On some mixed-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

The main purpose of this paper is the analysis of mixed-transmission problems for the anisotropic Stokes system in a compressible framework and in bounded Lipschitz domains with transversal Lipschitz interfaces in Rn, n≥2. Mixed problems and mixed-transmission problems for the anisotropic Navier-Stokes system in dimension n∈{2,3} are also considered. The anisotropy is introduced by an L∞-viscosity tensor coefficient, which satisfies an ellipticity condition in terms of symmetric matrices in Rn×n with null matrix traces. In the first part we use a variational approach to show the well-posedness of the analyzed linear problems for the Stokes system in L2-based Sobolev spaces. In the second part we show the existence and uniqueness of a weak solution of the mixed problem for the anisotropic compressible Navier-Stokes system with small data in L2-based Sobolev spaces in a bounded Lipschitz domain in Rn, n∈{2,3}. A mixed-transmission problem for the Navier-Stokes system in a Lipschitz domain with a transversal Lipschitz interface is also considered.