Sharp-Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
Sharp-Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
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简并迁移率 Cahn-Hilliard 方程的锐界面极限
DOI：
10.1137/140960189
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
E. Süli
中科院分区：
文献类型：
--
作者：
A. Lee;A. Münch;E. Süli
在这项工作中，尖锐的界面限制与多项式的双阱自由能和迁移率，消失在双阱的最小值的简并Cahn-Hilliard方程的推导。对于二次流动性的选择，领先的顺序尖锐的界面运动是不受纯表面扩散，如先前在文献中所声称的，但包含在同一顺序的非线性，多孔介质型体扩散的贡献。我们的分析揭示了有两种子情况：一种是序参数的解在最小值之间有界（Elliott和Garcke证明了第一迁移率的存在[SIAM J.Math.Anal.，27（1996），pp. 404- 423]），以及一个收敛于Cahn-Hilliard方程的经典定态解。体扩散的一致处理需要与多个内层相结合的指数大项和指数小项的匹配。此外，领先的顺序锐界面运动敏感地依赖于流动性的选择。渐近分析表明，例如，与一个双二次流动性，领先的顺序尖锐的界面运动是由表面扩散驱动。尖锐的接口模型得到证实，通过比较弛豫速率的扰动的径向对称的静止状态与相场模型得到的。
In this work, sharp-interface limits for the degenerate Cahn--Hilliard equation with a polynomial double-well free energy and a mobility that vanishes at the minima of the double well are derived. For the choice of a quadratic mobility, the leading order sharp-interface motion is not governed by pure surface diffusion, as has been previously claimed in the literature, but contains a contribution from nonlinear, porous-medium-type bulk diffusion at the same order. Our analysis reveals that there are two subcases: One, where the solution for the order parameter is bounded between the minima (proven to exist for the first mobility by Elliott and Garcke [SIAM J. Math. Anal., 27 (1996), pp. 404--423]), and one where it converges to the classical stationary solution of the Cahn--Hilliard equation. Consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers. Moreover, the leading order sharp-interface motion depends sensitively on the choice of mobility. The asymptotic analysis shows that, for example, with a biquadratic mobility, the leading order sharp-interface motion is driven only by surface diffusion. The sharp-interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model.