# A finite difference method for inhomogeneous incompressible Navier-Stokes equations

## TL;DR

This paper analyzes a fully discrete finite difference scheme combining explicit and implicit methods for inhomogeneous incompressible Navier-Stokes equations, proving convergence to weak solutions under certain conditions.

## Contribution

It introduces a novel convergence proof for a finite difference scheme applied to inhomogeneous Navier-Stokes equations, including a new compactness method.

## Key findings

- Proven strong convergence of the scheme to a weak solution.
- Established existence of weak solutions for the inhomogeneous system.
- Developed a new Aubin-Lions-Simon type compactness technique.

## Abstract

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.