  We adapt the Halperin-Mazenko formalism to analyze two-dimensional active
nematics coupled to a generic fluid flow. The governing hydrodynamic equations
lead to evolution laws for nematic topological defects and their corresponding
density fields. We find that $\pm 1/2$ defects are propelled by the local fluid
flow and by the nematic orientation coupled with the flow shear rate. In the
overdamped and compressible limit, we recover the previously obtained active
self-propulsion of the +1/2 defects. Non-local hydrodynamic effects are
primarily significant for incompressible flows, for which it is not possible to
eliminate the fluid velocity in favor of the local defect polarization alone.
For the case of two defects with opposite charge, the non-local hydrodynamic
interaction is mediated by non-reciprocal pressure-gradient forces. Finally, we
derive continuum equations for a defect gas coupled to an arbitrary
(compressible or incompressible) fluid flow.
