Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter.

*(English)*Zbl 0863.35009This paper is concerned with the Cahn-Hilliard equation:
\[
u_t=\Delta(-\varepsilon^2\Delta u+ F'(u)),\quad x\in\Omega,\quad t>0,\tag{CH}
\]

\[ {\partial\over\partial n} u={\partial\over\partial n} (-\varepsilon^2\Delta u+ F'(u))=0,\quad x\in\partial\Omega,\quad t>0, \] where \(\Omega\) is a \(C^3\) bounded domain in \(\mathbb{R}^2\), \(\partial/\partial n\) is the exterior normal derivative, \(\varepsilon\) is a positive small parameter, and \(F\in C^2(\mathbb{R})\) is a double, equal well potential (namely, \(F(-1)=F(1)=0\), \(F(u)>0\) for \(u\neq\pm1\)). The Cahn-Hilliard equation (CH) serves as a model for phase separation and coarsening phenomena in binary alloys at a temperature at which only two different concentration phases can coexist. The solution \(u^\varepsilon\) of (CH) is approximately equal to 1 in a subregion \(\Omega^+\) of \(\Omega\) and is approximately equal to \(-1\) in another subregion \(\Omega^-\) of \(\Omega\). These two subregions are separated by a thin region containing the interface \(\Gamma_\varepsilon\equiv\{u^\varepsilon=0\}\). R. L. Pego [Proc. R. Soc. Lond., Ser. A 422, No. 1863, 261-278 (1989; Zbl 0701.35159)], by making use of the method of matched asymptotic expansions, derived formally the following law of motion, the Hele-Shaw problem, for the interface \(\Gamma=\lim_{\varepsilon\to0}\Gamma_\varepsilon\) (assuming that \(\tau=\varepsilon t\)): \[ \Delta\mu=0,\quad x\in\Omega\backslash\Gamma,\quad \tau\geq 0,\quad \mu=\alpha_1\kappa,\quad x\in\Gamma,\quad \tau\geq 0,\tag{HS} \]

\[ {\partial\mu\over\partial n}=0,\quad x\in\partial\Omega,\quad \tau\geq 0,\quad V=\alpha_2\Biggl[{\partial\mu\over\partial n}\Biggr]_\Gamma,\quad\tau>0, \] where \(\mu\) is an auxiliary function, \(\alpha_1\) and \(\alpha_2\) are positive constants, \(\kappa\) is the mean curvature of \(\Gamma\), \(V\) is the velocity in the direction normal to \(\Gamma\) and \([\partial\mu/\partial n]_\Gamma\) is the jump of the normal derivatives of \(\mu\) across \(\Gamma\). In the case when \(\Gamma\cap\partial\Omega\neq\emptyset\), the above system is supplemented with the condition that \(\Gamma\) meets \(\partial\Omega\) orthogonally, which follows from the Neumann boundary condition in (CH). The connection between (HS) and (CH) was shown rigorously by N. D. Alikakos, P. W. Bates and the first author [Arch. Ration. Mech. Anal. 128, No. 2, 165-205 (1994; Zbl 0828.35105)]. Notice that in the two-dimensional case circles contained in \(\Omega\) or circular arcs intersecting \(\partial\Omega\) orthogonally are equilibria to (HS). In a series of papers of N. D. Alikakos and G. Fusco [see for example, Commun. Partial Differ. Equations 19, No. 9-10, 1397-1447 (1994; Zbl 0814.35042)], it was shown that the solutions of (CH) whose interfaces are close to circles contained in \(\Omega\) evolve superslowly in such a way that \(\Gamma\) moves towards the closest point on \(\partial\Omega\).

The purpose of the present paper is to prove that there exist equilibria of (CH) with the property that the stationary interfaces \(\Gamma_\varepsilon\) tend as \(\varepsilon\to0\) to a circular arc \(\Gamma\) intersecting \(\partial\Omega\) orthogonally. The proof depends on the theory of functions of bounded variations and the result of R. V. Kohn and P. Sternberg [Proc. R. Soc. Edinb., Sect. A 111, No. 1/2, 69-84 (1989; Zbl 0676.49011)].

\[ {\partial\over\partial n} u={\partial\over\partial n} (-\varepsilon^2\Delta u+ F'(u))=0,\quad x\in\partial\Omega,\quad t>0, \] where \(\Omega\) is a \(C^3\) bounded domain in \(\mathbb{R}^2\), \(\partial/\partial n\) is the exterior normal derivative, \(\varepsilon\) is a positive small parameter, and \(F\in C^2(\mathbb{R})\) is a double, equal well potential (namely, \(F(-1)=F(1)=0\), \(F(u)>0\) for \(u\neq\pm1\)). The Cahn-Hilliard equation (CH) serves as a model for phase separation and coarsening phenomena in binary alloys at a temperature at which only two different concentration phases can coexist. The solution \(u^\varepsilon\) of (CH) is approximately equal to 1 in a subregion \(\Omega^+\) of \(\Omega\) and is approximately equal to \(-1\) in another subregion \(\Omega^-\) of \(\Omega\). These two subregions are separated by a thin region containing the interface \(\Gamma_\varepsilon\equiv\{u^\varepsilon=0\}\). R. L. Pego [Proc. R. Soc. Lond., Ser. A 422, No. 1863, 261-278 (1989; Zbl 0701.35159)], by making use of the method of matched asymptotic expansions, derived formally the following law of motion, the Hele-Shaw problem, for the interface \(\Gamma=\lim_{\varepsilon\to0}\Gamma_\varepsilon\) (assuming that \(\tau=\varepsilon t\)): \[ \Delta\mu=0,\quad x\in\Omega\backslash\Gamma,\quad \tau\geq 0,\quad \mu=\alpha_1\kappa,\quad x\in\Gamma,\quad \tau\geq 0,\tag{HS} \]

\[ {\partial\mu\over\partial n}=0,\quad x\in\partial\Omega,\quad \tau\geq 0,\quad V=\alpha_2\Biggl[{\partial\mu\over\partial n}\Biggr]_\Gamma,\quad\tau>0, \] where \(\mu\) is an auxiliary function, \(\alpha_1\) and \(\alpha_2\) are positive constants, \(\kappa\) is the mean curvature of \(\Gamma\), \(V\) is the velocity in the direction normal to \(\Gamma\) and \([\partial\mu/\partial n]_\Gamma\) is the jump of the normal derivatives of \(\mu\) across \(\Gamma\). In the case when \(\Gamma\cap\partial\Omega\neq\emptyset\), the above system is supplemented with the condition that \(\Gamma\) meets \(\partial\Omega\) orthogonally, which follows from the Neumann boundary condition in (CH). The connection between (HS) and (CH) was shown rigorously by N. D. Alikakos, P. W. Bates and the first author [Arch. Ration. Mech. Anal. 128, No. 2, 165-205 (1994; Zbl 0828.35105)]. Notice that in the two-dimensional case circles contained in \(\Omega\) or circular arcs intersecting \(\partial\Omega\) orthogonally are equilibria to (HS). In a series of papers of N. D. Alikakos and G. Fusco [see for example, Commun. Partial Differ. Equations 19, No. 9-10, 1397-1447 (1994; Zbl 0814.35042)], it was shown that the solutions of (CH) whose interfaces are close to circles contained in \(\Omega\) evolve superslowly in such a way that \(\Gamma\) moves towards the closest point on \(\partial\Omega\).

The purpose of the present paper is to prove that there exist equilibria of (CH) with the property that the stationary interfaces \(\Gamma_\varepsilon\) tend as \(\varepsilon\to0\) to a circular arc \(\Gamma\) intersecting \(\partial\Omega\) orthogonally. The proof depends on the theory of functions of bounded variations and the result of R. V. Kohn and P. Sternberg [Proc. R. Soc. Edinb., Sect. A 111, No. 1/2, 69-84 (1989; Zbl 0676.49011)].

Reviewer: S.Sakaguchi (Ehime)

##### MSC:

35B25 | Singular perturbations in context of PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

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\textit{X. Chen} and \textit{M. Kowalczyk}, Commun. Partial Differ. Equations 21, No. 7--8, 1207--1233 (1996; Zbl 0863.35009)

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##### References:

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[2] | DOI: 10.1080/03605309408821059 · Zbl 0814.35042 |

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[5] | DOI: 10.1007/BF00695274 · Zbl 0780.35117 |

[6] | DOI: 10.1512/iumj.1983.32.32003 · Zbl 0486.49024 |

[7] | Quart.Appl.Math. 46 pp 301– (1988) |

[8] | Proc.Roy.Soc,Edin pp 69– (1989) |

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[11] | Proc.Roy.Soc. 422 pp 261– (1989) · Zbl 0701.35159 |

[12] | DOI: 10.1007/BF00253122 · Zbl 0647.49021 |

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