  This paper is build around the stationary anisotropic Stokes and
Navier-Stokes systems with an $L^\infty$-tensor coefficient satisfying an
ellipticity condition in terms of symmetric matrices in ${\mathbb R}^{n\times
n}$ with zero matrix traces. We analyze, in $L^2$-based Sobolev spaces, the
non-homogeneous boundary value problems of Dirichlet-transmission type for the
anisotropic Stokes and Navier-Stokes systems in a compressible framework in a
bounded Lipschitz domain with a Lipschitz interface in ${\mathbb R}^n$, $n\ge
2$ ($n=2,3$ for the nonlinear problems). The transversal interface intersects
the boundary of the Lipschitz domain. First, we use a mixed variational
approach to prove well-posedness results for the linear anisotropic Stokes
system. Then we show the existence of a weak solution for the nonlinear
anisotropic Navier-Stokes system by implementing the Leray-Schauder fixed point
theorem and using various results and estimates from the linear case, as well
as the Leray-Hopf and some other norm inequalities. Explicit conditions for
uniqueness of solutions to the nonlinear problems are also provided.
