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Preface
Global dynamics of boundary droplets
1. | Department of Mathematics, Michigan State University, East Lansing, MI 48824 |
2. | Department of Mathematics, Michigan State University, East Lansing, MI 48823, United States |
References:
[1] |
N. D. Alikakos, P. W. Bates and Xinfu Chen, The convergence of solutions of the Cahn-Hilliard equation to the solution of Hele-Shaw model, Arch. Rat. Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
N. D. Alikakos, P. W. Bates, Xinfu Chen and G. Fusco, Mullins-Sekerka motion of small droplets on a fixed boundary, J. Geo. Anal., 10 (2000), 575-596.
doi: 10.1007/BF02921987. |
[3] |
N. D. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Diff. Eqns., 90 (1991), 81-135.
doi: 10.1016/0022-0396(91)90163-4. |
[4] |
N. D. Alikakos, Xinfu Chen and G. Fusco, Motion of a droplet by surface tension along the boundary, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.
doi: 10.1007/s005260000052. |
[5] |
N. D. Alikakos and G. Fusco, Slow dynamics for the cahn-hilliard equation in higher space dimensions. I. Spectral estimates, Communications in Partial Differential Equations, 19 (1994), 1397-1447.
doi: 10.1080/03605309408821059. |
[6] |
N. D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of the bubble, Arch. Rat. Mech. Anal., 141 (1998), 1-61.
doi: 10.1007/s002050050072. |
[7] |
N. D. Alikakos and G. Fusco, Some aspects of the dynamics of the Cahn-Hilliard equation, Resenhas, 1 (1994), 517-530. |
[8] |
N. D. Alikakos, G. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Diff. Eqns., 126 (1996), 106-167.
doi: 10.1006/jdeq.1996.0046. |
[9] |
S. 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.
doi: 10.1016/0001-6160(79)90196-2. |
[10] |
P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-filed equations, and time scales for coarsening, Phys. D, 43 (1990), 335-348.
doi: 10.1016/0167-2789(90)90141-B. |
[11] |
P. W. Bates and P. C. Fife, The dynamis of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008.
doi: 10.1137/0153049. |
[12] |
P. W. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns., 160 (2000), 283-356.
doi: 10.1006/jdeq.1999.3660. |
[13] |
P. W. Bates, Kening Lu and Chongchun Zeng, Approximately invariant manifolds and global dynamics of spike states, Inventiones Mathematicae, 174 (2008), 355-433.
doi: 10.1007/s00222-008-0141-y. |
[14] |
P. W. Bates, Kening Lu and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Am. Math. Soc., 135 (1998). |
[15] |
P. W. Bates and J. P. Xun, Metastable patterns for the Cahn-Hilliard equation. I, J. Diff. Eqns., 111 (1994), 421-457.
doi: 10.1006/jdeq.1994.1089. |
[16] |
P. W. Bates and J. P. Xun, Metastable patterns for the Cahn-Hilliard equation. II. Layer dynamics and slow invariant manifold, J. Diff. Eqns., 117 (1995), 165-216.
doi: 10.1006/jdeq.1995.1052. |
[17] |
L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. R. Soc. London Ser. A, 439 (1992), 669-682.
doi: 10.1098/rspa.1992.0176. |
[18] |
L. Bronsard and R. V. Kohn, On the slowness of the phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997.
doi: 10.1002/cpa.3160430804. |
[19] |
L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Eqns., 90 (1991), 211-237.
doi: 10.1016/0022-0396(91)90147-2. |
[20] |
G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94.
doi: 10.1093/imamat/44.1.77. |
[21] |
J. W. Cahn, On the spinodal decompostion, Acta. Metall., 9 (1961), 795-801.
doi: 10.1016/0001-6160(61)90182-1. |
[22] |
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. |
[23] |
J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal., 86 (1984), 317-351.
doi: 10.1007/BF00280031. |
[24] |
J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t=\epsilon^2 u_{x x}-f(u)$, Comm. Pure. Appl. Math., 42 (1989), 523-576.
doi: 10.1002/cpa.3160420502. |
[25] |
J. Carr and R. L. Pego, Invariant manifolds for metastable patterns in $u_t=\epsilon^2 u_{x x}-f(u)$, Proc. R. Soc. Edinb. Sect. A, 116 (1990), 133-160.
doi: 10.1017/S0308210500031425. |
[26] |
X. Chen and M. Kowalczyk, Existence of equilibria for the Cahn-Hilliard equation via local minimizers of perimeter, Comm. PDE, 21 (1996), 1207-1233.
doi: 10.1080/03605309608821223. |
[27] |
E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850. |
[28] |
G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dyn. Diff. Eqns., 1 (1989), 75-94.
doi: 10.1007/BF01048791. |
[29] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[30] |
M. Kowalczyk, Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold, Duke Math. J., 98 (1999), 59-111.
doi: 10.1215/S0012-7094-99-09802-2. |
[31] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[32] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299. |
[33] |
N. C. Owen and P. Sternberg, Gradient flow and front propagation with boundary contact energy, Proc. Roy. Soc. Lond. Ser. A, 437 (1992), 715-728.
doi: 10.1098/rspa.1992.0088. |
[34] |
P. Sternberg, The effect of a singular perturbatoin on nonconvex variational problems, Arch. Rat. Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and Xinfu Chen, The convergence of solutions of the Cahn-Hilliard equation to the solution of Hele-Shaw model, Arch. Rat. Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
N. D. Alikakos, P. W. Bates, Xinfu Chen and G. Fusco, Mullins-Sekerka motion of small droplets on a fixed boundary, J. Geo. Anal., 10 (2000), 575-596.
doi: 10.1007/BF02921987. |
[3] |
N. D. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Diff. Eqns., 90 (1991), 81-135.
doi: 10.1016/0022-0396(91)90163-4. |
[4] |
N. D. Alikakos, Xinfu Chen and G. Fusco, Motion of a droplet by surface tension along the boundary, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.
doi: 10.1007/s005260000052. |
[5] |
N. D. Alikakos and G. Fusco, Slow dynamics for the cahn-hilliard equation in higher space dimensions. I. Spectral estimates, Communications in Partial Differential Equations, 19 (1994), 1397-1447.
doi: 10.1080/03605309408821059. |
[6] |
N. D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of the bubble, Arch. Rat. Mech. Anal., 141 (1998), 1-61.
doi: 10.1007/s002050050072. |
[7] |
N. D. Alikakos and G. Fusco, Some aspects of the dynamics of the Cahn-Hilliard equation, Resenhas, 1 (1994), 517-530. |
[8] |
N. D. Alikakos, G. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Diff. Eqns., 126 (1996), 106-167.
doi: 10.1006/jdeq.1996.0046. |
[9] |
S. 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.
doi: 10.1016/0001-6160(79)90196-2. |
[10] |
P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-filed equations, and time scales for coarsening, Phys. D, 43 (1990), 335-348.
doi: 10.1016/0167-2789(90)90141-B. |
[11] |
P. W. Bates and P. C. Fife, The dynamis of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008.
doi: 10.1137/0153049. |
[12] |
P. W. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns., 160 (2000), 283-356.
doi: 10.1006/jdeq.1999.3660. |
[13] |
P. W. Bates, Kening Lu and Chongchun Zeng, Approximately invariant manifolds and global dynamics of spike states, Inventiones Mathematicae, 174 (2008), 355-433.
doi: 10.1007/s00222-008-0141-y. |
[14] |
P. W. Bates, Kening Lu and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Am. Math. Soc., 135 (1998). |
[15] |
P. W. Bates and J. P. Xun, Metastable patterns for the Cahn-Hilliard equation. I, J. Diff. Eqns., 111 (1994), 421-457.
doi: 10.1006/jdeq.1994.1089. |
[16] |
P. W. Bates and J. P. Xun, Metastable patterns for the Cahn-Hilliard equation. II. Layer dynamics and slow invariant manifold, J. Diff. Eqns., 117 (1995), 165-216.
doi: 10.1006/jdeq.1995.1052. |
[17] |
L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. R. Soc. London Ser. A, 439 (1992), 669-682.
doi: 10.1098/rspa.1992.0176. |
[18] |
L. Bronsard and R. V. Kohn, On the slowness of the phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997.
doi: 10.1002/cpa.3160430804. |
[19] |
L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Eqns., 90 (1991), 211-237.
doi: 10.1016/0022-0396(91)90147-2. |
[20] |
G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94.
doi: 10.1093/imamat/44.1.77. |
[21] |
J. W. Cahn, On the spinodal decompostion, Acta. Metall., 9 (1961), 795-801.
doi: 10.1016/0001-6160(61)90182-1. |
[22] |
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. |
[23] |
J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal., 86 (1984), 317-351.
doi: 10.1007/BF00280031. |
[24] |
J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t=\epsilon^2 u_{x x}-f(u)$, Comm. Pure. Appl. Math., 42 (1989), 523-576.
doi: 10.1002/cpa.3160420502. |
[25] |
J. Carr and R. L. Pego, Invariant manifolds for metastable patterns in $u_t=\epsilon^2 u_{x x}-f(u)$, Proc. R. Soc. Edinb. Sect. A, 116 (1990), 133-160.
doi: 10.1017/S0308210500031425. |
[26] |
X. Chen and M. Kowalczyk, Existence of equilibria for the Cahn-Hilliard equation via local minimizers of perimeter, Comm. PDE, 21 (1996), 1207-1233.
doi: 10.1080/03605309608821223. |
[27] |
E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850. |
[28] |
G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dyn. Diff. Eqns., 1 (1989), 75-94.
doi: 10.1007/BF01048791. |
[29] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[30] |
M. Kowalczyk, Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold, Duke Math. J., 98 (1999), 59-111.
doi: 10.1215/S0012-7094-99-09802-2. |
[31] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[32] |
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299. |
[33] |
N. C. Owen and P. Sternberg, Gradient flow and front propagation with boundary contact energy, Proc. Roy. Soc. Lond. Ser. A, 437 (1992), 715-728.
doi: 10.1098/rspa.1992.0088. |
[34] |
P. Sternberg, The effect of a singular perturbatoin on nonconvex variational problems, Arch. Rat. Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
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