We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier–Stokes/Cahn–Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier–Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end, we show maximal \(L^2\)-regularity for the Stokes part of the linearized system and use maximal \(L^p\)-regularity for the linearized Cahn–Hilliard system.

我们在时间上局部地针对具有不同密度的两种粘性不可压缩流体的两相流显示了扩散界面模型的适定性。该模型导致不均匀的Navier–Stokes / Cahn–Hilliard系统具有混合物的螺线管速度场，但是在Navier–Stokes类型方程中流体混合物的密度可变。通过适当的线性化和收缩映射参数，我们及时地证明了强解的存在。为此，对于线性化系统的Stokes部分，我们显示最大\（L ^ 2 \） -正则性，对于线性化Cahn-Hilliard系统使用最大\（L ^ p \） -正则性。