Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result
Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result
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Stokes/Cahn-Hilliard 系统的尖锐界面限制第一部分：收敛结果。
DOI：
10.4171/ifb/457
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发表时间：
2020
影响因子：
1
通讯作者：
A. Marquardt
中科院分区：
文献类型：
--
作者：
H. Abels;A. Marquardt
本文考虑了一类耦合Stokes/Cahn\textendash Hilliard系统在二维有界光滑区域上的锐界面极限，即我们考虑了当扩散界面厚度的参数趋于零时解的极限行为。我们表明，在足够短的时间内，Stokes/Cahn\textendash Hilliard系统的解收敛到尖锐界面模型的解，其中界面的演化由Mullins\textendash Sekerka系统控制，该系统具有附加的对流项，耦合到两个\textendash相位固定Stokes系统，该系统具有Young Laplace定律，用于对应力张量的额外贡献的跳跃，代表毛细应力。我们通过估计精确解和近似解之间的差异证明了收敛结果。为此，我们利用X所示的谱估计的修改。Chen的线性化Cahn-Hilliard算子。耦合项的处理需要仔细估计，使用后者的频谱估计和近似解的适当结构的改进，这将在本贡献的第二部分中构建。
We consider the sharp interface limit of a coupled Stokes/Cahn\textendash Hilliard system in a two dimensional, bounded and smooth domain, i.e., we consider the limiting behavior of solutions when a parameter $\epsilon>0$ corresponding to the thickness of the diffuse interface tends to zero. We show that for sufficiently short times the solutions to the Stokes/Cahn\textendash Hilliard system converge to solutions of a sharp interface model, where the evolution of the interface is governed by a Mullins\textendash Sekerka system with an additional convection term coupled to a two\textendash phase stationary Stokes system with the Young-Laplace law for the jump of an extra contribution to the stress tensor, representing capillary stresses. We prove the convergence result by estimating the difference between the exact and an approximate solutions. To this end we make use of modifications of spectral estimates shown by X.\ Chen for the linearized Cahn-Hilliard operator. The treatment of the coupling terms requires careful estimates, the use of the refinements of the latter spectral estimate and a suitable structure of the approximate solutions, which will be constructed in the second part of this contribution.
Stokes/CahnâHilliard 系统的尖锐接口限制，第二部分：近似解
影响因子：
1.3
作者：
H. Abels;A. Marquardt
通讯作者：
A. Marquardt