I'm solving system of transport fluid equations for density (n) and for momentum (nu) with sources and electromagnetics equation for that sources. My computational domain consists of regions with zero density and some regions with zero density can appear in the process. The question is, how to apply advection across the entire system, i.e. how to define the velocity in the zero density regions.

Interesting. My question would be is Continuum modelling valid for gases with near "zero" density. Maybe you have to define regions where it is valid, solve using continuum equations there, set up boundary conditions, and then solve in the regions where continuum approximations aren't valid using say the boltzmann equation...

I think the Navier-Stokes equations are usually valid for a Knudsen Number greater than 0.001

Kn= (mean free path of molecules) / (characteristic length)

Maybe there are other techniques. If there are, I would be interested too...