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：本文从数学角度分析了一种基本完全离散的有限差异方法,该方法适用于封闭域上的不相容(非连续密度和粘度)不压缩的导航-斯托克斯系统。拟议方法包括一个版本的Lax-Friedrichs运输方程清晰计划,以及一个版本的Ladyzhenskaya纳维-斯托克斯方程隐含计划。在初始密度剖面严格离0美元很远的条件下,该方法被证明在任意的时间间隔内非常接近于一个薄弱的解决方案(直至一个子序列),这可被视为系统存在一个薄弱解决方案的证明。结果包含一种新的Aus-Lion-Simon型紧凑方法,在密度和速度产品的强标和弱标之间存在内推不平等。</s>