On the Cahn–Hilliard equation with no-flux and strong anchoring conditions
On the Cahn–Hilliard equation with no-flux and strong anchoring conditions
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DOI：
10.1007/s00030-023-00854-y
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发表时间：
2023-05
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影响因子：
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通讯作者：
Shibin Dai;Toai Luong
中科院分区：
文献类型：
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作者：
Shibin Dai;Toai Luong
Cahn-Hilliard方程是描述两组分混合物相分离过程的常用模型。我们研究了Cahn-Hilliard方程与齐次强锚定条件(即齐次Dirichlet条件)在两相相对浓度上的耦合。此外，我们采用无通量边界条件来保持质量守恒。通过将问题解释为Cahn-Hilliard自由能的梯度流，证明了该模型弱解的存在唯一性。利用极小运动格式和时间离散化方法，我们证明了近似解收敛于Cahn-Hilliard方程的弱解。最后，我们证明了弱解满足一个能量耗散不等式。
The Cahn–Hilliard equation is a common model to describe phase separation processes of a mixture of two components. We study the Cahn–Hilliard equation coupled with the homogeneous strong anchoring condition (i.e., homogeneous Dirichlet condition) on the relative concentrationuof the two phases. Moreover, we adopt no-flux boundary condition to keep conservation of mass. With a specific quartic form of the double-well potential, we prove the existence and uniqueness of the weak solution to this model by interpreting the problem as a gradient flow of the Cahn–Hilliard free energy. Utilizing the minimizing movement scheme and time discretization method, we show that the approximation solutions converge to the weak solution of the Cahn–Hilliard equation. Finally, we prove that the weak solution satisfies an energy dissipation inequality.