This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (non-constant density and viscosity) incompressible Navier-Stokes system on a bounded domain. The proposed method consists of a version of Lax-Friedrichs explicit scheme for the transport equation and a version of Ladyzhenskaya's implicit scheme for the Navier-Stokes equations. Under the condition that the initial density profile is strictly away from $0$, the scheme is proven to be strongly convergent to a weak solution (up to a subsequence) within an arbitrary time interval, which can be seen as a proof of existence of a weak solution to the system. The results contain a new Aubin-Lions-Simon type compactness method with an interpolation inequality between strong norms of the velocity and a weak norm of the product of the density and velocity.

翻译：本文对应用于有界区域上非均匀（非恒定密度和黏度）不可压缩Navier-Stokes系统的一种基本全离散有限差分方法进行数学分析。所提出的方法由运输方程的Lax-Friedrichs显式格式和Navier-Stokes方程的Ladyzhenskaya隐式格式组成。在初始密度分布严格远离$0$的条件下，该方法被证明可在任意时间区间内强收敛至一个弱解（在子序列意义下），这可作为该系统存在弱解的一个证明。研究结果包含一种新的Aubin-Lions-Simon型紧致性方法，该方法利用速度的强范数与密度和速度乘积的弱范数之间的插值不等式。