We consider the sharp interface limit of a coupled Stokes/Cahn–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–Hilliard system converge to solutions of a sharp interface model, where the evolution of the interface is governed by a Mullins–Sekerka system with an additional convection term coupled to a two–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.

中文翻译：

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Stokes/Cahn-Hilliard 系统的锐界面极限。第一部分：收敛结果

我们在二维有界光滑域中考虑耦合 Stokes/Cahn-Hilliard 系统的锐界面极限，即，当参数 $\epsilon>0$ 对应于扩散界面趋于零。我们表明，在足够短的时间内，Stokes/Cahn-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-Sekerka 系统控制，附加对流项耦合到两个–具有杨拉普拉斯定律的相位平稳斯托克斯系统，用于对应力张量的额外贡献的跳跃，代表毛细管应力。我们通过估计精确解和近似解之间的差异来证明收敛结果。为此，我们对线性化 Cahn-Hilliard 算子使用 X. Chen 所示的谱估计的修改。耦合项的处理需要仔细估计，使用后一个谱估计的改进和近似解的合适结构，这将在本贡献的第二部分构建。