论文标题
stokes/cahn-hilliard系统的尖锐接口限制，第一部分：收敛结果
Sharp Interface Limit of a Stokes/Cahn-Hilliard System, Part I: Convergence Result
论文作者
论文摘要
我们在二维，有界和光滑的域中考虑了耦合的Stokes/cahn \ TextEndEndash Hilliard系统的尖锐界面限制，即，当参数$ε> 0 $对应于与扩散界面的厚度相对应时，我们考虑了解决方案的限制行为。 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,代表毛细管应力。我们通过估计精确解决方案和近似解决方案之间的差异来证明收敛结果。为此，我们利用X. \ Chen为线性化的Cahn-Hilliard运算符所示的光谱估计值的修改。耦合项的处理需要仔细的估计，使用后一个光谱估计的改进以及近似溶液的合适结构，该结构将在此贡献的第二部分中构造。
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 $ε>0$ 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.
