Sharp interface limits for diffuse interface models
Sharp interface limits for diffuse interface models
复制标题
漫反射界面模型的尖锐界面限制
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Stefan Schaubeck
中科院分区：
文献类型：
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作者：
Stefan Schaubeck
在本文中，我们严格证明了 Cahn-Larche 系统收敛于改进的 Hele-Shaw 问题。为了证明，我们通过匹配渐近展开的方法构造了 Cahn-Larche 系统的近似解。然后我们可以证明Cahn-Larche系统的近似解收敛于修正的Hele-Shaw问题的解。
对于改进的 Hele-Shaw 问题，我们证明了经典解的存在性
足够小的时间间隔[0;T]。通过将系统简化为单一演化
距离函数方程，我们展示断言。此外，我们证明了一个
线性化 Hele-Shaw 问题经典解的存在性结果
高阶展开式。
通过与 Cahn-Larche 系统相同的方法，我们显示了清晰的界面
对流 Cahn-Hilliard 方程到界面演化方程的极限。对于 Cahn-Larche 系统，这里的主要问题是近似解的构造。
最后，我们得到“模型 H”中的表面张力项，其迁移率常数足够快地收敛到 0，但通常不会收敛到界面的平均曲率。
In this thesis we rigorously prove that the Cahn-Larche system converges to a modified Hele-Shaw problem. For the proof we construct an approximate solution of the Cahn-Larche system by the method of matched asymptotic expansions. Then we can show that the approximate solutions of Cahn-Larche system converge to the solution of the modified Hele-Shaw problem.
For the modified Hele-Shaw problem we prove the existence of a classical solution in
a sufficiently small time interval [0;T]. By reducing the system to a single evolution
equation for the distance function, we show the assertion. Furthermore, we prove an
existence result for classical solution to a linearized Hele-Shaw problem used in the
higher order expansions.
By the same methods as for the Cahn-Larche system, we show the sharp interface
limit of a convective Cahn-Hilliard equation to an evolution equation for the interface. Here and for the Cahn-Larche system the main problem is the construction of the approximate solutions.
Finally, we obtain that the surface tension term in the “model H” with mobility constant converging sufficiently fast to 0 does generally not converge to the mean curvature of the interface.