This paper proves existence of a global weak solution to the inhomogeneous (i.e., non-constant density) incompressible Navier–Stokes system with mass diffusion. The system is well-known as the Kazhikhov–Smagulov model. The major novelty of the paper is to deal with the Kazhikhov–Smagulov model possessing the non-constant viscosity without any simplification of higher order nonlinearity. Every global weak solution is shown to have a long time behavior that is consistent with mixing phenomena of miscible fluids. The results also contain a new compactness method of Aubin–Lions–Simon type.

中文翻译：

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质量扩散非齐次不可压缩纳维-斯托克斯系统全局弱解的存在性

本文证明了具有质量扩散的非均匀（即非恒定密度）不可压缩纳维-斯托克斯系统的全局弱解的存在。该系统被称为 Kazhikhov-Smagulov 模型。本文的主要新颖之处在于处理具有非恒定粘度的 Kazhikhov-Smagulov 模型，而无需对高阶非线性进行任何简化。每个全局弱解都显示出具有与可混溶流体的混合现象一致的长时间行为。结果还包含一种新的 Aubin–Lions–Simon 型紧致性方法。