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Spectral analysis for transition front solutions in Cahn-Hilliard systems
1. | Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States, United States |
References:
[1] |
N. D. Alikakos, S. I. Betelu, and X. Chen, Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities, European J. of Applied Mathematics, 17 (2006), 525-556.
doi: 10.1017/S095679250600667X. |
[2] |
N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906.
doi: 10.1512/iumj.2008.57.3181. |
[3] |
J. Alexander, R. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves, J. Reine Angew. Math., 410 (1990), 167-212. |
[4] |
J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math., 52 (1999), 839-871.
doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. |
[5] |
F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model, Mathematical Modeling and Numerical Analysis, 40 (2006), 653-687.
doi: 10.1051/m2an:2006028. |
[6] |
J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.
doi: 10.1016/0001-6160(61)90182-1. |
[7] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
[8] |
D. de Fontaine, "A Computer Simulation of the Evolution of Coherent Composition Variations in Solid Solutions,'' Ph. D. thesis, Northwestern University, 1967. |
[9] |
D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions I. Stability criteria, J. Phys. Chem. Solids, 33 (1972), 297-310.
doi: 10.1016/0022-3697(72)90011-X. |
[10] |
D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions II. Fluctuations and kinetics, J. Phys. Chem. Solids, 34 (1973), 1285-1304.
doi: 10.1016/S0022-3697(73)80026-5. |
[11] | |
[12] |
D. J. Eyre, Systems of Cahn-Hilliard equations, SIAM J. Appl. Math., 53 (1993), 1686-1712.
doi: 10.1137/0153078. |
[13] |
M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Annals of Mathematics Studies, Princeton University Press, 1983. |
[14] |
M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals, Invent. Math., 72 (1983), 285-298.
doi: 10.1007/BF01389324. |
[15] |
C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.
doi: 10.1137/S0036141092226053. |
[16] |
R. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), 797-855.
doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1. |
[17] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture notes in mathematics, Springer-Verlag, 840, 1981. |
[18] |
P. Howard, Pointwise estimates and stability for degenerate viscous shock waves, J. Reine Angew. Math., 545 (2002), 19-65.
doi: 10.1515/crll.2002.034. |
[19] |
P. Howard, Local tracking and stability for degenerate viscous shock waves, J. Differential Eqns., 186 (2002), 440-469. |
[20] |
P. Howard, Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form, Commun. Math. Phys., 269 (2007), 765-808.
doi: 10.1007/s00220-006-0102-5. |
[21] |
P. Howard, Asymptotic behavior near planar transition fronts for the Cahn-Hilliard equation, Phys. D, 229 (2007), 123-165.
doi: 10.1016/j.physd.2007.03.018. |
[22] |
P. Howard, Spectral analysis of stationary solutions of the Cahn-Hilliard equation, Advances in Differential Equations, 14 (2009), 87-120. |
[23] |
P. Howard and B. Kwon, Stability for transition front solutions in Cahn-Hilliard Systems,, in preparation., ().
|
[24] |
P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves, Discrete and Continuous Dynamical Systems, 10 (2004), 837-855.
doi: 10.3934/dcds.2004.10.837. |
[25] |
J. J. Hoyt, Spinodal decomposition in ternary alloys, Acta Metall., 37 (1989), 2489-2497.
doi: 10.1016/0001-6160(89)90047-3. |
[26] |
J. Kim and K. Kang, A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility, Applied Numerical Mathematics, 59 (2009), 1029-1042.
doi: 10.1016/j.apnum.2008.04.004. |
[27] |
R. V. Kohn and X. Yan, Coarsening rates for models of multicomponent phase separation, Interfaces Free Bound., 6 (2004), 135-149.
doi: 10.4171/IFB/94. |
[28] |
J. E. Morral and J. W. Cahn, Spinodal decomposition in ternary systems, Acta Metall., 19 (1971), 1037-1045.
doi: 10.1016/0001-6160(71)90036-8. |
[29] | |
[30] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics IV: Analysis of Operators," Academic Press, 1978. |
[31] |
V. Stefanopoulos, Heteroclinic connections for multiple-well potentials: The anisotropic case, Proc. Royal Soc. Edinburgh Sect. A, 138 (2008), 1313-1330.
doi: 10.1017/S0308210507000145. |
[32] |
K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871. See also the errata for this paper: Indiana U. Math. J., 51 (2002), 1017-1021.
doi: 10.1512/iumj.2002.51.2410. |
show all references
References:
[1] |
N. D. Alikakos, S. I. Betelu, and X. Chen, Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities, European J. of Applied Mathematics, 17 (2006), 525-556.
doi: 10.1017/S095679250600667X. |
[2] |
N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906.
doi: 10.1512/iumj.2008.57.3181. |
[3] |
J. Alexander, R. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves, J. Reine Angew. Math., 410 (1990), 167-212. |
[4] |
J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math., 52 (1999), 839-871.
doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. |
[5] |
F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model, Mathematical Modeling and Numerical Analysis, 40 (2006), 653-687.
doi: 10.1051/m2an:2006028. |
[6] |
J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.
doi: 10.1016/0001-6160(61)90182-1. |
[7] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
[8] |
D. de Fontaine, "A Computer Simulation of the Evolution of Coherent Composition Variations in Solid Solutions,'' Ph. D. thesis, Northwestern University, 1967. |
[9] |
D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions I. Stability criteria, J. Phys. Chem. Solids, 33 (1972), 297-310.
doi: 10.1016/0022-3697(72)90011-X. |
[10] |
D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions II. Fluctuations and kinetics, J. Phys. Chem. Solids, 34 (1973), 1285-1304.
doi: 10.1016/S0022-3697(73)80026-5. |
[11] | |
[12] |
D. J. Eyre, Systems of Cahn-Hilliard equations, SIAM J. Appl. Math., 53 (1993), 1686-1712.
doi: 10.1137/0153078. |
[13] |
M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Annals of Mathematics Studies, Princeton University Press, 1983. |
[14] |
M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals, Invent. Math., 72 (1983), 285-298.
doi: 10.1007/BF01389324. |
[15] |
C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.
doi: 10.1137/S0036141092226053. |
[16] |
R. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), 797-855.
doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1. |
[17] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture notes in mathematics, Springer-Verlag, 840, 1981. |
[18] |
P. Howard, Pointwise estimates and stability for degenerate viscous shock waves, J. Reine Angew. Math., 545 (2002), 19-65.
doi: 10.1515/crll.2002.034. |
[19] |
P. Howard, Local tracking and stability for degenerate viscous shock waves, J. Differential Eqns., 186 (2002), 440-469. |
[20] |
P. Howard, Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form, Commun. Math. Phys., 269 (2007), 765-808.
doi: 10.1007/s00220-006-0102-5. |
[21] |
P. Howard, Asymptotic behavior near planar transition fronts for the Cahn-Hilliard equation, Phys. D, 229 (2007), 123-165.
doi: 10.1016/j.physd.2007.03.018. |
[22] |
P. Howard, Spectral analysis of stationary solutions of the Cahn-Hilliard equation, Advances in Differential Equations, 14 (2009), 87-120. |
[23] |
P. Howard and B. Kwon, Stability for transition front solutions in Cahn-Hilliard Systems,, in preparation., ().
|
[24] |
P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves, Discrete and Continuous Dynamical Systems, 10 (2004), 837-855.
doi: 10.3934/dcds.2004.10.837. |
[25] |
J. J. Hoyt, Spinodal decomposition in ternary alloys, Acta Metall., 37 (1989), 2489-2497.
doi: 10.1016/0001-6160(89)90047-3. |
[26] |
J. Kim and K. Kang, A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility, Applied Numerical Mathematics, 59 (2009), 1029-1042.
doi: 10.1016/j.apnum.2008.04.004. |
[27] |
R. V. Kohn and X. Yan, Coarsening rates for models of multicomponent phase separation, Interfaces Free Bound., 6 (2004), 135-149.
doi: 10.4171/IFB/94. |
[28] |
J. E. Morral and J. W. Cahn, Spinodal decomposition in ternary systems, Acta Metall., 19 (1971), 1037-1045.
doi: 10.1016/0001-6160(71)90036-8. |
[29] | |
[30] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics IV: Analysis of Operators," Academic Press, 1978. |
[31] |
V. Stefanopoulos, Heteroclinic connections for multiple-well potentials: The anisotropic case, Proc. Royal Soc. Edinburgh Sect. A, 138 (2008), 1313-1330.
doi: 10.1017/S0308210507000145. |
[32] |
K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871. See also the errata for this paper: Indiana U. Math. J., 51 (2002), 1017-1021.
doi: 10.1512/iumj.2002.51.2410. |
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