Sharp interface limits for diffuse interface models for two-phase flows of viscous incompressible fluids

我们寻求对两种粘性流体（如油和水）流动的所谓扩散界面模型的尖锐界面限制的严格理解。在扩散界面模型中，考虑了宏观不混相流体在小长度尺度上的部分混合和大质量颗粒的扩散。这些模型能够描述由于碰撞或液滴形成而导致的界面拓扑奇点之外的两相流。对于理论和数值目的，更深入地理解极限ε→0依赖于迁移率mε的标度是有意义的。在这里，迁移率是佩莱特数的倒数，控制着扩散的强度。特别地，我们想要严格地理解迁移系数ε ε为ε→0时对极限系统和收敛速率的影响。通过这种方式，我们希望澄清文献中的数值观察和形式推导，并获得新的见解。

We seek a rigorous understanding of sharp interface limits of so-called diffuse interface models for the flow of two viscous fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ε > 0 and diffusion of the mass particles are taken into account. These models are capable of describing such two-phase flows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ε → 0 in dependence of the scaling of the mobility mε is of interest. Here the mobility is the inverse of the Peclet number and controls the strength of the diffusion. In particular, we want to understand the influence of the scaling of the mobility coeffcient mε as ε → 0 on the limit system and on the convergence rates rigorously. This way we want to clarify numerical observations and formal derivations in the literature and obtain new insights.