  We study the hydrodynamics of compressible flows of active liquid crystals in
the Beris-Edwards hydrodynamics framework, using the Landau-de Gennes
$Q$-tensor order parameter to describe liquid crystalline ordering. We prove
the existence of global weak solutions for this active system in three space
dimensions by the three-level approximations and weak convergence argument. New
techniques and estimates are developed to overcome the difficulties caused by
the active terms.
