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November  2015, 20(9): 3185-3213. doi: 10.3934/dcdsb.2015.20.3185

New asymptotic analysis method for phase field models in moving boundary problem with surface tension

1. 

Department of Scientific Computing, Florida State University, Tallahassee, FL 32306-4120, United States, United States

Received  October 2013 Revised  May 2015 Published  September 2015

In this paper, we give an asymptotic analysis of the phase field Allen-Cahn and Cahn-Hilliard models of free surfaces with surface tension. Unlike the traditional approach that approximates the solution by the so-called matched asymptotic expansion involving outer expansion, inner expansion and matching, our new approach utilizes a uniform double asymptotic expansion to expand the whole phase field function directly. Although the main result is not new, we would like to emphasize that we derive the result under a uniform double asymptotic expansion. Thus, in this paper the detailed structure of the phase field functions in the equilibrium state is obtained, and the consistency of the phase field models with the corresponding sharp interface models is discussed, including the free surface Allen-Cahn model, Cahn-Hilliard model, and the Allen-Cahn model with volume constraint. The explicit asymptotic expansion of the phase field function reveals rich details of its structures. Moreover, it nicely explains some unusual phenomena we observed in numerical experiments. The theory introduced in this paper can be applied to guide the future modeling and simulation of other moving boundary problems by phase field models.
Citation: Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
References:
[1]

N. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.

[2]

S. Allen and J. 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.

[3]

J. F. Blowey and C. M. Elliott, Curvature dependent phase boundary motion and parabolic double obstacle problems, Degenerate Diffusions, Springer New York, 47 (1993), 19-60. doi: 10.1007/978-1-4612-0885-3_2.

[4]

K. Brakke, The Motion of a Surface by Its Mean Curvature, Vol. 20. Princeton: Princeton University Press, 1978.

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-1-4757-4338-8.

[6]

G. Caginalp, The Limiting Behavior of a Free Boundary in the Phase Field Model, Carnegie-Mellon Research Report 82-5, 1982.

[7]

G. Caginalp, Mathematical models of phase boundaries, Material Instability in Continuum Problems and Related Mathematical Problems, Oxford Science Publications, (1988), 35-52.

[8]

G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A, 39 (1989), 5887-5896. doi: 10.1103/PhysRevA.39.5887.

[9]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. J. of Applied Mathematics, 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[10]

G. Caginalp and P. C. Fife, Elliptic problems involving phase boundaries satisfying a curvature condition, IMA J. Appl. Math., 38 (1987), 195-217. doi: 10.1093/imamat/38.3.195.

[11]

J. Cahn and J. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

[12]

G. Caginalp and Y. Nishiura, The existence of traveling waves for phase field equations and convergence to sharp interface models in sigular limit, Quart. J. Appl. Math., 49 (1991), 147-162.

[13]

J. Cahn and A. Novick-Cohen, Limiting motion for an Allen-Cahn/Cahn-Hilliard system, Free boundary problems theory and applications (Zakopane 1995), 363 (1996), 89-97. doi: 10.2307/1513327.

[14]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Diff. Geom., 44 (1996), 262-311.

[15]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.

[16]

X. Chen, Rigorous verifications of formal asymptotic expansions, Proceedings of the International Conference onAsymptotics in Nonlinear Diffusive Systems, Tohoku Math. Publ., 8 (1998), 9-33.

[17]

X. Chen and C. Elliott, Asymptotics for a parabolic double obstacle problem, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 444 (1994), 429-445. doi: 10.1098/rspa.1994.0030.

[18]

X. Chen, C. M. Elliott, A. Gardiner and J. J. Zhao, Convergence of numerical solutions to the Allen-Cahn equation, Appl. Anal., 69 (1998), 47-56.

[19]

P. G. Ciarlet, The Finite Element Method for Ellipic Problems, Elsevier, 1978.

[20]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65. doi: 10.1007/BF01385847.

[21]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 58 (1975), 842-850.

[22]

Q. Du, C. Liu, R. Ryham and X. Wang, Phase field modeling of the spontaneous curvature effect in cell membranes, Comm. Pure. Appl. Anal., 4 (2005), 537-548. doi: 10.3934/cpaa.2005.4.537.

[23]

Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membrane, Journal of Computational Physics, 198 (2004), 450-468. doi: 10.1016/j.jcp.2004.01.029.

[24]

Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, Journal of Computational Physics, 212 (2006), 757-777. doi: 10.1016/j.jcp.2005.07.020.

[25]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of pahse transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. doi: 10.1137/0728069.

[26]

C. M. Elliott, Approximation of curvature dependent interface motion, The state of the art in numerical analysis (York, 1996), Inst. Math. Appl. Conf. Ser. New Ser., Oxford Univ. Press, New York, 63 (1997), 407-440.

[27]

C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. doi: 10.1093/imamat/38.2.97.

[28]

C. M. Elliott and D. A. French, A nonconforming finite-element method for the two dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989), 884-903. doi: 10.1137/0726049.

[29]

C. M. Elliott, D. A. French and F. A. Milner, A second order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575-590. doi: 10.1007/BF01396363.

[30]

C. M. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[31]

L. Evans, H. Soner and P. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.

[32]

X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94 (2003), 33-65. doi: 10.1007/s00211-002-0413-1.

[33]

P. Fife, Dynamics of Internal Layers and Diffusive Interfaces, Vol. 53. Philadelphia: Society for Industrial and Applied Mathematics, 1988. doi: 10.1137/1.9781611970180.

[34]

D. A. French and S. Jensen, Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems, IMA J. Numer. Anal., 14 (1994), 421-442. doi: 10.1093/imanum/14.3.421.

[35]

G. Fusco, A geometric approach to the dynamics of $u_t = \epsilon^2u_{x x} +f(u)$ for small $\epsilon$, Problems involving change of type., Springer Berlin Heidelberg, 359 (1990), 53-73. doi: 10.1007/3-540-52595-5_85.

[36]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461.

[37]

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

[38]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.

[39]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.1090/S0002-9947-1995-1672406-7.

[40]

R. H. Nochetto and C. Verdi, Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., 74 (1996), 105-136. doi: 10.1007/s002110050210.

[41]

R. H. Nochetto and C. Verdi, Convergence past singularities for a fully discrete approximation of curvature-driven interfaces, SIAM J. Numer. Anal., 34 (1997), 490-512. doi: 10.1137/S0036142994269526.

[42]

R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, roc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.

[43]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.

[44]

Y. Sun and C. Beckermann, Sharp interface tracking using the phase-field equation, J. Compput. Phys., 220 (2007), 626-653. doi: 10.1016/j.jcp.2006.05.025.

[45]

J. Van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, J. Sta. Phys., 20 (1979), 197-244. (Original (1893) is written in Dutch and above is English translation by J.S. Rowlinson.) doi: 10.1007/BF01011513.

[46]

X. Wang, Astmptotic analysis of phase field formulations Of bending elasticity models, SIAM J. Math. Anal., 39 (2008), 1367-1401. doi: 10.1137/060663519.

show all references

References:
[1]

N. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.

[2]

S. Allen and J. 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.

[3]

J. F. Blowey and C. M. Elliott, Curvature dependent phase boundary motion and parabolic double obstacle problems, Degenerate Diffusions, Springer New York, 47 (1993), 19-60. doi: 10.1007/978-1-4612-0885-3_2.

[4]

K. Brakke, The Motion of a Surface by Its Mean Curvature, Vol. 20. Princeton: Princeton University Press, 1978.

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-1-4757-4338-8.

[6]

G. Caginalp, The Limiting Behavior of a Free Boundary in the Phase Field Model, Carnegie-Mellon Research Report 82-5, 1982.

[7]

G. Caginalp, Mathematical models of phase boundaries, Material Instability in Continuum Problems and Related Mathematical Problems, Oxford Science Publications, (1988), 35-52.

[8]

G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A, 39 (1989), 5887-5896. doi: 10.1103/PhysRevA.39.5887.

[9]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. J. of Applied Mathematics, 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[10]

G. Caginalp and P. C. Fife, Elliptic problems involving phase boundaries satisfying a curvature condition, IMA J. Appl. Math., 38 (1987), 195-217. doi: 10.1093/imamat/38.3.195.

[11]

J. Cahn and J. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

[12]

G. Caginalp and Y. Nishiura, The existence of traveling waves for phase field equations and convergence to sharp interface models in sigular limit, Quart. J. Appl. Math., 49 (1991), 147-162.

[13]

J. Cahn and A. Novick-Cohen, Limiting motion for an Allen-Cahn/Cahn-Hilliard system, Free boundary problems theory and applications (Zakopane 1995), 363 (1996), 89-97. doi: 10.2307/1513327.

[14]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Diff. Geom., 44 (1996), 262-311.

[15]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.

[16]

X. Chen, Rigorous verifications of formal asymptotic expansions, Proceedings of the International Conference onAsymptotics in Nonlinear Diffusive Systems, Tohoku Math. Publ., 8 (1998), 9-33.

[17]

X. Chen and C. Elliott, Asymptotics for a parabolic double obstacle problem, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 444 (1994), 429-445. doi: 10.1098/rspa.1994.0030.

[18]

X. Chen, C. M. Elliott, A. Gardiner and J. J. Zhao, Convergence of numerical solutions to the Allen-Cahn equation, Appl. Anal., 69 (1998), 47-56.

[19]

P. G. Ciarlet, The Finite Element Method for Ellipic Problems, Elsevier, 1978.

[20]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65. doi: 10.1007/BF01385847.

[21]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 58 (1975), 842-850.

[22]

Q. Du, C. Liu, R. Ryham and X. Wang, Phase field modeling of the spontaneous curvature effect in cell membranes, Comm. Pure. Appl. Anal., 4 (2005), 537-548. doi: 10.3934/cpaa.2005.4.537.

[23]

Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membrane, Journal of Computational Physics, 198 (2004), 450-468. doi: 10.1016/j.jcp.2004.01.029.

[24]

Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, Journal of Computational Physics, 212 (2006), 757-777. doi: 10.1016/j.jcp.2005.07.020.

[25]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of pahse transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. doi: 10.1137/0728069.

[26]

C. M. Elliott, Approximation of curvature dependent interface motion, The state of the art in numerical analysis (York, 1996), Inst. Math. Appl. Conf. Ser. New Ser., Oxford Univ. Press, New York, 63 (1997), 407-440.

[27]

C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. doi: 10.1093/imamat/38.2.97.

[28]

C. M. Elliott and D. A. French, A nonconforming finite-element method for the two dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989), 884-903. doi: 10.1137/0726049.

[29]

C. M. Elliott, D. A. French and F. A. Milner, A second order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575-590. doi: 10.1007/BF01396363.

[30]

C. M. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[31]

L. Evans, H. Soner and P. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.

[32]

X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94 (2003), 33-65. doi: 10.1007/s00211-002-0413-1.

[33]

P. Fife, Dynamics of Internal Layers and Diffusive Interfaces, Vol. 53. Philadelphia: Society for Industrial and Applied Mathematics, 1988. doi: 10.1137/1.9781611970180.

[34]

D. A. French and S. Jensen, Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems, IMA J. Numer. Anal., 14 (1994), 421-442. doi: 10.1093/imanum/14.3.421.

[35]

G. Fusco, A geometric approach to the dynamics of $u_t = \epsilon^2u_{x x} +f(u)$ for small $\epsilon$, Problems involving change of type., Springer Berlin Heidelberg, 359 (1990), 53-73. doi: 10.1007/3-540-52595-5_85.

[36]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461.

[37]

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

[38]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.

[39]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.1090/S0002-9947-1995-1672406-7.

[40]

R. H. Nochetto and C. Verdi, Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., 74 (1996), 105-136. doi: 10.1007/s002110050210.

[41]

R. H. Nochetto and C. Verdi, Convergence past singularities for a fully discrete approximation of curvature-driven interfaces, SIAM J. Numer. Anal., 34 (1997), 490-512. doi: 10.1137/S0036142994269526.

[42]

R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, roc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.

[43]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.

[44]

Y. Sun and C. Beckermann, Sharp interface tracking using the phase-field equation, J. Compput. Phys., 220 (2007), 626-653. doi: 10.1016/j.jcp.2006.05.025.

[45]

J. Van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, J. Sta. Phys., 20 (1979), 197-244. (Original (1893) is written in Dutch and above is English translation by J.S. Rowlinson.) doi: 10.1007/BF01011513.

[46]

X. Wang, Astmptotic analysis of phase field formulations Of bending elasticity models, SIAM J. Math. Anal., 39 (2008), 1367-1401. doi: 10.1137/060663519.

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