18
    • 0013027276 scopus 로고    scopus 로고
    • Models for phase separation and their mathematics
    • P.C. Fife, Models for phase separation and their mathematics, Electronic J. Differ. Equations, 2000, 48, 1-26, 2000.
    • (2000) Electronic J. Differ. Equations, 2000 , vol.48 , pp. 1-26
    • Fife, P.C.1
  • 19
    • 0003174115 scopus 로고    scopus 로고
    • Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation
    • P.C. Fife, H. Kielhoefer, S. Maier-Paape and T. Wanner, Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation, Physica D, 100, 1-4, 257-278, 1997.
    • (1997) Physica D , vol.100 , Issue.1-4 , pp. 257-278
    • Fife, P.C.1    Kielhoefer, H.2    Maier-Paape, S.3    Wanner, T.4
  • 20
    • 0040619949 scopus 로고
    • Slow motion in one-dimensional Cahn-Morral systems
    • C.P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26, 1, 21-34, 1995.
    • (1995) SIAM J. Math. Anal , vol.26 , Issue.1 , pp. 21-34
    • Grant, C.P.1
  • 21
    • 84971124145 scopus 로고
    • Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments
    • M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. R. Soc. Edinb., Sect. A, 125, 2, 351-370, 1995.
    • (1995) Proc. R. Soc. Edinb., Sect. A , vol.125 , Issue.2 , pp. 351-370
    • Grinfeld, M.1    Novick-Cohen, A.2
  • 22
    • 0001343091 scopus 로고    scopus 로고
    • The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor
    • M. Grinfeld and A. Novick-Cohen, The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor, Trans. Am. Math. Soc., 351, 6, 2375-2406, 1999.
    • (1999) Trans. Am. Math. Soc , vol.351 , Issue.6 , pp. 2375-2406
    • Grinfeld, M.1    Novick-Cohen, A.2
  • 23
    • 0001448409 scopus 로고
    • Toward a nonequilibrium thermodynamics of two-phase materials
    • M.E. Gurtin, Toward a nonequilibrium thermodynamics of two-phase materials, Arch. Rat. Mech. Anal., 100, 3, 275-312, 1988.
    • (1988) Arch. Rat. Mech. Anal , vol.100 , Issue.3 , pp. 275-312
    • Gurtin, M.E.1
  • 25
    • 0001228050 scopus 로고
    • On semi- and subanalytic geometry. (Sur la géométrie semi- et sous-analytique)
    • S. Lojasiewicz, On semi- and subanalytic geometry. (Sur la géométrie semi- et sous-analytique), Ann. Inst. Fourier, 43, 5, 1575-1595, 1993.
    • (1993) Ann. Inst. Fourier , vol.43 , Issue.5 , pp. 1575-1595
    • Lojasiewicz, S.1
  • 26
    • 10844251567 scopus 로고    scopus 로고
    • Numerical study of the Cahn-Hilliard equation in one, two and three dimensions
    • E.V.L. de Mello and O.T. da Silveira Filho, Numerical study of the Cahn-Hilliard equation in one, two and three dimensions, Physica A, 347, 429-443, 2005.
    • (2005) Physica A , vol.347 , pp. 429-443
    • de Mello, E.V.L.1    da Silveira Filho, O.T.2
  • 27
    • 0000520933 scopus 로고
    • Some global dynamical properties of a class of pattern formation equations
    • B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Commun. Partial Differ. Equations, 14, 2, 245-297, 1989.
    • (1989) Commun. Partial Differ. Equations , vol.14 , Issue.2 , pp. 245-297
    • Nicolaenko, B.1    Scheurer, B.2    Temam, R.3
  • 28
    • 0001410452 scopus 로고
    • Front migration in the nonlinear Cahn-Hilliard equation
    • R.L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422, 261-278, 1989.
    • (1989) Proc. Roy. Soc. London Ser. A , vol.422 , pp. 261-278
    • Pego, R.L.1
  • 29
    • 45149136070 scopus 로고
    • Thermodynamically consistent models of phase-field type for the kinetics of phase transitions
    • O. Penrose and P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43, 44-62, 1990.
    • (1990) Physica D , vol.43 , pp. 44-62
    • Penrose, O.1    Fife, P.C.2
  • 30
    • 33749607402 scopus 로고    scopus 로고
    • On a modified conjecture of De Giorgi
    • M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254, 4, 675-714, 2006.
    • (2006) Math. Z , vol.254 , Issue.4 , pp. 675-714
    • Röger, M.1    Schätzle, R.2
  • 32
    • 0033475056 scopus 로고    scopus 로고
    • Convergence of solutions to Cahn-Hilliard equation
    • P. Rybka and K.H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Commun. Partial Differ. Equations, 24, 5-6, 1055-1077, 1999.
    • (1999) Commun. Partial Differ. Equations , vol.24 , Issue.5-6 , pp. 1055-1077
    • Rybka, P.1    Hoffmann, K.H.2
  • 33
    • 0000137102 scopus 로고
    • Asymptotics for a class of non-linear evolution equations, with applications to geometric problems
    • L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math., 118, 525-571, 1983.
    • (1983) Ann. of Math , vol.118 , pp. 525-571
    • Simon, L.1
  • 34
    • 0034482446 scopus 로고    scopus 로고
    • Unexpectedly linear behavior for the Cahn-Hilliard equation
    • E. Sander and T. Warmer, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM J. Appl. Math., 60, 6, 2182-2202, 2000.
    • (2000) SIAM J. Appl. Math , vol.60 , Issue.6 , pp. 2182-2202
    • Sander, E.1    Warmer, T.2
  • 35
    • 0000698557 scopus 로고    scopus 로고
    • On the stationary Cahn-Hilliard equation: Interior spike solutions
    • J. Wei and M. Winter, On the stationary Cahn-Hilliard equation: Interior spike solutions, J. Differ. Equations, 148, 2, 231-267, 1998.
    • (1998) J. Differ. Equations , vol.148 , Issue.2 , pp. 231-267
    • Wei, J.1    Winter, M.2
  • 36
    • 0032374539 scopus 로고    scopus 로고
    • On the stationary Cahn-Hilliard equation: Bubble solutions
    • J. Wei and M. Winter, On the stationary Cahn-Hilliard equation: bubble solutions, SIAM J. Math. Anal., 29, 6, 1492-1518, 1998.
    • (1998) SIAM J. Math. Anal , vol.29 , Issue.6 , pp. 1492-1518
    • Wei, J.1    Winter, M.2
  • 39
    • 84972948188 scopus 로고
    • Asymptotic behavior of solution to the Cahn-Hilliard equation
    • S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation, Appl. Anal., 23, 165-184, 1986.
    • (1986) Appl. Anal , vol.23 , pp. 165-184
    • Zheng, S.1


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