We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system in a bounded smooth domain in ${\mathbb{R}}^d$, $d=2,3$. This model arises from the Diffuse Interface theory of binary mixtures accounting for density variation, capillarity effects at the interface and partial mixing. We consider the case of initial density away from zero and concentration-depending viscosity with free energy potential equal to either the Landau potential or the Flory-Huggins logarithmic potential. In this setting, we prove the existence of global weak solutions in two and three dimensions, and the existence of strong solutions with bounded and strictly positive density. The strong solutions are local in time in three dimensions and global in time in two dimensions.

我们研究了有限光滑域中的非均匀不可压缩Navier-Stokes-Cahn-Hilliard系统 ${\mathbb{[R}}^d$， $d=2，3$。该模型源自二元混合物的扩散界面理论，该理论解释了密度变化，界面处的毛细作用和部分混合。我们考虑初始密度远离零且粘度随浓度变化的情况，其自由能势等于Landau势或Flory-Huggins对数势。在这种情况下，我们证明了在二维和三维中存在全局弱解，并且存在有界且严格为正密度的强解。有力的解决方案是在三个维度上是本地时间，在两个维度上是全局时间。