Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray--type solutions towards a vector field which satisfies the usual 2D Navier--Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat--type equation elsewhere. The method of proof uses weak compactness arguments.