Corner direction flows of a compressible Stokes system

We study a compressible Stokes flow directed by a vector field that is tangential along the sides of a non-convex vertex and that has a jump across a subset in the domain. The jump restriction is imposed because the tangential flow lines along the corner sides may intersect in the interior of the domain. The solution of the transport equation directed by the vector field has a jump discontinuity across the interface curve emanating from the non-convex vertex. We construct a lifting vector handling the pressure gradient in the momentum equation and deal with the contact singularity at the point that the interface curve meets the boundary. We split from the velocity solution vector the corner singularities by the Lamé system. We finally show existence and piecewise regularity for a compressible Stokes system.