We study a diffuse interface model describing the motion of two viscous fluids driven by surface tension in a Hele-Shaw cell. The full system consists of the Cahn–Hilliard equation coupled with the Darcy’s law. We address the physically relevant case in which the two fluids have different viscosities (unmatched viscosities case) and the free energy density is the Flory–Huggins logarithmic potential. In dimension two we prove the uniqueness of weak solutions under a regularity criterion, and the existence and uniqueness of global strong solutions. In dimension three we show the existence and uniqueness of strong solutions, which are local in time for large data or global in time for appropriate small data. These results extend the analysis obtained in the matched viscosities case by Giorgini et al. (Ann Inst Henri Poincaré Anal Non Linéaire 35:318–360, 2018). Furthermore, we prove the uniqueness of weak solutions in dimension two by taking the well-known polynomial approximation of the logarithmic potential.

中文翻译：

---

Hele-Shaw流的扩散接口模型的正确性

我们研究了一个扩散界面模型，该模型描述了两种由Hele-Shaw单元中的表面张力驱动的粘性流体的运动。整个系统由Cahn-Hilliard方程和达西定律组成。我们讨论了两种流体具有不同粘度的物理相关情况（无与伦比的情况），自由能密度是Flory-Huggins对数势。在第二维中，我们证明了正则性准则下弱解的唯一性，以及全局强解的存在性和唯一性。在第三个维度中，我们显示了强大的解决方案的存在和唯一性，对于大数据而言，它们在时间上是本地的，对于适当的小数据而言，在时间上是全局的。这些结果扩展了Giorgini等人在匹配粘度情况下获得的分析结果。（Ann Inst HenriPoincaréAnal NonLinéaire35：318–360，2018年）。此外，我们通过采用对数势的众所周知的多项式逼近来证明第二维弱解的唯一性。