We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effect. The PDE system couples the Navier–Stokes equations for the fluid velocity, a convective Cahn–Hilliard equation for the phase field variable with an advection–diffusion–reaction equation for the nutrient density. In the analysis, we consider a singular (e.g., logarithmic type) potential in the Cahn–Hilliard equation and prove the existence of global weak solutions in both two and three dimensions. Besides, in the two dimensional case, we establish a continuous dependence result that implies the uniqueness of global weak solutions. The singular potential guarantees that the phase field variable always stays in the physically relevant interval [−1, 1] during time evolution. This property enables us to obtain the well-posedness result without any extra assumption on the coefficients that has been made in the previous literature

我们分析了一个扩散界面模型，该模型描述了具有趋化效应的不可压缩两相流的动力学。PDE 系统将流体速度的 Navier-Stokes 方程、相场变量的对流 Cahn-Hilliard 方程与营养密度的对流-扩散-反应方程耦合。在分析中，我们考虑了 Cahn-Hilliard 方程中的奇异（例如，对数型）势，并证明了二维和三维的全局弱解的存在。此外，在二维情况下，我们建立了一个连续的依赖结果，这意味着全局弱解的唯一性。奇异势能保证相场变量在时间演化过程中始终保持在物理相关的区间 [-1, 1] 内。