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

Global dynamics of boundary droplets

Peter W. Bates 1, and  Jiayin Jin 2,

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.
Keywords: infinitely dimensional dynamical systems, bubble solution, Invariant manifold theory, global dynamics., mass-conserving Allen-Cahn Equation.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Peter W. Bates, Jiayin Jin. Global dynamics of boundary droplets. Discrete & Continuous Dynamical Systems, 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), 165-205. doi: 10.1007/BF00375025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. D. Alikakos and G. Fusco, Some aspects of the dynamics of the Cahn-Hilliard equation, Resenhas, 1 (1994), 517-530.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. W. Cahn, On the spinodal decompostion, Acta. Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. D. Alikakos and G. Fusco, Some aspects of the dynamics of the Cahn-Hilliard equation, Resenhas, 1 (1994), 517-530.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. W. Cahn, On the spinodal decompostion, Acta. Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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