Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (i.e., non-constant density and viscosity) incompressible Navier–Stokes system on a bounded domain. The proposed method is classical in the sense that it consists of a version of the 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 zero, 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.