Simple bifurcation problem


I have a velocity field V on an annulus. It might look like this:

If you propagate a particle with this velocity field, it is quickly transpored to one of the two border components of the annulus (e.g. hole in the middle or the outer circle).
I also have a quantity ρ(t) that gets transported accoring to a simple transport equation:
dρ/dt = -div(V*ρ)
What I am interested in is the following: Given an initial distribution ρ_0 how much will end up in each border component for t → ∞?
Can this be solved with fenics?