We study a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effects. This model also takes into account some significant mechanisms such as active transport and nonlocal interactions of Oono's type. The system under investigation couples the Navier–Stokes equations for the fluid velocity, a convective Cahn–Hilliard equation with physically relevant singular potential for the phase-field variable and an advection-diffusion-reaction equation for the nutrient density. For the initial boundary value problem in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$, we first prove the existence and uniqueness of global strong solutions that are strictly separated from the pure states ±1 over time. Then we prove the continuous dependence with respect to the initial data and source term for the strong solution in energy norms. Finally, we show the propagation of regularity for global weak solutions.

我们研究了描述具有趋化效应的不可压缩两相流的动力学的扩散界面模型。该模型还考虑了一些重要的机制，例如 Oono 类型的主动运输和非局部相互作用。所研究的系统将流体速度的 Navier-Stokes 方程、具有物理相关奇异势的对流 Cahn-Hilliard 方程与相场变量的物理相关奇异势和营养密度的平流-扩散-反应方程相结合。对于光滑有界域中的初始边值问题$\mathrm{\Omega }\subset {\mathbb{电阻}}^2$，我们首先证明了随着时间的推移与纯状态±1严格分离的全局强解的存在性和唯一性。然后我们证明了关于能量范数强解的初始数据和源项的连续依赖性。最后，我们展示了全局弱解的规律性传播。