2014, 34(1): 1-17. doi: 10.3934/dcds.2014.34.1

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

Received  November 2012 Revised  April 2013 Published  June 2013

We establish the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions and give the dynamics for the center of the bubble.
Citation: Peter W. Bates, Jiayin Jin. Global dynamics of boundary droplets. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 1-17. doi: 10.3934/dcds.2014.34.1
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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1093/imamat/44.1.77.

[21]

J. W. Cahn, On the spinodal decompostion,, Acta. Metall., 9 (1961), 795. 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. 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. 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. 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. 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. 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.

[28]

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations,, J. Dyn. Diff. Eqns., 1 (1989), 75. doi: 10.1007/BF01048791.

[29]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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. 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. doi: 10.1007/BF00251230.

[32]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285.

[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. 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1093/imamat/44.1.77.

[21]

J. W. Cahn, On the spinodal decompostion,, Acta. Metall., 9 (1961), 795. 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. 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. 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. 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. 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. 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.

[28]

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations,, J. Dyn. Diff. Eqns., 1 (1989), 75. doi: 10.1007/BF01048791.

[29]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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. 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. doi: 10.1007/BF00251230.

[32]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285.

[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. 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. doi: 10.1007/BF00253122.

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