Weak solutions and convergent numerical schemes of Brenner-Navier-Stokes equations

Abstract

Lately, there has been some interest in modifications of the compressible Navier-Stokes equations to include diffusion of mass. In this paper, we investigate possible ways to add mass diffusion to the 1-D Navier-Stokes equations without violating the basic entropy inequality. As a result, we recover a general form of Brenner's modification of the Navier-Stokes equations. We consider Brenner's system along with another modification where the viscous terms collapse to a Laplacian diffusion. For each of the two modifications, we derive a priori estimates for the PDE, suffciently strong to admit a weak solution; we propose a numerical scheme and demonstrate that it satisfies the same a priori estimates. For both modifications, we then demonstrate that the numerical schemes generate solutions that converge to a weak solution (up to a subsequence) as the grid is refined.