Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity

We consider numerical approximations of incompressible Newtonian fluids having variable, possibly discontinuous, density and viscosity. Since solutions of the equations with variable density and viscosity may not be unique, numerical schemes may not converge. If the solution is unique, then approximate solutions computed using the discontinuous Galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution. If the solution is not unique, a subsequence of these approximate solutions will converge to a solution.